Segmentation of Stochastic Images using ... - Jacobs University

Segmentation of Stochastic Images using
Stochastic Partial Differential Equations
by
Torben Pätz
A thesis submitted in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy in Mathematics
Thesis Supervisor:
Second Referee:
External Referee:
Thesis committee:
Prof. Dr. Tobias Preusser
Jacobs University Bremen
Prof. Dr. Marcel Oliver
Jacobs University Bremen
Prof. Dr. Joachim Weickert
Saarland University, Saarbrücken
Date of Defense: January 13, 2012
School of Engineering and Science
Jacobs University Bremen

Acknowledgement
I would like to thank my advisor Prof. Dr. Tobias Preusser for the guidance during my PhD studies
and for giving me the opportunity to work in the rapidly growing field of stochastic modeling. Furthermore,
Prof. Preusser’s connection to Fraunhofer MEVIS enabled me to work in an inspiring
environment of people working on image processing problems.
Furthermore, I would like to thank Prof. Dr. Marcel Oliver and Prof. Dr. Joachim Weickert for
being members of my dissertation committee.
Special thanks go to the Jacobs University Bremen for the financial support during my PhD studies.
Without the tuition waiver for the complete time of my PhD studies and the scholarship during the
first two years, I would not be able to finish my studies successfully. Furthermore, special thanks go
to Fraunhofer MEVIS for the student assistance contract during the first two years of my PhD studies
and the possibility to use the infrastructure of the institute for my studies.
I am grateful to all my colleagues at the "Modeling and Simulation" group at Fraunhofer MEVIS.
Especially, I would like to thank Sabrina Haase, Hanne Tiesler, and Dr. Ole Schwen for reading and
commenting on a draft of this thesis.
I am also thankful to the QuocMesh collective from the work group of Prof. Dr. Martin Rumpf at
the Institute of Numerical Simulation of the Rheinische Friedrich-Wilhelms-Universität Bonn. All
implementations of the methods from this thesis are done in QuocMesh and without this excellent
finite element library this would be much more work. Especially, I would like to thank Dr. Ole
Schwen for answering all my questions concerning QuocMesh.
I am also grateful to Prof. Dr. Robert M. Kirby from the University of Utah in Salt Lake City,
USA, for the possibility to stay two weeks at the Scientific Computing and Imaging Institute in Salt
Lake City. Furthermore, I would like to thank Prof. Kirby and Prof. Joshi for the fruitful discussions
during my stay in Salt Lake City.
iii

Abstract
The task of segmentation, the separation of an image into foreground and background, is typically
performed on noisy images, and it is a great challenge to get satisfactory segmentation results. The
noise in the images depends on the acquisition modality (e.g. digital camera, MR, CT, ultrasound),
the acquisition parameters (acquisition time, sound frequency, magnetic field strength) and extrinsic
parameters (illumination, reflection). The acquisition step itself is a kind of physical measurement
(photon density, time-of-flight of the waves, spin, absorption) and – according to good scientific
practice – has to be equipped with information about the measurement error. This allows to estimate
the reliability of the measurement. The last step of quantifying the measurement error is typically
omitted in image processing. Neglecting the error leads to a loss of information about the influence
of the input error to the result of the image processing steps. This is important in medical application,
where radiologists generate decisions about the patients’ treatment based on the information
extracted from the images. For example, the further treatment for cancer patients is based on the volume
of the lesions segmented in the noisy images. It is important to equip the extracted information
with a reliability estimate or, and this is the aim of the presented work, to be able to compute the
probability density function for the extracted information depending on the estimation or modeling
of the input noise.
A possibility to model the image noise is to perceive a pixel inside the image as a random variable.
These images are called stochastic images. Doing this, the segmentation acts on images containing
random variables as pixels. This is contrary to the classical image processing task, where every
pixel has a deterministic value. Applying segmentation methods based on partial differential equations
(PDEs) on these stochastic images leads to stochastic PDEs (SPDEs), i.e. PDEs with stochastic
coefficients or right hand side. The discretization of SPDEs is an active and fast proceeding research
field and new methods for an efficient and elegant discretization are available in the literature.
In this thesis, the focus is on intrusive methods for the discretization of SPDEs, because classical
sampling strategies like Monte Carlo simulation or stochastic collocation are time-consuming. The
approximation of random variables uses the Wiener-Askey (or generalized) polynomial chaos and
the discretization of the SPDEs uses the recently developed generalized spectral decomposition and
finite difference schemes for random variables.
This thesis investigates the random walker segmentation, Ambrosio-Tortorelli segmentation, a regularization
of the Mumford-Shah functional, and the level set based segmentation methods geodesic
active contours, gradient-based segmentation, and Chan-Vese segmentation for stochastic extensions.
Furthermore, a sensitivity analysis of the classical segmentation approaches uses the stochastic
framework by making segmentation parameters random variables and investigating the influence of
the stochastic parameters on the segmentation result.
The result of the presented work is a framework carrying the probability distribution of the stochastic
gray values, i.e. the random variables, through all steps of the segmentation pipeline. This yields
segmentation results containing, for each pixel, a probability of belonging to the object or to the
background. Furthermore, this stochastic segmentation identifies regions where the image noise has
an important impact on the segmentation result and regions, which are robust in the presence of
noise. In addition, the visualization of the resulting stochastic images/contours is investigated.
v

Contents
Acknowledgement
Abstract
Notation
iii
v
ix
1 Introduction 1
2 Image Segmentation and Limitations 7
2.1 Mathematical Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Random Walker Segmentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 Mumford-Shah and Ambrosio-Tortorelli Segmentation . . . . . . . . . . . . . . . . 12
2.4 Level Sets for Image Segmentation . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.5 Why is Classical Image Processing not Enough? . . . . . . . . . . . . . . . . . . . . 21
2.6 Work Related to the Stochastic Framework . . . . . . . . . . . . . . . . . . . . . . . 23
3 SPDEs and Polynomial Chaos Expansions 25
3.1 Basics from Probability Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2 Stochastic Partial Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3 Polynomial Chaos Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.4 Relation to Interval Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4 Discretization of SPDEs 37
4.1 Sampling Based Discretization of SPDEs . . . . . . . . . . . . . . . . . . . . . . . 37
4.2 Stochastic Finite Difference Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.3 Stochastic Finite Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.4 Generalized Spectral Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.5 Adaptive Grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5 Stochastic Images 47
5.1 Polynomial Chaos for Stochastic Images . . . . . . . . . . . . . . . . . . . . . . . . 47
5.2 Generation of Stochastic Images from Samples . . . . . . . . . . . . . . . . . . . . 48
5.3 Comparison of the Space from [130] and the Space Used in this Thesis . . . . . . . . 52
5.4 Visualization of Stochastic Images . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
6 Segmentation of Stochastic Images Using Elliptic SPDEs 57
6.1 Random Walker Segmentation on Stochastic Images . . . . . . . . . . . . . . . . . 57
6.2 Ambrosio-Tortorelli Segmentation on Stochastic Images . . . . . . . . . . . . . . . 67
7 Stochastic Level Sets 79
7.1 Derivation of a Stochastic Level Set Equation . . . . . . . . . . . . . . . . . . . . . 79
7.2 Discretization of the Stochastic Level Set Equation . . . . . . . . . . . . . . . . . . 83
7.3 Reinitialization of Stochastic Level Sets . . . . . . . . . . . . . . . . . . . . . . . . 84
7.4 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
vii

7.5 Segmentation of Stochastic Images Using Stochastic Level Sets . . . . . . . . . . . 88
8 Segmentation of Classical Images Using Stochastic Parameters 97
8.1 Random Walker Segmentation with Stochastic Parameter . . . . . . . . . . . . . . . 98
8.2 Ambrosio-Tortorelli Segmentation with Stochastic Parameters . . . . . . . . . . . . 101
8.3 Gradient-Based Segmentation with Stochastic Parameter . . . . . . . . . . . . . . . 104
8.4 Geodesic Active Contours with Stochastic Parameters . . . . . . . . . . . . . . . . . 105
9 Summary, Discussion, and Conclusion 107
9.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
9.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
9.3 Outlook and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
List of Figures 111
List of Tables 117
A Publications Written During the Course of the Thesis 119
A.1 Publications Related to Stochastic Images . . . . . . . . . . . . . . . . . . . . . . . 119
A.2 Publications Related to Radiofrequency Ablation . . . . . . . . . . . . . . . . . . . 119
Bibliography 121
viii

Notation
u image function ⊗ tensor product
D image domain ∂D boundary of the domain D
R real numbers S ρ,k,m Kondratiev space
P i finite element hat function H univariate polynomial
I node set of a finite element grid Ψ multivariate polynomial
D c
(
Cantor measure
a
b)
binomial coefficient
BV functions with bounded variation ξ basic random variable
SBV special BV space (D c = 0) δ i j Kronecker delta
GSBV
generalized SBV space
∂ f
∂x
partial derivative
sign sign function ∂ t partial temporal derivative
H d d-dim. Hausdorff measure τ time step size
K
edge set (discontinuities) of an
image
h
spatial grid spacing
φ phase field or level set V finite element space
H 1 (D) Sobolev space H 1 over D S stochastic space, ⊂ L 2 (Ω)
| · | absolute value of real numbers ‖ · ‖ x x-norm
∆ Laplace operator F cumulative distribution function
tanh hyperbolic tangent N normal vector
κ
curvature of level sets or phase
fields
T
tangential vector (of level sets)
∗ convolution operator (·) ′ derivative of univariate function
E expected value W width of the tangential profile of
a phase field
Ω probability (event) space L p Lebesgue spaces
H m
Sobolev spaces
ix

Chapter 1
Introduction
The development of mathematical methods for image processing became a rapidly growing research
field during the last decades. The fast progress in the speed of widely available computer systems
allowed the numerical implementation of complex models. A specialty is the development of segmentation
algorithms based on partial differential equations (PDEs). The aim of a segmentation
algorithm is the decomposition of an image into the object and the background. Typically, detecting
edges inside an image or meeting a homogenization criterion for the object and the background lead
to a segmentation. Widely used segmentation approaches are the random walker segmentation [59],
the Mumford-Shah segmentation [107] and the related Ambrosio-Tortorelli regularization [14], and
active contour methods based on level set formulations [30,31,82,138]. Besided these segmentation
methods, which will be investigated in this thesis, there are other segmentation methods like region
growing [127], watersheds [136], snakes [76], and graph cuts [25].
Many applications use segmentation methods, e.g. quality control, machine vision, and medical
image processing. For example, the further treatment for cancer patients bases on the segmented
volume of the lesions from images. Fig. 1.1 shows a computed tomography (CT) image of a lung
lesion and the corresponding segmentation mask.
Typically, the segmentation methods act on noisy images (see Figs. 1.1 and 1.2). The image noise
depends on the image acquisition modality (e.g. digital camera, MR, CT, ultrasound), the acquisition
parameters (acquisition time, sound frequency, magnetic field strength), and extrinsic parameters
(illumination, reflection). The acquisition itself is a physical measurement (photon density, time-offlight
of the waves, spin, absorption), and it is good scientific practice to equip this measurement
with information about the measurement error. This last step of quantifying the measurement error
is typically omitted in image processing, leading to a loss of information about the influence of the
input error to the result of the image processing steps. Furthermore, image processing operators,
especially segmentation operators, do not have the ability to propagate this error information to the
result. This is e.g. important in medical application, where physicians decide about the patients’
treatment based on the information extracted from the images.
Figure 1.1: Left: CT image of a lung lesion (the small roundish structure in the middle of the image).
Right: The segmentation mask computed via region growing [127].
1

Chapter 1 Introduction
Figure 1.2: Noisy images from an ultrasound device (left) showing a structure in the forearm and a
computed tomography (right) of a vertebra in a human spine.
The aim of this thesis is to provide a representation for images containing error information and
to provide a framework for the error propagation of image processing operators.
The representation of images containing error information is based on a concept presented by
Preusser et al. [130]. This thesis identifies pixels by random variables. We call images containing
random variables as pixels stochastic images. The discretization of stochastic images uses the
generalized polynomial chaos developed by Xiu and Karniadakis [160] to approximate the random
variables at the pixels in a numerically meaningful way. This way of image representation is possible
when information about the distribution of the gray value for a pixel is available. Repeated acquisitions
of the same scene with the same imaging device or the usage of noise models can generate
this information. The repeated acquisitions are only possible in rare situations, where a still scene is
available and the repeated acquisition is ethically maintainable. Typically, the generation of medical
images violates these conditions, because the human under investigation is alive and acquisition devices
like computed tomography use high-energy radiation. Thus, for medical applications there is
only a limited area for the application of these methods. For other areas like quality control, it is easy
to generate samples of the same typically still scene. A possibility to overcome the need for multiple
samples is the application of noise models in combination with a single image. However, the
available image sample has to be as close as possible to the expected value of multiple acquisitions
to get meaningful results. This is hard to achieve. Nevertheless, we present a possibility to generate
a stochastic image from a sinogram of a computed tomography.
Replacing the classical images in the PDE based operators by their stochastic counterparts
achieves error propagation for image processing operators, but leads to stochastic partial differential
equations (SPDEs). The numerical solution of SPDEs is a rapidly growing field, because these equations
arise in the modeling of physical processes with uncertain parameters like heat propagation [24]
or fluid dynamics [84, 93, 109]. Uncertain parameters are e.g. the thermal conductivity or the speed,
because it is impossible to estimate these parameters exactly, but sometimes information about the
probability density function (PDF) is available for these parameters. In the classical modeling with
PDEs, one uses the expected value of the parameters for the calculation, yielding results that seem
accurate, but lose the information about the distribution of the input parameters. It is a great advantage
to have this information also in the output of such a calculation. The simplest method to
get information about this distribution is to perform a Monte Carlo simulation [101], i.e. to perform
deterministic calculations with a parameter sampled from the known input distribution. This is timeconsuming,
due to the high number of runs needed to achieve a sufficient precision. To overcome this
problem, methods have been developed ranging from stochastic collocation [158], a technique to use
2

Figure 1.3: This thesis combines findings from image processing with findings about SPDEs to yield
segmentation algorithms acting on stochastic images.
special sampling points in the random space, to the stochastic finite element method (SFEM) [54],
a discretization with finite elements in the random and spatial dimensions. Furthermore, the generalized
spectral decomposition (GSD) [113] allows breaking down the huge equation system of the
SFEM into a series of smaller systems.
The main goal of this thesis is to investigate, whether it is possible to combine the results from the
SPDE context with the results from image processing (see Fig. 1.3), especially for the task of image
segmentation. In other words: Which methods from image processing benefit from a stochastic
modeling of the input or model parameters, and how can we interpret the results from stochastic
modeling? The combination of stochastic images and SPDE, in the way presented in this thesis, was
never done before in the literature, although this approach yields new insights for the PDE based
segmentation of images:
• It determines regions where the segmentation is reliable also in the presence of image noise
and regions where the image noise has a great impact on the result of a segmentation.
• It quickly investigates the influence of parameters on the segmentation results, when using
SFEM or similar techniques for the computation.
The development of SPDE methods for image segmentation is based on existing PDE segmentation
methods. However, there are various methods proposed in the literature (see [144] for a review),
and we limit the analysis for a stochastic extension to a few of them. Namely, these are random
walker segmentation [59], Mumford-Shah segmentation [107] with the related Ambrosio-Tortorelli
approximation [14], gradient-based segmentation [29], geodesic active contours [30, 82], and Chan-
Vese segmentation [31]. The latter three are based on a level set formulation.
Random walker segmentation [59] in contrast to other segmentation methods based on PDEs is
a supervised segmentation method, meaning that the user influences the segmentation result by interactive
input. For random walker segmentation, the user input consists of defining seed regions,
i.e. regions where the user specifies whether they belong to the object or not. The idea of the random
walker segmentation is to start random walks from the unseeded pixels and to give every pixel
a probability to belong to the object dependent on the fraction of random walks reaching the seed
region of the object. The direction the random walker chooses is dependent on the image gradient
between neighboring pixels, i.e. the probability to walk from one pixel to another is higher, when the
image gradient between the pixels is low. An implementation of the random walker algorithm uses a
different strategy, because it is unnecessary to compute random walks for every pixel to compute the
probabilities. Doyle and Snell [45] showed the equivalence to a Dirichlet problem. This reduces the
complexity to the solution of an elliptic PDE with an unknown for every unseeded pixel.
3

Chapter 1 Introduction
Ambrosio-Tortorelli segmentation [14] is a regularization of the segmentation approach proposed
by Mumford and Shah [107]. The idea is to compute a smooth representation of the image and
the corresponding edges, respectively a phase field approximation of the edges. For the Ambrosio-
Tortorelli model, the author developed a stochastic extension [1, 3], allowing to propagate information
about the measurement error to the result, the smooth image and the phase field.
Level set based segmentation is based on the evolution of a contour, represented by a level set
function, i.e. the contour is given as the zero level set of a higher-dimensional function. A speed
function controls the evolution of the contour. A typical choice for the speed function is to make it
dependent on the image gradient [29, 96]. Caselles et al. [30] and simultaneously Kichenassamy et
al. [82] developed improvements by adding a term that forces the contour to stay at edges. Furthermore,
Chan and Vese [31] developed a segmentation method that is able to segment objects without
sharp edges to the background. Instead of using gradient information, they proposed a functional
that segments homogeneous regions in the image.
Besides the development of stochastic segmentation algorithms, the investigation of pre- and postprocessing
steps is essential to end up with a complete framework for error propagation in image processing.
For example, it is necessary to develop a technique to acquire stochastic images, i.e. images
whose pixels are random variables when image samples are available. This step benefits from techniques
available in the literature [41, 130, 141] or from the modeling of the noise distribution. In
addition, this thesis investigates the visualization of the stochastic segmentation results.
Furthermore, it is possible to change the perspective and use the segmentation methods developed
for stochastic images for a sensitivity analysis of the segmentation methods with respect to
the segmentation parameters. The sensitivity analysis uses segmentation parameters that are random
variables. The segmentation result is a stochastic image that contains information about the influence
of the segmentation parameters. Thus, the stochasticity comes from the parameters and not from the
input image, but the equations are nearly the same.
Structure of the Thesis
The thesis has the following structure: Chapter 2 presents segmentation methods for images based
on PDEs. In particular, these are random walker segmentation, Ambrosio-Tortorelli segmentation,
and methods based on level sets. Besides the presentation of these classical methods, this chapter
discusses the drawbacks, especially for the propagation of errors. Furthermore, we review related
work and highlight the differences between the related work and the methods proposed here.
Chapter 3 contains an introduction into SPDEs and provides a theoretical background for the treatment
of SPDEs. Furthermore, it presents the polynomial chaos expansion, a widely used tool for the
approximation of random variables. The polynomial chaos expansion is the key for the numerical
treatment of SPDEs and random variables, because this expansion converts the abstract idea of
random variables into a series expansion with deterministic coefficients. A computer can work with
these coefficients, which enables the development of numerical methods for random variables. At the
end of this chapter we highlight the advantages of the polynomial chaos over interval arithmetic [64].
Chapter 4 investigates the discretization of SPDEs based on the polynomial chaos. The presented
methods range from sampling based methods like Monte Carlo simulation and stochastic collocation
to methods based on the polynomial chaos approximation for random variables. For the polynomial
chaos, this chapter presents a finite difference method as well as SFEM and the GSD method.
After the presentation of discretization methods for random variables and SPDEs in the previous
chapters, Chapter 5 presents stochastic images. The concept of stochastic images is crucial for this
thesis, because all methods developed in this thesis act on stochastic images. The main idea is to
replace a pixel from a classical image by a random variable. Using the notion from stochastics, a
stochastic image is a random field indexed by the position of the pixels inside the image. Besides
4

the presentation of the stochastic images, Section 5.2 describes a possibility to generate stochastic
images from image samples. This stochastic image generation is based on the method presented by
Desceliers et al. [41], who applied an empirical Karhunen-Loève expansion on the centered covariance
matrix. Section 5.4 investigates the visualization of 2D and 3D stochastic images.
Chapter 6 generalizes two segmentation algorithms based on elliptic PDEs to stochastic segmentation
methods acting on stochastic images. Section 6.1 deals with the extension of the random walker
segmentation to stochastic images. The idea of the random walker segmentation is to prescribe a
set of seed points for the objects and the background. Then a random walk starts at every unseeded
point and the probability that the random walker goes from one point to another is dependent on
the image intensity difference. In a stochastic image, this difference is a stochastic quantity and the
probabilities for the walk of the random walker are stochastic quantities. The discretization of this
method is based on the solution of a diffusion equation, because diffusion is the limit process of an
infinite number of random walks.
Section 6.2 investigates a stochastic extension of the Ambrosio-Tortorelli model for segmentation.
The author presented this work at the European Conference on Computer Vision (ECCV) 2010 [3]
and received the “ECCV 2010 Best Student Paper Award”. The idea is to replace all quantities in
the Ambrosio-Tortorelli approach by their stochastic counterparts, yielding to two coupled SPDEs
as the stochastic Euler-Lagrange equations for the computation of an energy minimizer. Using the
GSD for the solution of the discretized SPDEs, the Ambrosio-Tortorelli method segments stochastic
images computed from samples acquired via devices like digital camera or ultrasound imaging.
Chapter 7 presents the last segmentation method for stochastic images investigated in this thesis,
the segmentation of stochastic images with stochastic level sets. First, this chapter presents the
extension of level sets to stochastic level sets, i.e. level sets evolving under an uncertain velocity.
This extension is based on a parabolic approximation of the original level set equation. Having the
stochastic level set extension at hand, it is possible to develop methods for the segmentation based
on stochastic level sets. A method where the speed for the stochastic level set evolution is based
on the image gradient and stochastic extensions of the geodesic active contour approach developed
simultaneously by Caselles et al. [30] and Kichenassamy et al. [82] and the Chan-Vese approach [31].
Chapter 8 deals with a sensitivity analysis of segmentation methods with respect to parameter
changes. The sensitivity analysis uses the stochastic framework developed in the previous chapters,
but applies it on a single deterministic input image. The stochasticity comes from the segmentation
parameters that are random variables. With this modeling, we investigate the influence of the
parameters on the result with the same segmentation framework developed for stochastic images.
Chapter 9 contains a summary of the thesis along with a discussion. Furthermore, the chapter
draws conclusions and gives directions for future work.
5

Chapter 2
Image Segmentation and Limitations
In this chapter, we give a short review of the research in mathematical image processing and segmentation
related to the work in this thesis. We focus on PDE based methods for image processing,
because these methods have advantages over other image processing methods:
• They are based on a continuous formulation of images, but the discretization based on finite
differences or finite elements naturally leads to regular grids, characteristic for digital images.
• It is possible to show existence and uniqueness of solutions of PDE based methods using
well-known results from functional analysis.
• Later, we will see that PDE based methods extend naturally to stochastic images, the object
under investigation in this thesis.
The application of PDE models in image processing is a rapidly growing field of research. Many
authors (see [17,130] for an overview) presented methods based on PDEs to solve problems arising in
image processing like denoising, restoration, segmentation, registration, flow extraction, etc. Since
we are interested in segmentation, the presentation focuses on results important for segmentation.
Image segmentation, the separation of an image into object and background, is a repeatedly investigated
problem in image processing. The literature divides the proposed methods into three
categories, based on the user interaction necessary to perform the segmentation:
Automatic segmentation: The user defines segmentation parameters at the beginning only, but
has no possibility to refine the segmentation result.
Semi-automatic segmentation: The user defines initial contours and parameters to optimize the
segmentation result, but again has no chance to refine the result.
Interactive segmentation: The user interactively refines the segmentation result. Thus, this
method computes a segmentation result based on the user input and allows user interaction
afterwards to get new input for the next iteration step.
PDE based image segmentation methods are in all of these segmentation categories. The random
walker segmentation [59] is an interactive segmentation approach, where the user interactively refines
the segmentation result. The level set based segmentation methods [29, 96, 138] are semiautomatic,
because the user has to define an initial contour as the starting point for the algorithm, but
has no chance to influence the segmentation result during the run of the algorithm. The Mumford-
Shah approach [107] is fully automatic. The user defines parameters only, but has no possibility to
define initial contours or to modify the result locally afterwards.
We organized this chapter as follows: First, we present basic definitions needed for the presentation
of PDE based segmentation algorithms. Afterwards, we present five segmentation algorithms:
random walker segmentation, Ambrosio-Tortorelli segmentation, and the level set based segmentation
methods gradient-based segmentation, geodesic active contours and Chan-Vese segmentation.
At the end, we present limitations of classical segmentation algorithms to motivate further investigations
to extend these classical methods and draw conclusions.
7

Chapter 2 Image Segmentation and Limitations
✘ ✘✘ ✘ ✘✘✘ u(x j ) = u j
✘ ✘ ✘ ✘✘ ✘ ✘ x i
❳ ❳❳
❳ ❳ ❳
supp Pi (x)
Figure 2.1: Sketch of the ingredients of a digital image. At every intersection of the regular grid lines
a pixel is located and for every pixel the corresponding FE basis function has its support
in the elements around this pixel.
2.1 Mathematical Images
Before we start with the presentation of segmentation methods, we give a short overview over the
notation and basic definitions for mathematical image processing. The primary object is the image:
Definition An image is a function u from the image domain D ⊂ IR d ,d ∈ {2,3}, into the real numbers,
i.e. u : D → IR. In what follows, the image domain D is a rectangular domain.
Mathematical images are defined on a continuous space, i.e. they have an infinite number of values.
An image acquired by a digital imaging device, e.g. a digital camera or advanced devices like CT [66]
or MR [91], is called a digital image and the image intensities are known on a finite point set only:
Definition A digital image (see Fig. 2.1) is a set of image intensities at the intersections of regular
grid lines, called pixels. We denote the pixel value of the ith pixel of the digital image u by u i . The
set of all pixels of a digital image is denoted by I and called the image grid.
The link between this continuous definition and the pixel representation of digital images is the usage
of an interpolation rule. Let us denote by P i the bilinear (2D) or trilinear (3D), basis function of the
i-th pixel belonging to the multi-linear finite element space of the grid I . Then a digital image is
interpolated at every point x in the image domain D by using the interpolation
u(x) = ∑ u i P i (x) . (2.1)
i∈I
Remark 1. In what follows, we deal with gray value images only. This is not a strong restriction,
because color images are typically composed of three color channels and it is possible to apply the
methods presented in the following on these color channels separately when there is no coupling
between the channels.
Until now, we have no regularity assumptions on the image u, but to show existence and uniqueness
of solutions of image processing methods, we have to restrict the analysis to images with a prescribed
regularity. For the methods used in this thesis, the space of functions of bounded variation and
generalizations of this space are important.
8

2.2 Random Walker Segmentation
Definition The space of functions of bounded variation is
{
∫
}
BV(D) = u ∈ L 1 (D) : |Du|dx < ∞
D
. (2.2)
Following [17] and using the Lebesgue decomposition theorem (see [32]) the derivative of a BVfunction
decomposes into three parts, the absolutely continuous part ∇udx, the jump part D j u and
the Cantor part D c u. This leads us to the definition of the class of special BV-functions:
Definition The class of BV-function for which the Cantor part vanishes, i.e. D c u = 0, is called the
space of special functions of bounded variation (SBV).
Based on the space SBV we define the space of generalized functions of bounded variation (GSBV):
Definition The space GSBV consists of all functions u ∈ L 1 (D) satisfying
i.e. the truncated function belongs to SBV for all T .
∀T > 0 : u T = sign(u)max(T,|u|) ∈ SBV , (2.3)
In particular, the spaces SBV and GSBV are useful, when we introduce Mumford-Shah segmentation
and the related Ambrosio-Tortorelli approximation.
2.2 Random Walker Segmentation
Random walker segmentation performs on a single image u : D → IR defined on the image domain D.
Since random walker segmentation is based on pixel values only, we need no additional assumptions
about the smoothness of the images. The main idea of random walker segmentation is that the user
prescribes a set of seed points for the object and the background. From the remaining unseeded
points, random walks start and the percentage of random walks reaching the object seeds is the
probability of the pixel to belong to the object.
Before we begin to introduce random walker segmentation, we have to define notation for the
graph representation of the image. A graph G is a pair G = (V,E) containing vertices or nodes
v ∈ V and edges e ∈ E ⊂ V ×V. We denote an edge connecting the vertices v i and v j by e i j and
identify e i j with e ji , because we are interested in nondirectional graphs only. Every edge has a
weight w(e i j ) =: w i j that describes the costs for using the edge. A graph containing edge weights is
a weighted graph. The summation of all edge weights for a node i,
d i =
∑
{ j∈V :e i j ∈E}
w(e i j ) , (2.4)
is the degree of the node i.
For random walker segmentation, we identify the image u with a graph G. The pixels of the digital
image are the graph nodes and every pixel (respectively node) is connected to the neighboring nodes
by a weighted edge. Fig. 2.2 shows a graph corresponding to a 3 × 3 image. For random walker
segmentation the graph weights are
w(e i j ) = exp ( −β(g i − g j ) 2) , (2.5)
where (g i − g j ) 2 is the normalized difference between the image intensities at position i and j:
(g i − g j ) 2 =
The parameter β is the only free parameter that the user chooses.
(u i − u j ) 2
max {k,l∈V :ekl ∈E}(u k − u l ) 2 . (2.6)
9

Chapter 2 Image Segmentation and Limitations
✈
✈
✈
✈
✈
✈
✈✘ ✘ ✘ v k
✎☞
w 7
✍✌ jk
✘✘✘
✈ v j
✘ ✘ ✘ e i j
✘✈
✘ ✘ v i
Figure 2.2: The graph generated from a 3 × 3 image contains 9 nodes and 12 edges. The edges e mn
connect the nodes (the black dots) v l . Every edge e mn has a weight w mn describing the
costs for traveling along this edge.
2.2.1 Relation to the Dirichlet Problem
The simulation of an infinite number of random walks is equivalent to solving a combinatorial Dirichlet
problem [45,59]. Therefore, we review the Dirichlet problem in this section and show its relation
to random walks. The Dirichlet integral is
R[u] = 1 ∫
|∇u| 2 dx . (2.7)
2
D
A minimizer of the Dirichlet integral is a harmonic function, i.e. a function satisfying ∆u = 0 and the
prescribed boundary conditions. The Dirichlet integral presented above is only useful for graphs with
equal weights, for different weights we have to use a Dirichlet integral that respects these weights:
R w [u] = 1 ∫
w(x)|∇u| 2 dx . (2.8)
2
D
A minimizer of (2.8) is a function that satisfies ∇ · (w∇u) = 0. To compute the minimizer of the
discrete Dirichlet problem, i.e. to find a minimizer of the discrete version of (2.8), we introduce the
combinatorial Laplacian matrix:
⎧
⎨ d i if i = j
L i j = −w i j if v i and v j are adjacent nodes
(2.9)
⎩
0 otherwise .
Using the combinatorial Laplacian matrix, the graph discrete version of (2.8) is
R[x] = 1 2 xT Lx , (2.10)
where x is a vector containing all nodes/pixels of the graph resp. the image, i.e. x = (v 1 ,...,v n ).
The user prescribes seed points V M = V O ∪V B for the object V O and background V B , see Fig. 2.3.
These points act as boundary conditions for the Dirichlet problem, because the probability that a
random walk starting at a seed point reaches it is one. The unseeded points V U are the degrees of
freedom. Reordering the nodes according to the set they belong to, (2.10) is written in block form
R[x U ] = 1 [
x
T
2 M xU] [ ][ ]
T L M B xM
B T L U x U
= 1 (2.11)
(
x
T
2 M L M x M + 2xUB T T x M + xUL T )
U x U .
In (2.11) L M is the part of the matrix L describing the dependencies between the seed points V M .
L U is the part with the dependencies between the unseeded pixels. Finally, B, respectively B T , is the
10

2.2 Random Walker Segmentation
Figure 2.3: Left: Definition of the seed regions for the object (yellow) and the background (red).
Middle: The probability that a random walker reaches an object seed. Black denotes
probability zero, white probability one. Right: Random walker segmentation result of the
ultrasound image. As input we used the seed regions from the left image and β = 200.
part of the matrix describing the coupling between the seeded and unseeded pixels. Differentiation
of (2.11) yields a minimizer of (2.11) given by the solution of
L U x U = −B T x M . (2.12)
Remark 2. For 2D-images, the matrices L U and B are band matrices with five bands used only. The
numerical solution of the system benefits from the use of numerical methods that make use of this
special matrix structure, e.g. , it is necessary to store five bands as single vectors only. Furthermore,
arithmetic operations for matrices with band structure can be implemented efficiently [57].
As already mentioned, the random walker segmentation is an interactive segmentation method. Due
to the fast calculation of the random walker result, the user interactively defines new seed regions or
eliminates unwanted seed regions to get an optimal segmentation result. Fig. 2.4 shows three steps
of the computer/user interaction for the refinement of a segmentation result.
Another, more mathematical, motivation for the derivation of (2.12) is that we have to solve
−∇ · (w∇u) = 0 in D
u = 1 on V O
u = 0 on V B .
(2.13)
Transforming this PDE to homogeneous boundary conditions [124], applying the reordering of the
nodes, and using the combinatorial Laplacian, we end up with (2.12). Eq. (2.12) is a system of linear
equations solvable by using iterative methods, e.g. the method of conjugate gradients.
Remark 3. The presented segmentation method, the random walker segmentation, sounds like a
stochastic method for segmentation that includes randomness and uncertainty, but this is false. Using
the equivalence to the Dirichlet problem presented above, the method computes deterministic weights
for an elliptic PDE on a graph. Thus, all “randomness” is lost. The “randomness” comes from the
interpretation of the result: Every pixel gets a value between zero and one, and we interpret these
values as probabilities for reaching the seed region from this specific pixel.
The random walker segmentation result is a probability for every pixel for belonging to the object (see
middle of Fig. 2.3). Typically, the threshold 0.5 distinguishes between the object and the background.
A pixel having a probability above 50% is assigned to the object and a pixel with a probability below
50% to the background. Fig. 2.3 shows a random walker segmentation result.
11

Chapter 2 Image Segmentation and Limitations
Figure 2.4: From left to right: Three steps of the interactive random walker segmentation. We show
the seeds and the image to segment in the upper row and the segmentation corresponding
to this particular choice of the seeds in the lower row. The addition of seed regions for
the object and the background yield an iterative refinement of the segmentation.
2.3 Mumford-Shah and Ambrosio-Tortorelli Segmentation
The minimization of a functional, as seen in the random walker segmentation, is a common technique
for segmentation problems. The next method, the Mumford-Shah segmentation, bases on
the minimization of a functional, too. The Mumford-Shah functional is not as easy as the random
walker functional, because the Mumford-Shah functional involves two unknowns, the image and an
additional edge set. This leads to a couple of mathematical problems for the theoretical proof of
existence and uniqueness of minimizers and it is hard to discretize the Mumford-Shah functional
directly. We avoid the numerical problems by introducing the Ambrosio-Tortorelli approximation
that Γ-converges to the Mumford-Shah functional [14].
Mumford and Shah [107] proposed to minimize the functional
∫
∫
E MS (u,K) := (u − u 0 ) 2 dx + µ |∇u| 2 dx + νH d−1 (K) , (2.14)
D\K
where u 0 : D → IR is the initial image, u : D → IR is an image that is smooth and differentiable in D\K,
K ⊂ D a set of discontinuities, µ,ν are nonnegative constants, and H d−1 (K) is the d −1-dimensional
Hausdorff measure of the edge set K. The aim is to find an image u and a set K such that the functional
is minimal. Roughly speaking, the minimizer u must be an image, which is close to the initial u 0 away
from the edges (then ∫ D\K (u − u 0) 2 dx is small) and smooth away from the edges (then ∫ D\K |∇u|2 dx
is small). Moreover, the length of the edge set K must be small (then H d−1 (K), measuring the length
of the edge set, is small). The direct minimization of the Mumford-Shah energy is difficult due to
the different nature of u and K: u is a function and K is a set. In addition, the proof of existence of a
D\K
12

2.3 Mumford-Shah and Ambrosio-Tortorelli Segmentation
minimizer is a challenging problem, cf. [35]. Since the functional is not differentiable, the estimation
of minimizers based on the Euler-Lagrange equations is impossible. Instead, researchers proposed
regularized approximations (see [17]). The following paragraph summarizes one of these methods,
proposed by Ambrosio and Tortorelli [14].
Remark 4. All components of the Mumford-Shah functional are essential to get a segmentation of
the image u, i.e. it is impossible to omit one of the components to end up with a mathematically and
numerically easier problem. If we omit the first component, we have no control over the difference
between the image and the smooth approximation and u = 0, K = /0 minimize the remaining parts. We
obtain another trivial solution if we omit the second component: Now u = u 0 and K = /0 minimize the
functional. When omitting the last component, K = D minimizes the functional. Thus, the Mumford-
Shah functional contains the minimal number of components necessary for segmentation, and it is
essential to discretize them well to get meaningful numerical solutions.
2.3.1 Ambrosio-Tortorelli Segmentation
As already mentioned, the Ambrosio-Tortorelli segmentation [14] is a kind of regularization of the
Mumford-Shah functional. Ambrosio-Tortorelli segmentation uses a function φ : D → IR, the phase
field, instead of the edge set K. The phase field is a smooth indicator function of the edge set K. It
is zero on the edge set K and goes smoothly to one away from the edge set. An additional variable ε
controls the width of the transition zone. When ε goes to zero, the phase field goes to the characteristic
function of the edge set. In a following section, we will recap that the Ambrosio-Tortorelli
energy converges in the Γ-sense to the Mumford-Shah energy [14].
The idea of the Ambrosio-Tortorelli segmentation for a given initial image u 0 is to find a phase
field φ and a smooth image u minimizing the energy
where
E AT [u,φ] := Efid,u ε [u] + Eε reg,u[u,φ] + Ephase ε [φ] , (2.15)
E ε fid,u [u] = ∫
D
∫
Ereg,u[u,φ] ε =
∫
Ephase ε [φ] =
D
D
1
2 (u − u 0) 2 dx
µ ( φ 2 )
+ k ε |∇u| 2 dx
(
νε|∇φ| 2 + ν 4ε (1 − φ)2) dx .
(2.16)
The first energy, the fidelity energy, ensures closeness of the smoothed image to the original u 0 . The
second energy, the regularization energy, measures smoothness of u apart from areas where φ is
small (the edges), and enforces φ to be small close to edges. The parameter k ε ensures coerciveness
of the differential operator and existence of solutions, because φ 2 may vanish. The third energy,
the phase energy, drives the phase field towards one and ensures small edge sets via the term |∇φ| 2 .
The parameter ε controls the scale of the detected edges, µ the amount of detected edges, and ν the
behavior of the phase field. k ε is a small regularization parameter.
The relation between the first two components of the Ambrosio-Tortorelli and the Mumford-Shah
energy are obvious. The third component, the phase energy, is a combination of a term forcing φ to
be one and the term ∫ D ε|∇φ|2 . In the limit ε → 0, it can be shown to be equal to H d−1 (K) by using
the co-area formula [105].
A minimizer of this energy is an image that is flat away from edges and a phase field, which
is close to zero at edges only. To obtain a minimizer of an energy, a widely used technique is to
solve the Euler-Lagrange equations resulting from this energy. For the computation of the Euler-
Lagrange equations, we have to compute the first variation of the above energies using the Gâteaux
13

Chapter 2 Image Segmentation and Limitations
Figure 2.5: Left: The initial (noisy) US image treated as input for the Ambrosio-Tortorelli approach.
Middle: The smooth Ambrosio-Tortorelli approximation of the initial image. Right: The
corresponding phase field, i.e. the approximation of the edge set of the smoothed image.
derivatives [17]. In the following θ : D → IR is a test function. For the fidelity energy, we get:
d
dε E ∫
f id[u + εθ]
∣ = 2(u + u 0 )θ dx . (2.17)
ε=0
For the other energies, we get similar results. The Euler-Lagrange equations of (2.15) are
D
−∇ · (µ(φ 2 + k ε )∇u ) + u = u 0
( 1
−ε∆φ +
4ε + µ )
2ν |∇u|2 φ = 1
4ε . (2.18)
This is a system of two coupled elliptic PDEs. We seek u,φ ∈ H 1 (D) as the weak solutions of these
Euler-Lagrange equations. An implementation solves both equations alternately, letting either u or φ
vary alternatingly until they reach a fixed point as the joint solution of both equations. Fig. 2.5 shows
an exemplary result of the Ambrosio-Tortorelli segmentation approach.
2.3.2 Γ-Convergence
As already stated, the Ambrosio-Tortorelli functional approximates the Mumford-Shah functional.
We show that the Ambrosio-Tortorelli functional converges in a variational sense towards the
Mumford-Shah functional. This variational convergence is called Γ-convergence [14]:
Definition The sequence of functionals F n : X → IR Γ-converges to the functional F if
1. For every x ∈ X and for every sequence x n converging to x ∈ X,
F(x) ≤ liminf
n→∞ F n(x n ) . (2.19)
2. For every x ∈ X there exists a sequence x n converging to x ∈ X such that
F(x) ≥ limsupF n (x n ) . (2.20)
n→∞
The proof of the Γ-convergence of a function sequence consists of two steps: First, we have to prove
(2.19) for all sequences and then we have to construct a sequence fulfilling (2.20). This last step is
the challenging task when proving Γ-convergence [14]. Using the definition of Γ-convergence and
the space GSBV introduced in Section 2.1, the following theorem from [14, 17] holds.
14

2.3 Mumford-Shah and Ambrosio-Tortorelli Segmentation
Theorem 2.1. Define Ẽ AT : L 1 (D) × L 1 (D) → IR + by
{
EAT (u,φ) if (u,φ) ∈ H
Ẽ AT (u,φ) :=
1 (D) × H 1 (D),0 ≤ φ ≤ 1
+∞ otherwise
(2.21)
and G : L 1 (D) × L 1 (D) → IR + by
{
EMS (u) if u ∈ GSBV(D) and φ = 1 almost everywhere
G(u,φ) =
+∞ otherwise.
(2.22)
If k ε = o(ε), then Ẽ AT Γ-converges to G(u,φ) for ε → 0.
The convergence of the Ambrosio-Tortorelli energy towards the Mumford-Shah energy enables us
to use the coupled pair of PDEs obtained as Euler-Lagrange equations of the Ambrosio-Tortorelli
energy and to solve this with very small ε. The result is a phase field that is close to the characteristic
function of the edge set of the Mumford-Shah functional.
2.3.3 Edge Continuity and Edge Consistency
The classical Mumford-Shah model and the Ambrosio-Tortorelli approximation lack a step linking
edges. This step is necessary to enforce the detection of closed contours in the images. Otherwise,
the appearance of partially detected, breaking up contours is possible, see Fig. 2.5. For example,
Erdem et al. [49] introduced such a step for the Ambrosio-Tortorelli model. The idea is to use a
modified diffusion coefficient in the image equation. This modified coefficient does not depend on
the phase field exclusively, but contains information about the continuity and directional consistency
of the detected edges. To be more precise, Erdem et al. [49] proposed to use the equation
−∇ · (µ((cφ) 2 + k ε )∇u ) + u = u 0 , (2.23)
instead of the first equation of (2.18). The additional factor c is the product of the two factors from
the directional consistency c dc and the edge continuity c h , i.e.
c = c dc · c h . (2.24)
If c < 1, the diffusivity decreases, allowing to form new edges in the image, whereas c > 1 leads to
an increased diffusivity, allowing to smooth away unwanted edges.
Directional Consistency
The directional consistency tries to judge the quality of the detected edges based on information
from surrounding pixels. The idea is that an edge is reliable if the gradients of the image for pixels
in directions perpendicular to the edge are in parallel. For inaccurately detected edges, e.g. due to
noise, these gradients are typically not aligned. To do so, Erdem et al. [49] introduced
(c dc ) i = ζ dc
i
+ 1 − ζ i
dc
(2.25)
φ i
for all pixels i ∈ I , where ζi
dc measures the alignment of the gradients. This factor increases the
diffusion, if the image gradients around the detected edge are not aligned. As the feedback measure
for the alignment of the gradients they proposed to use
( ( ))
1
ζi
dc = exp ε dc |η s | ∑ ∇v
j∈η s i · ∇v j − 1 , (2.26)
15

Chapter 2 Image Segmentation and Limitations
where ∇v i and ∇v j are the normalized gradients at position i respectively j, i.e. ∇v k = ∇u k /|∇u k |.
Eq. (2.26) is close to one if the gradients are aligned (than the scalar product ∇v i · ∇v j is close to
one) and close to zero if the gradients are not aligned. The set η s contains s pixels in the direction
perpendicular to the image gradient and the parameter ε dc controls the influence of the gradient.
For the numerical experiments we used ε dc = 0.25 and four pixels in directions perpendicular to the
image gradient, i.e. |η s | = 4.
Edge Continuity
To avoid the breaking up of edges, Erdem et al. [49] proposed to use an additional feedback measure,
which lowers the diffusivity around detected edges, to allow a growth of the detected edges. Using a
simplified version of the original model [49], the feedback measure is
(c h ) i =
1
1 + φ i − φ 2
i
, (2.27)
where φ is the phase field. We use a slight modification of the above feedback measure by adding an
additional scale factor α, allowing us to weight the deviation of the phase field from 0 and 1:
1
(c h ) i =
1 + α ( )
φ i − φi
2 . (2.28)
In the numerical experiments, we set α = 10. Thus, the diffusivity decreases in regions, where the
phase field is away from zero and one, i.e. regions, where the edge detection is in an intermediate state
between smoothing the structure away and building up a sharp edge. Fig. 2.6 shows a comparison
between Ambrosio-Tortorelli segmentation with and without the additional factor c.
2.4 Level Sets for Image Segmentation
The Ambrosio-Tortorelli segmentation approach introduces a new quantity besides the image and a
smooth representation: the phase field. This phase field approximates the edge set in the Ambrosio-
Tortorelli approach, but even with the above modifications there is no guarantee to obtain a connected
phase field in the end.
Level set based segmentation methods use another viewpoint. They place a closed curve somewhere
in the image and try to adjust this initial curve to the edges, respectively the object boundaries
in the image. This approach assures that the final segmentation result is a closed contour. For the
representation of the contour, there are methods for an explicit [76] or an implicit [29,31,40,96,138]
representation available in the literature. A famous method for the explicit representation is the snake
model [76], but explicit representations have drawbacks: parametrization of the curve, distribution
of the nodes describing the curve, dependence of the result on the parametrization, etc. To avoid
these shortcomings we focus on the implicit representation based on level sets in the following.
Dervieux and Thomasset [40] and Osher and Sethian [121, 138] developed the level set method.
The main idea is the implicit representation of a curve by embedding a curve C 0 ⊂ IR n into a higherdimensional
function φ : IR n+1 → IR and to identify the zero level set of φ with the curve, i.e.
C 0 = {x ∈ IR n : φ(0,x) = 0} . (2.29)
It is possible to describe the motion of the curve by the motion of the level sets of φ. At any time t > 0,
we get the curve C(t) back from the level set representation via C(t) = {x ∈ IR n : φ(t,x) = 0}. Using
this concept, we describe the motion of the curve using the level set equation [138]
φ t + F|∇φ| = 0 , (2.30)
16

2.4 Level Sets for Image Segmentation
input image without edge linking edge linking
image
phase field
n/a
Figure 2.6: Comparison of the Ambrosio-Tortorelli model (left) and the extended model using the
edge linking procedure (right). Data set provided by PD Dr. Christoph S. Garbe.
where F : IR n+1 → IR is the speed in the normal direction. For the discretization of (2.30) Osher
and Sethian [121] developed numerical methods based on Hamilton-Jacobi equations. The use
of simple finite difference approximations, like central differences, fails due to the hyperbolic nature
of (2.30) [138]. In addition, Sethian [138] developed efficient methods, like the Narrow Band
Method, where the equation is solved in the surrounding of the zero level set only.
Due to numerical reasons, level set methods use signed distance functions, i.e. functions that satisfy
|∇φ| = 1, as level set function. Since the function loses this attribute during the evolution of
the curve, we reinitialize the signed distance function from time to time. For this purpose, methods
are available ranging from iterative methods, e.g. solving φ t = sign(φ)(1 − |∇φ|) to steady state
(see [138]), to efficient – every grid point is only visited once – Fast Marching methods [138].
Besides the application for image segmentation, other research fields like computer-aided design,
flow simulations, or optimal path planning [138] use level sets. Furthermore, it is possible to use a
level set approach for the simulation of phase change problems. In this context, the author investigated
the simulation of the phase change during radio-frequency ablation [10, 12, 13].
2.4.1 Phase Field Models
Phase fields, like the phase field in the Ambrosio-Tortorelli approach, have a close relation to level
sets. In fact, the literature [55,137] refers to level set methods as “sharp interface approach”, because
level sets know the position of the interface precisely due to the implicit tracking. On the other hand,
one refers to phase fields as “diffusive interface approach”, because phase fields are constant away
from the interface and vary smoothly near the interface. Phase field methods treat the transition
zone around the interface as a zone with mixed content of the regions separated by the interface.
17

Chapter 2 Image Segmentation and Limitations
Phase fields are frequently used for interface tracking (see [23, 143] and the references therein), also
the image processing community uses phase fields for segmentation purposes [15, 123]. In contrast
to the phase field used in the Ambrosio-Tortorelli approach, the phase fields needed in this context
differentiate between object and background, i.e. they describe closed contours. They have a value of
−1 in the object, 0 at the interface, +1 on the background and vary smoothly between these values.
Typically, the phase field is ±1 away from the interface and changes smoothly inside a small layer
with thickness ε around the interface in a tangential profile. A phase field equation [143] like
(
φ t + F|∇φ| + u e · ∇φ = b ∆φ + φ(1 − φ 2 )
)
ε 2 (2.31)
controls the evolution of the diffusive phase field interface. In this equation, φ is the phase field, u e an
external advection, b the interface speed depending on the curvature, ε the thickness of the diffusive
interface, and F again the speed in the normal direction. In contrast to the level set approach (2.30),
this is a parabolic equation, which avoids the numerical difficulties arising in the discretization of
the level set equation. When the curvature-depending interface speed vanishes, i.e. when b = 0, this
equation is kept parabolic by adding a counter term introduced by Folch [51]. Following [143] this
counter term leads to the parabolic equation
(
φ t + F|∇φ| + u e · ∇φ = b ∆φ + φ(1 − φ 2 ( ))
)
∇φ
ε 2 − |∇φ|∇ ·
, (2.32)
|∇φ|
where b is a purely numerical parameter, because the curvature at the end cancels out the Laplacian
and the term φ(1−φ 2 )
. It is easier to discretize (2.32) than (2.30) using central differences. Sun et
ε 2
al. [143] developed a phase field equation based on nonlinear preconditioning [56]
(
φ t + a|∇φ| + u e · ∇φ = b ∆φ + 1 ε (1 − |∇φ|2 ) √ ( ) ( ))
φ
∇φ
2tanh √ − |∇φ|∇ ·
, (2.33)
2ε |∇φ|
which is an integrated reinitialization scheme for the phase field. The phase field φ in this equation
becomes a signed distance function. Thus, this equation is a parabolic level set equation with integrated
reinitialization. Again, it is possible to discretize (2.33) using simple difference schemes. This
connection between phase fields and level sets allows us to use known segmentation algorithms from
the level set context and embed them into this nonlinear preconditioned phase field equation. The
discretization of the parabolic phase field equations is easier in the stochastic context, cf. Chapter 7.
2.4.2 Gradient-Based Segmentation
The idea of the level set propagation is useful to segment objects inside an image. The simplest
approach for segmentation based on level sets is to use a speed F in the level set equation that
depends on characteristics of the image. Popular is F = F(|∇u|), i.e. to stop the evolution on edges
inside the image (see [29, 96, 138] and the references therein). Caselles et al. [29] proposed to use
with
g u =
φ t + g u |∇φ| = 0 (2.34)
1
(1 − εκ) , (2.35)
1 + |∇G σ ∗ u|
where u is the image, G σ a Gaussian smoothing filter with width σ, κ the curvature of the level set
function and ε a small scale parameter that controls the influence of the curvature smoothing term.
Although this idea sounds simple, the method achieves good results when a high gradient separates
the objects from the background (see Fig. 2.7). One major drawback of this method is the need for
18

2.4 Level Sets for Image Segmentation
Figure 2.7: Segmentation of a medical image based on a level set propagation with gradient-based
speed function. The time increases from left to right and the zero level set (red line)
approximates the boundary of the object (a liver mask) at the end.
finding a stopping criterion. The evolution speed g u is always positive, even close to edges. Thus, it
is possible that the zero level set passes the edge. A typical stopping criterion is to stop the evolution
when the difference between the level sets of subsequent time steps is small. This occurs when the
level set reached the boundary of the object and the speed dropped down. Using methods that are
more sophisticated, it is possible to stop the zero level set at the edge. Thus, these methods have a
convergent solution. The next section presents one of these methods, geodesic active contours.
Remark 5. It is also possible to formulate the gradient-based segmentation based on the phase field
model presented in the last section. This yields the equation
(
φ t + g u |∇φ| = ε ∆φ + φ(1 − φ 2 )
)
ε 2 . (2.36)
2.4.3 Geodesic Active Contours
Caselles et al. [30] and simultaneously Kichenassamy et al. [82] developed geodesic, or minimal
distance, active contours. They minimize an energy B that depends on the curve C and on the
parametrization of the curve C(q) : [0,1] → IR 2 :
∫ 1
B(C) = α |C ′ (q)| 2 dq + β
0
∫ 1
0
g u (|∇u(C(q))|) 2 dq , (2.37)
where g u is the edge indicator from the last section. They computed a minimizer of this energy
by using a level set representation of the curve and computing the Euler-Lagrange equations of the
resulting energy. This leads to a level set equation with an additional advection term that forces the
zero level set to stay in regions with high gradient:
φ t = −α∇g u · ∇φ − βg u |∇φ| + εκ|∇φ| . (2.38)
The user chooses the parameters α,β and ε. For given parameters and an initial level set we solve
to steady state. Fig. 2.8 shows a typical geodesic active contours segmentation result.
2.4.4 Chan-Vese Segmentation
The segmentation methods presented so far are based on a high gradient that separates the object
from the background. When such a gradient is not present, the methods fail to segment the object.
19

Chapter 2 Image Segmentation and Limitations
Figure 2.8: Segmentation using geodesic active contours. Left: The initial image. Right: Solution of
the geodesic active contour method initialized with small circles inside the object.
Chan and Vese [31] proposed a method that is independent of gradient information. Instead, they
proposed to segment homogeneous regions inside the image. To be more precise, Chan and Vese [31]
proposed to minimize the functional
∫
∫
F(c 1 ,c 2 ,C) = µ · Length(C) + ν · Area(inside(C)) + λ 1 |u 0 − c 1 | 2 dx + λ 2 |u 0 − c 2 | 2 dx .
inside(C)
The corresponding Euler-Lagrange equation is
( ( ) )
∇φ
φ t = δ(φ) µ∇ · − ν − λ 1 (u 0 − c 1 ) 2 + λ 2 (u 0 − c 2 ) 2
|∇φ|
outside(C)
(2.39)
, (2.40)
where δ is the Dirac δ-function [42]. This equation is a parabolic PDE that contains a curvature
smoothing term, a term penalizing the segmented area, and two terms penalizing variations from
the mean value of the segmented object and the background. Instead of δ, we use a regularized
δ-function δ ε for the discretization given by the derivative of the Heaviside approximation
H ε = 1 (
1 + 2 ( z
) )
2 π arctan . (2.41)
ε
By using H ε from above, δ ε is
1
δ ε (x) =
πε + π . (2.42)
ε
x2
The mean value of the object and the background can be computed using the Heaviside function:
∫
D
c 1 (φ) =
u ∫
0(x)H ε (φ(x))dx
∫
D
D H , resp. c 2 (φ) =
u 0(x)(1 − H ε (φ(x)))dx
∫
ε(φ(x))dx
D (1 − H . (2.43)
ε(φ(x)))dx
The user chooses λ 1 ,λ 2 , µ,ν. The advantage of the Chan-Vese model is that it does not need edges
in the image to segment objects. In fact, the model is independent of gradient information. Instead,
it tries to separate homogeneous regions in the image. Fig. 2.9 shows a typical result of Chan-Vese
segmentation on an image without edges.
This concludes the presentation of classical segmentation algorithms based on PDEs. The presented
segmentation algorithms range from interactive, nearly parameter free algorithms, like random
walker segmentation, over semi-automatic with a moderate number of variables, like the level
set based algorithms, to automatic methods like Mumford-Shah segmentation, where no user interaction
is necessary. All these segmentations are able to produce accurate results on a wide range
20

2.5 Why is Classical Image Processing not Enough?
Figure 2.9: Segmentation of an object without sharp edges using the Chan-Vese approach. In red, we
show the steady-state solution of the Chan-Vese segmentation method initialized with a
small circle inside the object.
of images and from the perspective of segmentation of single images, there is no need for new concepts.
Nevertheless, the approaches presented in this chapter have drawbacks regarding robustness
with respect to noise, reproducibility, and error propagation. The next section investigates this.
2.5 Why is Classical Image Processing not Enough?
In the last sections, we introduced five segmentation methods and showed that all these segmentation
methods perform well on some selected medical images. Besides the segmentation of single images,
a segmentation method has to fulfill other features not presented so far:
• It is unclear how robust the methods are with respect to image noise.
• The robustness of the methods for parameter changes and different initializations is unclear.
• Propagating error information through these algorithms is hard, i.e. if information about measurement
errors at the image acquisition is available, it is impossible to propagate this information
through the segmentation to get segmentation results containing the error information.
We organized this section as follows: First, we give an introduction to image noise, show how the
image noise influences the image quality for different acquisition modalities and how image noise
is modeled mathematically. Then, we investigate the noise robustness of the presented segmentation
methods, and finally, we discuss error propagation in classical image segmentation methods.
2.5.1 Image Noise
Image noise is a serious problem when dealing with medical images and images from digital cameras.
Different noise sources degrade the images. A principal problem of image acquisition devices is the
noise due to the random arrival of the photons. Light or X-ray emission is a stochastic process [44].
In addition, the instrumentation noise due to thermal effects in the acquisition device degrades the
image quality. Further sources of image noise are the quantization noise due to the conversion
from analog to digital signals and the compression process for the images, if any. Physical effects
influencing the path of the photons, like blurring, diffraction, and scattering, cause image noise, too.
21

Chapter 2 Image Segmentation and Limitations
Figure 2.10: A test pattern corrupted by uniform (left), Gaussian (middle), and speckle noise (right).
It is possible to reduce some of the noise sources by averaging the image values over a period.
When a signal is available for a period L, the expected value of a pixel is
1
E(a) = lim
L→∞ L
∫ L
0
a(x)dx . (2.44)
When the probability density function (PDF) of the process a is known the integral reduces to
E(a) =
∫ ∞
where ρ is the PDF. The variance of the stochastic process is (cf. [44])
σ 2 =
∫ ∞
With these quantities, the signal-to-noise ratio (SNR) [44] is
−∞
−∞
aρ(a)da , (2.45)
(a − E(a)) 2 ρ(a)da . (2.46)
SNR = |E(a)|2
σ 2 . (2.47)
One divides the noise sources into additive and multiplicative noise sources. Fig. 2.10 shows three
noise models. Additive noise is modeled via
g(x) = f (x) + n(x) , (2.48)
where g is the measured signal, f the true signal and n the noise. Multiplicative noise is modeled via
g(x) = f (x) + n(x) f (x) . (2.49)
The multiplicative noise depends on the image value. In what follows, we use the additive noise
model, because we are not directly interested in the noise modeling, but need a noise model as input
for the stochastic image processing framework. Once the noise is characterized, the noise is no
longer a free parameter and (2.49) can be expressed as
g(x) = f (x) + ñ(x) , (2.50)
and it is possible to use the additive model.
All these sources of noise influence the image quality and it is not well understood how the noise
influences the segmentation result, e.g. how the image noise influences the segmented object volume.
This is due to the construction of typical segmentation algorithms. They have no knowledge about the
noise that corrupted the image to segment and it is impossible to apply the segmentation algorithm
on noise realizations, apart from artificial test data corrupted with a known noise model. In the
next two sections, we present two problems related to image noise that cannot be investigated with
deterministic segmentation models, apart from using a sampling based approach.
22

2.6 Work Related to the Stochastic Framework
2.5.2 Robustness
Robustness of segmentation methods is desirable in two ways: The methods should be robust with
respect to the noise and with respect to the segmentation parameters.
Robustness with respect to the segmentation parameters, e.g. the β for random walker segmentation
or µ,ν, and ε for Ambrosio-Tortorelli segmentation, is necessary to get stable results. When the
segmentation result changes significantly for small parameter changes, the results are arbitrary, and
it is not recommendable to base e.g. medical diagnoses on such a segmentation result. It is possible
to investigate this kind of robustness by comparing the results of segmentations with slightly modified
parameters or by treating the segmentation parameters as random variables and investigate the
variance of the segmentation result. Chapter 8 of this thesis is about this.
Robustness with respect to noise is an essential property of a segmentation method for medical
images. The real, noise-free, image is not available, and it is a random choice which noise realization
the image at hand shows. It is desirable for segmentation methods to be robust with respect to the
noise realization, i.e. the segmentation result should not vary much for noise realizations. To investigate
this, it is possible to run the segmentation on noise realizations or to make image pixels random
variables, describing the process leading to the image noise. The first way is time-consuming. We
will see this later in the thesis, e.g. in Section 6.1. The second way is the fundamental idea of this
thesis. It needs a theoretical foundation, which the following chapters will present.
2.5.3 Error Propagation
Image processing widely neglects error propagation. Nearly all methods in image processing consider
the available data as the “truth”, but as we saw, the real data is not available. Instead, we have to
use an image corrupted by a random noise realization. When neglecting this error introduced by the
noise and other imaging artifacts we end up with results looking precise, but ignoring the influence
of the noise. It is desirable to have image processing methods that are able to deal with information
about the image noise, e.g. via the mentioned description of noise via the introduction of random
variables for the image pixels. The next chapters deal exactly with this new idea for the processing
of images and provide a theoretical background.
2.6 Work Related to the Stochastic Framework
In this section, we review work sounding similar to the work presented in this thesis and identify the
differences and similarities. A lot of authors presented methods for image segmentation that take
more or less stochasticity into account, e.g. via a modeling with Markov random fields or stochastic
annealing, but none of the methods mentioned can propagate stochastic information from the input
to the output of the segmentation process or model pixels as random variables.
Markov Random Fields for Image Segmentation
The literature [39, 65, 153] uses Markov random fields (MRFs) for image segmentation frequently.
MRFs are a possibility to model the noise in the input image, but the result of MRF segmentation
is a deterministic segmentation result along with a map to remove the noise from the input image.
Thus, this method tries to incorporate the noise via a stochastic modeling approach, but is not able
to propagate uncertainty information from the input to the output.
23

Chapter 2 Image Segmentation and Limitations
Random Level Set Functions
Stefanou et al. [141] presented a method to obtain a random level set, i.e. a polynomial chaos expansion
of a level set function. The proposed work relates to this work, but [141] obtains the polynomial
chaos expansion of the level set from a classical level set approach on the samples and an estimation
of the polynomial chaos coefficient afterwards. Computing the level set equation on the samples is
exactly the objective we want to overcome in this work by applying a stochastic level set equation
on a stochastic image, obtained from the samples. This results in a significant speed-up, because in
the stochastic framework the level set method has to be applied only once on the stochastic image,
not on every sample.
Stochastic Active Contours
The work presented in this thesis should not be confounded with the method presented by Juan et
al. [75] named “Stochastic Active Contour”. Although the title suggests a close relation between
the stochastic level set equation and the work presented in [75], they are opposed. Juan et al. [75]
proposed a technique to overcome drawbacks of the classical level set approach by adding fictive
noise to the image. This relates to the simulated annealing technique [83]. The idea of the stochastic
level set framework is the development of a stochastic active contour moving under an uncertain
velocity, obtained from the uncertain gray values of the stochastic pixels.
A Multiresolution Stochastic Level Set Method for Mumford-Shah Image Segmentation
Law et al. [89] use a method called “Stochastic Level Set Method” for the segmentation of objects.
This also should not be confounded with the stochastic level set work presented in this thesis, because
Law et al. proposed a method to overcome the drawback of the level set segmentation to run into local
minima. They developed a method that “jumps” out of these local minima to get a global minimum
of the solution. Again, this method is related to the simulated annealing technique [83].
Error-in-Variables Likelihood Functions for Motion Estimation
Nestares et al. [111,112] where able to compute confidence measures for error propagation in motion
estimation [71]. Their method allows to estimate the influence of the image noise on the computed
motion field. To achieve this, they combined Bayesian estimation [77] and likelihood functions to
solve a total-least squares problem [58]. Nevertheless, their investigations are restricted to independent,
identically distributed Gaussian random variables. Thus, their proposed framework can be used
in rare situations only.
Conclusion
We presented the basics for mathematical image processing and gave a short overview over segmentation
methods based on PDEs. All these methods produce good results on single images, but are
unable to deal with error propagation. Furthermore, the robustness of the methods with respect to
image noise and parameter changes is unknown. This highlights the need for error-aware methods
for segmentation and image processing in general. Before we are able to present these methods, we
have to provide background from stochasticity for the representation of random variables and about
SPDEs. This is the task of the next chapter.
24

Chapter 3
SPDEs and Polynomial Chaos Expansions
This chapter deals with the fundamentals required to develop stochastic images. First, we review
notation and results from probability theory. Afterwards, we introduce SPDEs and the polynomial
chaos expansion, the main ingredient for the numerical approximation of random variables.
3.1 Basics from Probability Theory
This section provides background from probability theory for the presentation of the stochastic images
and SPDEs. First, we introduce the basic ingredients, probability measures, probability spaces
and random variables.
Definition A probability space (Ω,A ,Π) is a triple consisting of a sample space Ω containing all
possible outcomes, a σ-algebra of events A ⊂ 2 Ω and a probability measure Π. The probability
measure Π is defined on the σ-algebra A and has the following properties:
• Π is non-negative: Π(A) ≥ 0 for all A ∈ A .
• The measure of the sample space Ω is one: Π(Ω) = 1.
• Π is countable additive, i.e. for a countable number of pairwise disjoint sets A i ⊂ A we have
Π(∪A i ) = ∑(Π(A i )).
On the probability space (Ω,A ,Π) we define functions from this space into the real numbers.
Definition A random variable f : Ω → IR is a function from the sample space Ω into the real numbers
that is measurable with respect to the σ-algebras A and B, where B is the Borel measure.
Random variables are an important object for the definition of stochastic images. In Chapter 5, we
will see that every pixel of a stochastic image is a random variable. For random variables, it is
possible to define the probability density function (PDF):
Definition The function ρ is called probability density function (PDF) of the random variable f if it
satisfies Π(a < f < b) = ∫ b
a ρ(x)dx for all a,b ∈ IR.
Having the probability density at hand, we define further properties of random variables. The most
important property of random variables is the expected value:
Definition The expected value or first moment of a random variable X : Ω → IR with PDF ρ is
∫
∫
∫
E(X) = X(ω)dω = xρ(x)dx = xdΠ . (3.1)
Ω
In (3.1) we used dΠ = f dx to characterize integration with respect to the PDF.
Knowing the probability density of a random variable allows us to transform the integral over the
sample space Ω into an easier computable integral over the real numbers weighted by the probability
density. Using this equality, it is also possible to compute higher-order moments of random variables:
IR
IR
25

Chapter 3 SPDEs and Polynomial Chaos Expansions
Definition The n-th central moment of a random variable X is
M n (X) = E((X − E(X)) n ) . (3.2)
A famous member of this class of moments is the second central moment, the variance:
Var(X) = E ( (X − E(X)) 2) . (3.3)
Later we have to evaluate the relation between random variables. A famous tool for this is the
covariance.
Definition The covariance of two random variables f ,g with finite second order moments is
Cov( f ,g) = E( f g) − E( f )E(g) . (3.4)
In what follows, it will be necessary to have a set of random variables indexed by a spatial position.
This motivates the following definition:
Definition A random field X is a collection of random variables indexed by a spatial position x ∈ IR n :
X = {X x |x ∈ IR n } . (3.5)
Random fields are elements of a tensor product space consisting of functions defined on the Cartesian
product Ω × D. A random field is a function taking two arguments, a random event and a spatial
position. We restrict the investigations to random fields satisfying smoothness assumptions.
Definition Let D ⊂ IR n be the spatial domain of the random field and Ω the sample space. The
tensor space L 2 (Ω) ⊗ H 1 (D) is the space of random fields satisfying u(ω,·) ∈ H 1 (D) almost sure
and u(·,x) ∈ L 2 (Ω), where H 1 (D) is the usual Sobolev space and
{
∫
}
L 2 (Ω) = f : Ω → IR : f (ω) 2 dω < ∞ . (3.6)
Ω
This is a strong limitation which is typically not satisfied for random fields arising in financial problems
[69, 108]. For image processing problems, this space is reasonable, because H 1 -regularity is
typically assumed for classical image processing tasks [17] and L 2 -regularity of the stochastic part
seems reasonable due to the limited energy an image acquisition device detects. Furthermore, the
restriction to random fields with finite variance, i.e. satisfying (3.6), allows a discretization of the
random fields using polynomial chaos expansions. Random fields will play a crucial role in the presentation
of SPDEs, because the locally varying random coefficients of the SPDEs are random fields.
3.2 Stochastic Partial Differential Equations
We introduce SPDEs following [18] and use an elliptic model equation. The deterministic equation is
−∇ · (a∇u) = f
on D
u = g on ∂D ,
(3.7)
where a is a diffusion coefficient, f a source term, g the boundary condition, and D the deterministic
domain this equation holds in. In this equation, we assumed that we perfectly know the diffusion
coefficient a and the right hand side f . In many applications, these quantities are not known exactly,
but a description of the quantities through random fields is possible (see e.g. [8]). Let D be a bounded
26

3.2 Stochastic Partial Differential Equations
domain in IR d , (Ω,A ,Π) a complete probability space, and a : Ω× ¯D → IR a stochastic function with
continuous and bounded covariance function that satisfies ∃a min ,a max ∈ (0,∞) with
P(ω ∈ Ω : a(x,ω) ∈ [a min ,a max ],∀x ∈ ¯D) = 1 , (3.8)
i.e. the diffusion coefficient is bounded away from zero and infinity for realizations ω ∈ Ω almost
sure. In addition, let f : Ω × ¯D → IR be a stochastic function that satisfies
∫ ∫
( ∫ )
f 2 (x,ω)dxdω = E f 2 (x,ω)dx < ∞ . (3.9)
Ω D
D
Then the elliptic SPDE analog to 3.7 reads
−∇ · (a(ω,·)∇u(ω,·)) = f (ω,·)
almost sure on D
u(·) = g(·) on ∂D .
Applying this concept to other PDEs yields parabolic and hyperbolic SPDEs.
3.2.1 Existence and Uniqueness of Solutions for Elliptic SPDEs
(3.10)
The proof of the existence and uniqueness of solutions for elliptic SPDEs is closely related to the
existence and uniqueness proof of the classical problem. The Lax-Milgram theorem [37] is applicable
in the stochastic context when we show continuity and coercivity of the related linear and
bilinear forms. The main difficulty of the proof is that the stochastic PDE requires the multiplication
of stochastic quantities, because the expression a∇u has to be well-defined. For this, we introduce
the Wick product [68, 155] and have to investigate conditions for its existence. Let us begin with
notation for the definition of the Wick product. The presentation is based on [150]. In the following
let {H α : α ∈ I}, where I is an index set, be an orthogonal basis of L 2 (Ω).
Definition The Wick product of two random variables f ,g : Ω → IR is the formal series
(
f g = ∑ f α g β H α+β (ξ ) = ∑ ∑ f α g β
)H γ (ξ ) , (3.11)
α,β
γ α+β=γ
whereas a random variable is expressed in the orthogonal basis via f = ∑ α f α H α (ξ ). The H α depend
on a vector ξ = (ξ 1 ,...) of basic random variables.
The Wick product is not well-defined for all second order random variables, i.e. L 2 (Ω) is not closed
under Wick multiplication (see [150]). Therefore, we introduce restrictions of the space L 2 (Ω) to
ensure a well-defined Wick multiplication.
Definition The Kondratiev-Hilbert spaces S ρ,k [85] are
{
}
(S ) ρ,k := f = ∑ α
f α H α : f α ∈ IR for α ∈ I and ‖ f ‖ ρ,k < ∞
where −1 ≤ ρ ≤ 1 and k ∈ IR. We define the norm ‖ · ‖ ρ,k via the scalar product
and the expression (2N) α via
, (3.12)
( f ,g) ρ,k := ∑ α
f α g α (α!) 1+ρ (2N) αk (3.13)
(2N) α :=
∞
∏
i=1
(2 d β (i)
1 β (i)
2
...β
(i)
d )α i
. (3.14)
The product ∏ ∞ i=1 is the product over all possible multi-indices β. Kondratiev spaces are separable
Hilbert spaces [150].
27

Chapter 3 SPDEs and Polynomial Chaos Expansions
Roughly speaking, a Kondratiev space S ρ,k is a subspace of L 2 (Ω), where the coefficients f α respect
a decay condition such that (3.13) stays finite.
The Kondratiev spaces from the previous definition are purely stochastic spaces. To add the spatial
dependencies we have to make the coefficients f α functions that depend on a spatial variable and
fulfill regularity assumptions.
Definition The Hilbert space (S ) ρ,k,m (D) is
{
}
(S ) ρ,k,m (D) := f (x) = ∑ α
f α (x)H α : f α ∈ H m (D) ∀α ∈ I
, (3.15)
and the scalar product is defined in the same way as the scalar product of the space S ρ,k , where the
scalar product of H m (D) ( f α ,g α ) H m replaces the expression f α g α .
After the definition of the basic spaces for the Wick product, we define a Banach space such that the
Wick product g → f g for every f from the Banach space is a continuous linear operator on (S ) −1,k,0
(see [150], Proposition 4).
Definition For D ⊂ IR d and l ∈ IR we define the space F l (D) via
{
F l (D) := f (x) = ∑ f α (x)H α : f α is measurable on D for every α and
α
(
) } (3.16)
‖ f ‖ l := esssup ∑ f α (x)|(2N) lα < ∞
x∈D α
and the space P l (D) via
P l (D) := { f ∈ F l (D) : ∃A > 0 such that (E( f )g,g) 0,D ≥ A‖g‖ 2 0,D ∀g ∈ L 2 (D) } . (3.17)
Using all the previous definitions it is possible to show existence and uniqueness of solutions of
SPDEs, because f ∈ F l ensures that the bilinear form is continuous and f ∈ P l the coerciveness of
the bilinear form. The existence and uniqueness result is originally by Vage [150]:
Theorem 3.1. Let D ⊂ IR d be an open set of finite diameter and suppose a ∈ P l (D) for some l ∈ IR.
Then there exists a constant K(a) ≤ 2l such that if k < K(a), (3.10) has a unique variational solution
u ∈ (S ) −1,k,1 for every f ∈ (S ) −1,k,0 and g ∈ (S ) −1,k,1 .
To sum up, we assure existence and uniqueness of solutions of SPDEs using the methods for PDEs,
when the Wick product a∇u is well-defined and the SPDE fulfills (3.8) and (3.9).
3.2.2 Parabolic SPDEs
We construct parabolic SPDEs from elliptic SPDEs in the same way as for classical PDEs. We have
to add time-dependence for the solution and incorporate an additional time derivative. We end up
with a prototype for parabolic SPDEs given by
u t (ω,x,t) − ∇ · (a(ω,x,t)∇u(ω,x,t)) = f (ω,x,t) almost sure on D × (0,T )
u(x,t) = 0 on ∂D × (0,T )
u(x,0) = u 0 on D × {0} .
(3.18)
Vage [150] proved existence and uniqueness of solutions for this kind of parabolic SPDEs. The
findings are condensed in the following theorem (cf. Theorem 4 in [150]).
Theorem 3.2. Let 0 < T < ∞, D ⊂ IR d be an open set of finite diameter, and a ∈ P l be given. Then
there exists a constant K(a) ≤ 2l such that if ρ = −1 and k < K(a), (3.18) has a unique solution
u ∈ W(0,T ) for any f ∈ L 2 (0,T ;((S ) −1,k,1
0
) ′ ) and u 0 ∈ (S ) −1,k,0 .
28

3.3 Polynomial Chaos Expansions
Figure 3.1: Relation between the stochastic spaces. We avoid the integration over Ω with respect to
the measure Π. Instead, we transform the integral into integration over a subset of IR (the
space Γ i ) with respect to the known PDF ρ of the basic random variables ξ i .
3.2.3 Doob-Dynkin Lemma
A famous result for the representation of the result of SPDEs is the Doob-Dynkin lemma. The
version given here is cited from [132]:
Lemma 1. Let (Ω,Σ) and (S,A ) be measurable spaces and f : Ω → S be a measurable function,
i.e. f −1 (A ) ⊂ Σ. Then a function g : Ω → IR is measurable relative to the σ-algebra f −1 (A ) if and
only if there is a measurable function h : S → IR such that g = h ◦ f .
This lemma ensures that the solution of an SPDE is representable in the same random variables as
the finite-dimensional input. This is due to the measurability of SPDEs when the coefficient is a
linear combination of a finite number of random variables, because random variables are measurable
by definition and the solution of an SPDE has to be continuous and thus is measurable. Furthermore,
the product of a measurable function is measurable.
Having the theory for existence and uniqueness of SPDE solutions at hand, we need a representation
of stochastic quantities compatible with numerical schemes to compute approximations of the
SPDE solutions. This approximation is based on the representation of random variables from (3.11).
3.3 Polynomial Chaos Expansions
The main contribution for the numerical treatment of SPDEs is the polynomial chaos expansion of
random variables. Based on the fundamental work of Wiener [156], who developed the polynomial
chaos for Gaussian processes, leading to a basis formed by Hermite-polynomials, Cameron and
Martin [27] proved that every random variable with a finite variance has a representation as Fourier-
Hermite series. Later, Xiu and Karniadakis [160] developed the Wiener-Askey polynomial chaos
or generalized polynomial chaos, which allows a representation of any random process with finite
second-order moments in the polynomial chaos with an optimal basis.
One main advantage of the representation of random variables in the polynomial chaos is the
simplification of the calculation of integrals over the stochastic part. For arbitrary random variables
with unknown probability density function, we have to calculate the integral over the abstract event
space Ω. The use of the polynomial chaos expansions allows us to transform this integral into an
integral over the real numbers by using the probability density function of the underlying random
29

Chapter 3 SPDEs and Polynomial Chaos Expansions
variables. Fig. 3.1 shows the situation. Instead of the direct computation of the integrals with the
random variable X and the measure Π, we transform the integration into integration over the real
numbers by using the polynomial chaos and the PDF ρ of the underlying random variables.
3.3.1 Wiener Chaos
In his seminal paper [156], Wiener developed the homogeneous (or Wiener) chaos formulated using
Hermite-polynomials in independent Gaussian random variables with zero mean and unit variance.
Let ˜ξ = (ξ 1 ,...) be a vector of independent Gaussian random variables with zero mean, unit variance
and PDFs ρ i , and V n (ξ i1 ,...,ξ in ) be Hermite-polynomials in n random variables. Cameron and
Martin [27] proved that a random variable X with finite second-order moments has the representation
X(ω) = a 0 V 0 +
∞
∑
i 1 =1
a i1 V 1 (ξ i1 (ω)) +
∞ ∞
∑ ∑
i 1 =1 i 2 =1
a i1 i 2
V 2 (ξ i1 (ω),ξ i2 (ω)) + ... . (3.19)
For notational convenience, this expression can be rewritten using multi-index notation
X(ω) = ∑ ∞ α=1 a αΨ α ( ˜ξ (ω)) . (3.20)
The functions V n and Ψ α have a one-to-one correspondence, i.e. every V n appears in the summation
over j, but has a different index. In what follows, we do not denote the dependence of ξ on ω
explicitly to ease notation when no integration over the stochastic space Ω is involved.
The Hermite-polynomials Ψ α form an orthogonal basis of the space L 2 (Ω), i.e.
∫
Ω
Ψ α ( ˜ξ (ω)
)
Ψ β ( ˜ξ (ω)
)dω = 〈Ψ α ,Ψ β 〉 = 〈(Ψ α ) 2 〉δ αβ . (3.21)
For a finite number of basic random variables ξ = (ξ 1 ,...,ξ n ) we simplify (3.21) by using (3.1). The
scalar product 〈 f ,g〉 is
∫
∫
〈Ψ α (ξ ),Ψ β (ξ )〉 = Ψ α (ξ (ω))Ψ β (ξ (ω))dω = Ψ α (x)Ψ β (x)dΠ, (3.22)
Ω
Γ
where Γ = supp(ξ ) ⊂ IR n . It follows from (3.22) that the weighting function w that is needed to get
orthonormal polynomials is
1
w(x) = √ . (3.23)
(2π) n e − 1 2 xT x
This weighting function is the key to understand the good approximation quality of the Hermiteexpansion,
because the weighting function for the Hermite-polynomials is the same as the PDF of
an n-dimensional Gaussian random variable, i.e. w(x) = ∏ i ρ i = ρ(x). Xiu and Karniadakis [160]
investigated this correspondence between the weighting functions for the orthogonal polynomial
basis and the density functions of random variables. Section 3.3.3 summarizes the findings. Thus,
the computation of the scalar product reduces to integration over a subset of IR n . For this, we use a
quadrature rule. Since we are integrating polynomials, the usage of a suitable quadrature rule leads
to exact results up to numerical inaccuracies.
3.3.2 Cameron-Martin Theorem
The Wiener chaos is an abstract representation for random variables, but it is unclear whether it converges
to the desired random variable. The Cameron-Martin theorem [27] fills this gap of knowledge.
We present the theorem in the version proposed in [27], but with the notation used in this thesis.
30

3.3 Polynomial Chaos Expansions
Theorem 3.3. The Wiener chaos representation of any random variable X ∈ L 2 (Ω) converges in the
L 2 (Ω)-sense to X. This means, if X is any functional for which
∫
|X(ω)| 2 dω < ∞ , (3.24)
Ω
then
∫
lim |X(ω) −∑ N N→∞
α=1 a αΨ α (ξ (ω))| 2 dω = 0 . (3.25)
Ω
The Fourier-Hermite coefficient a α is
∫
a α = X(ω)Ψ α (ξ (ω))dω . (3.26)
Ω
The Cameron-Martin theorem ensures that every random variable with finite variance has a representation
in the Wiener chaos, but gives no information about the convergence rate of the representation.
The convergence rate is important when the series expansion is cut after a finite number of terms.
This is necessary for numerical algorithms dealing with polynomial chaos expansions. In fact, [160]
showed that the convergence rate of the Wiener chaos is substantially less the optimal, exponential,
convergence rate. The development of other chaos types leads to expansions that have better convergence
properties. This is the topic of the next section, which introduces the generalized polynomial
chaos expansion, originally proposed by Xiu and Karniadakis [160].
3.3.3 Generalized Polynomial Chaos
Xiu and Karniadakis [160] generalized the idea of the representation of random variables in an orthogonal
basis formed by polynomials in random variables with known distribution. They proposed
to use polynomials whose weighting functions correspond to the PDF of the underlying random variables.
It turns out that these polynomials are the polynomials from the Askey-scheme [16]. Table 3.2
shows the correspondence between important random variables and the associated polynomials. To
summarize, a random variable with finite variance has a representation in the polynomial chaos by
X(ω) = ∑ ∞ α=1 a αΨ α (ξ ) , (3.27)
where the multi-dimensional polynomials are selected from the Askey-scheme [16]. The multidimensional
polynomials are constructed from one-dimensional polynomials via
ψ α = ∏ n i=1 H α i
(ξ i ) , (3.28)
whereas α is the index corresponding to the multiindex (α 1 ,...,α n ) and H αi , i = 1,...,n are polynomials
in one random variable. Fig. 3.1 shows the first one-dimensional polynomials for the Legendrechaos
and Fig. 3.3 the polynomials for the Hermite-chaos. We rescaled the Legendre- and Hermitepolynomials
to get an orthonormal basis of L 2 (Ω) with respect to the weighted scalar product, i.e.
〈Ψ α ,Ψ β 〉 = δ αβ , (3.29)
because the weighting functions for the random variables are 0.5 and 1 √
2π
exp −x2
2 , respectively.
Ernst et al. [50] proved that the polynomial chaos expansion converges in quadratic mean, i.e. in
the L 2 (Ω) sense [73], if and only if the basic random variables have finite moments of all orders and
the probability density of the basic random variables is continuous. Furthermore, the moment problem
(cf. [50]), i.e. the identification of the measure from the moments, has to be uniquely solvable.
Nouy [116] showed that multimodal random variables are hard to approximate in a onedimensional
polynomial chaos expansion. He solved this problem by introducing a special kind
31

Chapter 3 SPDEs and Polynomial Chaos Expansions
H 1 (x) = 1
H 2 (x) = √ 3x
H 3 (x) = √ 5 · (1.5 ∗ x 2 − 0.5)
H 4 (x) = √ 7 · (2.5x 3 − 1.5 ∗ x)
H 5 (x) = √ 9 · 1
8 (35x4 − 30x 2 + 3.0)
H 6 (x) = √ 11 · 1
8 (63x5 − 70x 3 + 15x)
H 7 (x) = √ 13 · 1
16 (231x6 − 315x 4 + 105x − 5)
H 8 (x) = √ 15 · 1
16 (429x7 − 693x 5 + 315x 3 − 35x)
H 9 (x) = √ 1
17 ·
128 (6435x8 − 12012x 6 + 6930x 4 − 1260x 2 + 35)
H 10 (x) = √ 1
19 ·
128 (12155x9 − 25740x 7 + 18018x 5 − 4620x 3 + 315x)
Table 3.1: The first ten one-dimensional Legendre-polynomials. The multi-dimensional polynomials
up to degree nine are based on these polynomials and (3.40).
of polynomial chaos expansion. In this expansion, one random variable acts as indicator function for
the modes of the approximated random variable. Then the multimodal random variable is approximated
on all modes independently. Wan and Karniadakis [152] introduced a similar approach called
multi-element polynomial chaos (MEPC). The idea of this method is to decompose the stochastic
space into smaller elements. Since we are approximating L 2 -functions in the stochastic space, we
need no coupling condition between the stochastic elements, i.e. the solutions may have jumps across
the elements. This allows for an efficient parallelization of the MEPC, because it is possible to perform
the computations for elements in the stochastic space on different machines and there is no
need for communication between the machines.
To use the polynomial chaos expansion in numerical schemes makes it necessary to cut of the
series expansion after a finite number of terms. This is done by choosing the number of random
variables used for the approximation, denoted by n and by prescribing the maximal polynomial
degree p in the expansion. As usual for a polynomial basis, the number of terms in the expansion is
( ) n + p
N = . (3.30)
p
Random variable Wiener-Askey chaos Support
Gaussian Hermite-Polynomials (−∞,∞)
Gamma Laguerre-Polynomials [0,∞)
Beta Jacobi-Polynomials [a,b]
Uniform Legendre-Polynomials [a,b]
Poisson Charlier-Polynomials discrete
Binomial Krawtchouk-Polynomials discrete
Table 3.2: Important distributions and the corresponding polynomials for the expansion.
32

3.3 Polynomial Chaos Expansions
H 1 (x) = 1
H 2 (x) = x
H 3 (x) = 1 √
2!
(x 2 − 1)
H 4 (x) = 1 √
3!
(x 3 − 3x)
H 5 (x) = 1 √
4!
(x 4 − 6x 2 + 3)
H 6 (x) = 1 √
5!
(x 5 − 10x 3 + 15x)
H 7 (x) = 1 √
6!
(x 6 − 15x 4 + 45x 2 − 15)
H 8 (x) = 1 √
7!
(x 7 − 21x 5 + 105x 3 − 105x)
H 9 (x) = 1 √
8!
(x 8 − 28x 6 + 210x 4 − 420x 2 + 105)
H 10 (x) = 1 √
9!
(x 9 − 36x 7 + 378x 5 − 1260x 3 + 945x)
Table 3.3: The first ten one-dimensional Hermite-polynomials. The construction of the multidimensional
polynomials up to degree 9 is based on these polynomials and (3.40).
Thus, it is necessary to select a vector of random variables ξ = (ξ 1 ,...,ξ n ) and the polynomial
degree p. Then, an approximation of a random variable in the polynomial chaos is
X(ω) ≈ ∑ N α=1 a αΨ α (ξ ) . (3.31)
The remaining part of this chapter deals with numerical methods for polynomial chaos expansions.
Although the presented material is valid for all polynomials from the Askey-scheme, the numerical
implementation is based on the Legendre-polynomials and uniform distributed random variables,
because the support of the Legendre-polynomials is compact. This is advantageous for algorithms,
especially when dealing with stochastic level sets. Chapter 4 discusses the combination of polynomial
chaos expansions and SPDEs. There the information presented for the polynomial chaos is
combined with finite element and finite difference schemes for the discretization of the equations.
3.3.4 Calculations in the Polynomial Chaos
To use the polynomial chaos in numerical schemes it is necessary to perform arithmetic operations
in the polynomial chaos. In this section, we review the development of the basic operations like
addition, subtraction, multiplication, division and the calculation of square roots. The presentation
is based on the work of Debusschere et al. [38]. For the remaining part of this section let
a = ∑ N α=1 a αΨ α (ξ ), b = ∑ N α=1 b αΨ α (ξ ), c = ∑ N α=1 c αΨ α (ξ ) (3.32)
be three polynomial chaos variables. We compute the sum and the difference of quantities in the
polynomial chaos by adding or subtracting the corresponding chaos coefficients, because the addition
or subtraction of polynomials results in a polynomial with the same degree at most:
c = a ± b = ∑ N α=1 a αΨ α (ξ ) ±∑ N α=1 b αΨ α (ξ ) = ∑ N α=1 (a α ± b α )Ψ α (ξ ) . (3.33)
33

Chapter 3 SPDEs and Polynomial Chaos Expansions
The multiplication of two polynomial chaos variables is more difficult. Since polynomials form
the basis, the naive multiplication of polynomial chaos variables results in a polynomial with twice
the degree of the factors. Thus, an additional projection step onto a polynomial with the same degree
as the factors of the multiplication is necessary. This projection step is done by using the Galerkin
or L 2 -projection, leading to a projection polynomial, whose error is orthogonal to the space spanned
by the polynomial chaos. The idea of the projection is to multiply the naive product c = a · b with an
element Ψ γ of the polynomial chaos basis, integrate over the stochastic dimensions and to compute
the coefficient of the multiplication one after another from this expression:
∫
Γ
N
∑
α=1
∫
c α Ψ α Ψ γ dΠ =
Γ
N
∑
α=1
N
∑
β=1
a α b β Ψ α Ψ β Ψ γ dΠ ⇒ c γ =
N
∑
α=1
N
∑
β=1
〈Ψ α Ψ β Ψ γ 〉
a α b β
〈(Ψ γ ) 2 . (3.34)
〉
} {{ }
C αβγ
Note that we omit denoting the dependence of Ψ α from ξ and ω to simplify the notation. The
quantity C αβγ is independent of the actual problem, it depends on the basis only. The values of C αβγ
can be precomputed in a lookup table. The next section describes the generation of this table.
The computation of the quotient of two random variables, a = c b
is possible, too. To do this, we
multiply the expression by b, yielding c = ab and use again the Galerkin projection for this equation:
c γ = ∑ N α=1∑ N β=1 C αβγb β a α = ∑ N α=0 A γαa α . (3.35)
This is a system of linear equations for the coefficients a α , which we solve by an iterative solver.
In a similar manner, we compute the square root b = √ a of a polynomial chaos variable. First, we
rewrite the equation in the form a = b 2 and then use the Galerkin projection to obtain
a γ = ∑ N α=1∑ N β=1 C αβγb α b β . (3.36)
This is a nonlinear system of equations for the unknown coefficients b α , which we solve using
Newton’s method to find a root of
f (b) = b 2 − a . (3.37)
The partial derivatives of this function are
∂ f α (b)
=
∂b β
N
∑
γ=1
C βγα b γ . (3.38)
As pointed out by Matthies and Rosic [98], it is possible to use a mild convergence criterion for
Newton’s method depending on the expected value and the variance of the polynomial chaos variable.
Using these building blocks, it is possible to construct numerical methods for nearly all possible
calculations, e.g. the exponential of a random variable in the polynomial chaos is
exp(a) = exp(a 1 )
(
1 +
K
∑
n=1
(
∑
N
α=2 a α Ψ α) )
n
n!
. (3.39)
With the methods from this section, it is also possible to construct finite difference schemes for
random variables. Chapter 4 investigates this further.
3.3.5 The Stochastic Lookup Table
We precompute the values of C αβγ in a lookup table to speed up the calculations in the polynomial
chaos. It is possible to replace the calculation of the multi-dimensional integrals ∫ Γ Ψα Ψ β Ψ γ dΠ
34

3.4 Relation to Interval Arithmetic
Figure 3.2: Sparsity structure of the stochastic lookup table for n = 5 random variables and a polynomial
degree p = 3. The gray dots indicate positions in the three-dimensional lookup
table C αβγ that contain nonzero entries.
by one-dimensional integration, because the basis functions are Ψ α = ∏ n j=1 H α j
(ξ j ) whereas α corresponds
to the multi-index (α 1 ,...,α n ) and H αi are polynomials in one random variable. Using
the product representation of the polynomials, we simplify the equation by using that the random
variables ξ i are statistically independent, i.e. E(ξ i ξ j ) = E(ξ i )E(ξ j ):
)
∫
〈Ψ α Ψ β Ψ γ 〉 =
dΠ
=
Γ
n
∏
m=1
(
n
∏
m=1
∫
H (i) α m
Γ m
n
H (i) α
(ξ m ))(
∏
m
m=1
(ξ m )H β
( j)
m
n
H ( j) β
(ξ m ))(
∏
m
m=1
(ξ m )H (k) γ
(ξ m )dΠ m .
m
H (k) γ
(ξ m )
m
(3.40)
In (3.40) Π m = ρ m Π m ,i = 1,...n denotes integration with respect to the probability measures of the
random variables ξ m ,m = 1,...n.
3.4 Relation to Interval Arithmetic
Interval arithmetic [64,78,102,104] is a possibility for reliable computations on a computer. Instead
of using a single fixed number, this concept is based on intervals of numbers to provide an upper and
a lower bound for the computation result. The result is considered to be uniformly distributed inside
this interval. Arithmetic operations for these reliability intervals are defined via the lower and upper
bounds of the intervals. Let x = [x, ¯x],y = [y,ȳ] be two intervals and ◦ one of the operations +,−,×,/.
Then the resulting interval is defined as
[x, ¯x] ◦ [y,ȳ] = [ min ( x ◦ y,x ◦ ȳ, ¯x ◦ y, ¯x ◦ ȳ ) ,max ( x ◦ y,x ◦ ȳ, ¯x ◦ y, ¯x ◦ ȳ )] . (3.41)
The definition of the new interval bounds based on the old interval bounds is useful when dealing
with monotonic functions only, e.g. computing the sine function of an interval fails, because
35

Chapter 3 SPDEs and Polynomial Chaos Expansions
Figure 3.3: PDFs of initial uniformly distributed input intervals (gray) and the PDFs of the results of
the polynomial chaos computation (black) for squaring an interval (left) and dividing an
interval by itself (right).
sin([30 ◦ ,150 ◦ ]) = [0.5,0.5] in the above arithmetic. Other problems are that there are no simple
methods to link realizations of intervals, e.g. the naive multiplication yields [−2,2] 2 = [−4,4], because
there is no information in the interval arithmetic calculus that the realizations of both intervals
must be the same. This is also problematic for division, because dividing an interval by itself should
result into an interval with zero width, but, e.g. 2x/x for x = [2,4] yields the interval [1,4], not the
desired interval [2,2]. Furthermore, the result is forced to be uniformly distributed inside the resulting
interval, which is not the case for nonlinear operations and the resulting intervals can become
arbitrarily large. Nevertheless, interval arithmetic is used in applications [70].
The polynomial chaos expansion can be thought as an extension of the interval arithmetic calculus.
The results of polynomial chaos calculations are not forced to be uniformly distributed. Instead, they
can have every distribution that can be represented in the chosen polynomial chaos basis. Results
that can not be represented in this basis are projected onto the basis using the Galerkin projection
introduced earlier. Fig. 3.3 shows the polynomial chaos result of the problematic operations squaring
an interval and dividing by an interval. In both cases the polynomial chaos expansions yields the
exact result, up to the machine precision. Furthermore, the problem of the huge resulting intervals
of interval arithmetic operations is solved by using polynomial chaos expansions, because events at
the tails of the intervals have a very low probability.
Conclusion
This chapter provided us with a possibility for the finite-dimensional approximation of arbitrary
random variables, the polynomial chaos expansion. Even if the representation is possible for second
order random variables only, this is sufficient for random variables arising in the image processing
context. The finite-dimensional approximations and the associated closed computations provide a
powerful toolbox for the discretization of SPDEs and a stochastic modeling of image processing
problems. With the presented theoretical background, it is also possible to prove existence and
uniqueness of solutions for the stochastic image processing models in the preceding chapters. It
remains to show the few, easy to verify, assumptions.
36

Chapter 4
Discretization of SPDEs
The discretization of SPDEs is an active research field [47, 159]. Besides the discretization based
on sampling approaches like Monte Carlo simulation or stochastic collocation, there are methods for
the intrusive computation of stochastic solutions in the literature. Intrusive means that we do not
generate the solutions based on sampling strategies and we cannot reuse deterministic algorithms for
the solution at the sampling points. This makes it necessary to develop new algorithms, but has the
advantage that these algorithms are more efficient than the classical sampling approaches. This thesis
focuses on intrusive methods. We present the sampling based approaches as well, but use them to
verify the correctness of the intrusive algorithms and implementations only. The intrusive methods
presented in this thesis range from the stochastic finite difference method based on polynomial chaos
expansions to the generalized spectral decomposition [113–115, 117, 118], a method allowing to
speed up the solution process of the stochastic finite element method (SFEM) [54].
4.1 Sampling Based Discretization of SPDEs
Since the mid of the 20 th century [100, 101] authors developed sampling based algorithms for the
simulation of stochastic processes, starting with the development of the Monte Carlo method. Later,
advanced sampling-based methods like the stochastic collocation method and improvements of these
methods e.g. the combination of stochastic collocation and polynomial chaos expansions or the use
of sparse grids based on a Smolyak construction [140] have been developed.
4.1.1 Monte Carlo Simulation
Monte Carlo simulation is the simplest technique for the discretization of random variables and
SPDEs. A set of samples is generated randomly from the known distribution of the random variables
via a pseudo random number generator like [97]. We can use the well-known deterministic
algorithms on these samples and compute from the results approximations to stochastic quantities
like expected value, variance, etc. using well-known formulas. For example, when we computed the
solution of R samples, approximate expected value and variance are
E(x) ≈ ¯x = 1 R ∑R i=1 x i and Var(x) ≈ 1
R − 1 ∑R i=1 (x i − ¯x) 2 . (4.1)
The main drawback of the Monte Carlo method is the slow convergence. In fact, Kendall [79] showed
for the Monte Carlo method that the convergence of the samples mean towards the expected value
is of order O(σ/ √ R). Despite the slow convergence rate, Monte Carlo methods are widely used (see
e.g. [88, 94, 133]) due to the simple implementation and the possibility to reuse deterministic code.
4.1.2 Stochastic Collocation
During the last years, a variety of stochastic collocation (SC) techniques was developed. These
techniques range from simple collocation techniques over sparse grid techniques to SC techniques
allowing to obtain polynomial chaos coefficient (see [159] for a more detailed review).
37

Chapter 4 Discretization of SPDEs
Figure 4.1: Comparison between a sparse grid (left) constructed via Smolyak’s algorithm and a full
tensor grid (right). The sparse grid contains significantly less nodes than the full tensor
grid whose number of nodes growth exponentially with the dimension, but has nearly the
same approximation order.
SC is a non-intrusive approach for the discretization of SPDEs. In the simplest form, SC uses
known points in the stochastic dimensions and performs runs of the deterministic problem for these
points. The points are chosen following a quadrature rule, e.g. Gauss quadrature or Clenshaw-Curtis
quadrature [33] where the points are selected based on the roots of the Chebyshev-polynomials [151].
We construct higher-dimensional SC from the one-dimensional SC using tensor grids. This simple
approach leads to a “curse of dimension” [119] when using the full tensor grids in higher dimension.
To overcome this, we use Smolyak’s algorithm [122, 140], resulting in a sparse grid containing
significant less nodes than the full tensor grid (see Fig. 4.1), but resulting in an approximation with
nearly the same approximation order. The orders differ only by a logarithmic term, see [52].
Following [158] it is possible to obtain a representation in the polynomial chaos from SC calculations.
Having the usual polynomial chaos expansion (cf. Section 3.3)
u(x,ω) = ∑ N α=0 a α(x)Ψ α (ξ (ω)) (4.2)
in mind, we get the coefficients of the polynomial chaos from the collocation samples via a projection
on the polynomial chaos
a α (x) = ∑ Q j=1 u(x,y( j) )Ψ α (y ( j) )w j , (4.3)
where y ( j) are the collocation points, w j the corresponding quadrature weights and Q the total number
of collocation samples. This collocation approach allows an easy comparison of results obtained
via SC and results from intrusive techniques presented in the following paragraphs. Besides the
calculation of a polynomial chaos representation, the usual usage of SC is the computation of a
Lagrange interpolation of the solution, i.e. to compute a representation like
where L j are the Lagrange-polynomials.
I u(x,ω) = ∑ Q j=1 u(x,y(i) )L j (ω) , (4.4)
4.2 Stochastic Finite Difference Methods
The sampling based approaches of the last section have the great advantage that calculations on
the samples use classical methods for the solution of PDEs like finite element or finite difference
38

4.3 Stochastic Finite Elements
methods. In the following, we present an approach, where we discretize the SPDE directly. To make
the approach more illustrative we demonstrate the method by using a parabolic SPDE
∂ t u(t,x,ω) − u xx (t,x,ω) = f (t,x,ω) . (4.5)
The temporal and spatial derivatives are determined using well-known approximations. Using the
explicit Euler scheme for the discretization of the time derivative, we get
u(t + τ,x,ω) = u(t,x,ω) + τ(u xx (t,x,ω) + f (t,x,ω)) . (4.6)
Discretizing the spatial derivative using central differences, the fully discrete equation is
( )
u(t,x + h,ω) − 2u(t,x,ω) + u(t,x − h,ω)
u(t + τ,x,ω) = u(t,x,ω) + τ
+ f (t,x,ω)
h 2
. (4.7)
The stochastic quantities in this equation are approximated by using a truncated polynomial chaos
expansion leading to a numerical scheme that needs methods for the addition and multiplication of
polynomial chaos expansions. Section 3.3 presents numerical methods for this task.
The main drawback of these methods is that computations in the polynomial chaos require the
solution of linear systems of equations. Furthermore, the construction of unstructured or adaptive
grids is complicated in comparison to the generation of adaptive grids for finite elements.
The advantage of stochastic finite difference methods is the simple possibility to parallelize explicit
stochastic finite difference schemes, because the computations on different nodes are independent.
4.3 Stochastic Finite Elements
It is well-known that the variational formulation of a deterministic PDE is
find u ∈ V such that a(u,v) = b(v) ∀v ∈ V , (4.8)
where a(·,·) is a bilinear form related to the PDE and b(·) a linear form related to the right hand side
of the PDE. The space V is the space of all admissible functions, e.g. the Sobolev space H0 1 for the
simple prototype equation −∇ · (k∇φ) = f in D, φ = 0 on ∂D.
For stochastic coefficients, right hand sides or boundary conditions, the bilinear and/or linear
form become stochastic quantities. Denote by a(u,v,ω), b(v,ω) the dependence of the forms on the
stochastic event ω ∈ Ω. The aim of the stochastic problem is to find a random field, i.e. an element of
the tensor product space V ⊗S , u ∈ V ⊗S , where S is the space of random functions, e.g. L 2 (Ω),
the space of all random variables with finite second order moments. The weak formulation of the
stochastic problem is:
find u ∈ V ⊗ S such that A(u,v) = B(v) ∀v ∈ V ⊗ S , (4.9)
where
∫
A(u,v) = a(u,v,ω)dω = E(a(u,v,ω)) (4.10)
Ω
and
∫
B(v) = b(v,ω)dω = E(b(v,ω)) . (4.11)
Ω
The weak formulation of an SPDE is simply the expectation of the deterministic problem (4.8).. To
ensure existence and uniqueness of a solution, we need the form A to be continuous and coercive and
the form B to be continuous on the space V ⊗S . Hence, coercivity and continuity are ensured if the
forms a and b are coercive, respectively continuous, for elementary events ω ∈ Ω almost sure and
such that the Wick product is well-defined (cf. Section 3.2.1).
39

Chapter 4 Discretization of SPDEs
4.3.1 Discretization of the Spaces V and S
We approximate the deterministic space V using the classical finite element approach. That means
every u ∈ V ⊗ S is approximated by
u(x,ω) ≈ ∑ n i=1 u i(ω)P i (x) , (4.12)
where u i ∈ S and {P i } i=1,...,n is a basis of a finite dimensional subspace V h ⊂ V . We identify the
space V h with IR n , because we have to store the coefficients for the basis elements only.
We approximate the stochastic space S in two steps. First, we choose a finite set of random
variables ζ =(ζ 1 ,...,ζ m ), span(ζ 1 ,...,ζ m )=S m ⊂S with finite variance and approximate u i ∈ S by
u i (ω) ≈ ∑ m k=1 uk i ζ k (ω) , (4.13)
where the coefficients u k i are deterministic coefficients for the random variables ζ k . Numerical calculations
cannot use the space S m . Hence, we approximate the space S m by using the generalized
polynomial chaos [160], cf. Section 3.3. We approximate the random variables ζ i with unknown
distribution in the polynomial chaos by the same number of random variables and a prescribed polynomial
degree p:
u ∈ S m,p ⊂ S p : u = ∑ N i=1 u iΨ i (ξ ) . (4.14)
The dimension of the space S m,p is N = ( )
m+p
m .
For the finite dimensional subspace IR n ⊗ S p , the problem (4.9) is rewritten as
E(v T Au) = E(v T b) ∀v ∈ IR n ⊗ S p , (4.15)
where A ∈ Sp
n×n is a stochastic matrix.
Using the polynomial chaos basis, i.e. the space S m,p for the stochastic space and V h for the
deterministic space we end up with a huge deterministic equation system to approximate the solution
of ∇ · (a∇u) = f given by
∑ N α=1
where the matrices M α,β and L α,β are
(M α,β ) i, j = E (Ψ α Ψ β ) ∫
(
L α,β ) i, j = ∑
k
(
L α,β ) U α = ∑ N α=1 Mα,β F α , (4.16)
(
∑E
γ
D
P i P j dx
Ψ α Ψ β Ψ γ) a k γ
∫
D
∇P i · ∇P j P k dx .
(4.17)
In (4.16) we used the notation F α = ( f i α) for the polynomial chaos representation of the quantities.
4.4 Generalized Spectral Decomposition
Selecting suitable subspaces of S m,p ⊗IR n and a special basis, which captures the dominant stochastic
effects, we achieve a significant speed-up of the solution process and an enormous reduction of
the memory requirements. In the generalized spectral decomposition (GSD) [113], we approximate
the solution u ∈ L 2 (Ω) ⊗ H 1 (D) by
u(x,ξ ) ≈ ∑ K j=1 λ j(ξ )V j (x) , (4.18)
where V j is a deterministic function, λ j a stochastic function and K the number of modes of the
decomposition. Thus, the GSD computes a solution where the deterministic and the stochastic basis
40

4.4 Generalized Spectral Decomposition
functions are not fixed a priori. With the flexible basis functions we find a solution having significant
fewer modes, i.e. K ≪ N, but nearly the same approximation quality.
Nouy [113] showed how to compute the modes of an optimal approximation in the energy norm
‖v‖ 2 A = E(vT Av) of the problem, i.e. such that
∥
∥
∥u −∑ K ∥∥
j=1 λ 2
∥
jU j = min ∥
∥u −∑ K ∥∥
A γ,V
j=1 γ 2
jV j . (4.19)
A
The next sections provide details about the GSD method, proofs for the optimality of the approximation
and implementation details. Further details about the GSD method can be found in [113].
4.4.1 Best Approximation
For deterministic linear systems of equations, it is possible to formulate an associated minimization
problem, whose solution is the same as the solution of the weak formulation. For the discrete version
of SPDEs, this minimization problem allows developing efficient methods for the solution of the
weak formulation.
The discrete version of the problem (4.15) is equivalent to the minimization problem
( 1
J (u) = min J (v), where J (v) = E
v∈IR n ⊗S p 2 vT Au − v b)
T
. (4.20)
This equivalence is well-known for deterministic problems, but holds for the expectation in stochastic
equations, too. Using this relation, the best approximation of order M is
J
(
∑
M
i=1 λ iU i
)
( )
M
= min J
V 1 ,...V M ∈IR ∑ n i=1 γ iV i
γ i ,...,γ M ∈S p
. (4.21)
It is well-known that in the deterministic setting the best approximation can be defined recursively:
Let (λ 1 ,...,λ M−1 ),(U 1 ,...,U M−1 ) be the best approximation of order M − 1. Then the best approximation
of order M is
J
(
∑
M
i=1 λ iU i
)
(
)
= min J γV +∑ M−1
V ∈IR n
j=1 λ iU i
γ∈S p
. (4.22)
This recursive definition is in general not true in the stochastic case (see the following calculations),
but numerical tests show that we achieve good approximations for stochastic operators. With the
recursive definition, we develop efficient numerical schemes for the solution of the minimization
problem. The functional decomposes into two parts when we use the recursive definition:
(
λ M U M +∑ M−1
i=1 λ iU i
)
J
)
1
= E(
2 (λ MU M ) T Au − (λ M U M ) T b
( 1
(
M−1
+ E
2 ∑
)
i=1 λ iU i ) T b
i=1
) T λ iU i Au − ( ∑ M−1
} {{ }
already minimized
. (4.23)
The second summand of the equation is minimized already due to the recursive definition of the
minimization. Introducing the residual values
ũ = u −∑ M−1
i=1 λ iU i and ˜b = b − ∑ M−1
i=1 Aλ iU i (4.24)
41

Chapter 4 Discretization of SPDEs
and performing an additional calculation for the first term results in
)
1
E(
2 (λ MU M ) T Au − (λ M U M ) T b
( 1
= E
2 (λ MU M ) T Aũ − (λ M U M ) T˜b 3
( ) )
+
2 (λ MU M ) T M−1
A ∑ i=1 λ iU i .
(4.25)
In the deterministic case, the product (λ M U M ) T A ( ∑ M−1
i=1 λ )
iU i is equal to zero because Ui are eigenvectors
of the operator A and therefore U ⊥ AU i . In the stochastic case, this is not true, but we neglect
this small error. However, the numerical results are reasonable. We introduce a functional J ˜ :
J ˜
1
(λ M U M ) = E(
2 (λ MU M ) T Aũ − (λ M U M ) T˜b
)
. (4.26)
With the functional J ˜ , we transformed the minimization problem (4.20) into a series of simpler
minimization problems. The next step is to find a method that allows an efficient solution of the
problem series given by (4.26).
Remark 6. The definition of the best approximation can be rewritten in matrix form as follows. Let
W = (U 1 ,...,U M ) ∈ IR n×M be the matrix of all coefficients and Λ = (λ 1 ,...,λ M ) T ∈ IR M ⊗ S p the
vector of stochastic functions. Then, (4.20) can equivalently be written
J (WΛ) =
min
W∈IR n×M
Λ∈IR M ⊗S p
J (WΛ) , (4.27)
and (4.26) can be written as
˜ J (U M λ M ) =
min
V ∈IR n
γ∈IR⊗S p
˜ J (V γ) . (4.28)
4.4.2 Stationary Conditions for the Functionals J and J ˜
The simultaneous minimization of the functional J or J ˜ for deterministic W and stochastic Λ,
respectively λ M and U M , is difficult due to the high dimension of the product space IR n ⊗ S p . A possibility
to avoid the simultaneous minimization is to fix either the deterministic W or the stochastic Λ.
For a fixed W, the stationary condition of J for Λ is
and for fixed U the stationary condition of
E ( Λ ∗T (W T AW)Λ ) = E ( Λ ∗T W T b ) ∀Λ ∗ ∈ IR M ⊗ S p , (4.29)
˜ J for λ is
E ( λ ∗ (Ui T AU i )λ ) = E ( λ ∗ Ui
T ˜b ) ∀λ ∗ ∈ S p . (4.30)
Having the optimal Λ (or λ M ) at hand, we try to find a better W (or U M ) by solving stationary
conditions for fixed Λ (or λ M ).
For a fixed Λ, the stationary condition of J for W is
and for fixed λ the stationary condition of
E ( Λ T (W ∗T AW)Λ ) = E ( Λ T W ∗T b ) ∀W ∗ ∈ IR n×M , (4.31)
E ( λ(U ∗T
i
˜ J for U is
AU i )λ ) = E ( λUi
∗T ˜b ) ∀U ∗ ∈ IR n . (4.32)
Iterating these stationary conditions leads to a solution of the coupled problem (4.27).
42

4.4 Generalized Spectral Decomposition
Algorithm 1 A GSD algorithm
1: u ← 0, ˜b ← b
2: for i = 1 to M do
3: λ i ← λ 0
4: for k = 1 to k max do
5: U i ← E(Aλi 2)−1
E(˜bλ i )
6: U i ← U i /‖U i ‖
7: λ i ← (Ui T AU i ) −1 Ui
T ˜b
8: end for
9: u ← u + λ i U i
10: ˜b ← ˜b − Aλ i U i
11: end for
4.4.3 An Algorithm for the GSD
Having all the primarily presented results at hand, we combine them for a first algorithm for the
numerical solution of an SPDE. The presented algorithm is a simplification of the power-type GSD
algorithm presented by Nouy [113] and given in pseudo-code in Algorithm 1. The algorithm uses
the presented iterative minimization of the functional J ˜ by iterating the stationary conditions (4.30)
and (4.32). The expression in line 5 of the algorithm is a direct consequence of (4.32), the expression
in line 7 a consequence of (4.30). As stated by Nouy [113], k max = 3,4 and M = 8 is sufficient for a
good approximation of the solution, i.e. in a numerical algorithm we perform k max inner iterations of
(4.30) and (4.32) to find a new stochastic and a new deterministic basis function. Furthermore, we
use a subspace spanned by M deterministic and M stochastic functions.
4.4.4 Relation to the Karhunen-Loève Expansion
The Karhunen-Loève expansion [95] is the classical way for the approximation of stochastic quantities.
The expansion minimizes the distance between the solution u and an approximation of order
M, i.e. we minimize the expression
∥
∥u −∑ M i=1 λ iU i
∥ ∥∥
2
=
∥ ∥ ∥∥u min −
V 1 ,...V M
∑ M ∥∥
∈IR n i=1 γ 2
iV i . (4.33)
γ i ,...,γ M ∈S p
The GSD minimizes the expression ∥ ∥u − ∑ M i=1 γ iV i
∥ ∥
2
A , where the norm is ‖v‖2 A = E(vT Av), although
the solution u is unknown before. Thus, the GSD computes the best approximation of a given order
of the unknown solution, whereas we measure the distance between solution and approximation in
the energy norm of the problem.
Remark 7. The possibility to compute the best approximation of an unknown solution is the great
advantage of the GSD, because the Karhunen-Loève expansion is able to compute an approximation
with fewer modes to a known quantity only.
4.4.5 Using the Polynomial Chaos Approximation with the GSD
We cannot implement the GSD algorithm presented above directly, because it is formulated for random
variables. Thus, an additional approximation step, e.g. the approximation of random variables in
the polynomial chaos, is necessary to end up with a useful algorithm. Using the notation introduced
earlier, we reformulate the steps 5 and 7 of the algorithm above. These steps are the complicated
43

Chapter 4 Discretization of SPDEs
Figure 4.2: Comparison of discretization methods with respect to implementational effort and speed.
steps, in which we have to generate and solve equation systems. We reformulate the remaining steps
in the same fashion. In step 5, we have to solve the system
Using the polynomial chaos, the matrix is
E(Aλ i λ j ) = ∑
α
E(Aλ 2
i )U i = E(˜bλ i ) . (4.34)
∑
β
E (˜bλ i
)
= ∑
α
(
∑λ i,α λ j,β A γ E Ψ α Ψ β Ψ γ) . (4.35)
γ
The generation of the system matrix benefits from the generation of the lookup tables presented in
Section 3.3.5. We generate the right hand side in a similar fashion:
( ) Ψ α Ψ β . (4.36)
∑
˜b α λ i,β E
β
The value E ( Ψ α Ψ β ) is extracted from the lookup table by setting Ψ γ = 1. To sum up, the generation
of the equation system requires the summation of values weighted by entries from the lookup table.
The generation of the matrix and the right hand side can be parallelized.
In Step 7, we have to solve the system
E ( Ui T AU i Ψ α) λ i = E ( Ui
T ˜bΨ α) . (4.37)
Using the polynomial chaos for (4.37) results in a summation for the matrix and the right hand side:
E ( Ui T AU i Ψ α) λ i = ∑∑Ui T A β U i λ i,γ E
(Ψ α Ψ β Ψ γ)
γ β
E ( U T
i
˜bΨ α) = ∑
β
Ui
T ˜b β E
(
Ψ α Ψ β ) .
(4.38)
To conclude, having a polynomial chaos approximation of the stochastic quantities, the GSD is
implemented efficiently using the lookup table from Section 3.3.3. Furthermore, to improve the
efficiency we skip the calculation as early as possible when the lookup table entry is zero.
Fig. 4.2 compares the discretization methods presented in this chapter with respect to the implementational
effort and the speed of the methods. The sampling based methods Monte Carlo simulation
and stochastic collocation are easy to implement due to the possibility to reuse existing code.
The drawback of these methods is the slow convergence of these methods towards the stochastic
solution. The intrusive methods Stochastic FEM and the GSD need a lot of implementational effort
because they cannot reuse existing deterministic code. The advantage of the these methods is the fast
calculation of the stochastic result compared to the sampling based approaches.
44

4.5 Adaptive Grids
Figure 4.3: Refinement of a rectangular element of a finite element mesh. A single element on a
coarser level splits up into four elements on the next finer level.
4.5 Adaptive Grids
To improve the efficiency of the GSD further, we combine the GSD with an adaptive grid approach
for the spatial dimensions. Classically, images are represented by a regular grid, see Section 2.1.
The discretization of stochastic images using regular image grids and the polynomial chaos will be
described in detail in Section 5.1. Using adaptive grids for the spacial discretization we are able to use
an optimal small basis in the stochastic dimensions through the GSD and a minimal set of nodes in
the spatial dimensions, which reduces the memory requirements due to the tensor product structure.
We adopt the adaptive grid approach from [129], which is based on rectangular elements and a
quadtree structure for the refinement of the elements. Fig. 4.3 shows the refinement of a single
element. The main idea is to start on the finest grid level and to coarsen an element if the error
indicator S(x) of every node x of the element is smaller than a threshold ι.
As error indicator, we used the gradient of the expected value of the solution, i.e.
S(x) = |∇(E(u(x)))| . (4.39)
The adaptive coarsening of rectangular elements leads to constrained or hanging nodes, i.e. nodes
that are not vertices of all neighboring elements, see Fig. 4.4. These nodes need special handling
when we assemble the FE-matrices, because these nodes are not usual degrees of freedom. Instead,
they are constrained by the nodes which lie on the edges of the face the node lies on (see Fig. 4.4).
For details about the assembling of the FE-matrices with hanging nodes, we refer to [120, 129].
The error indicator S leads to problematic situations, in which the constraining node of a hanging
node is also a hanging node on the next coarser level. Fig. 4.5 shows such a situation. To avoid this,
the error indicator has to be saturated, as pointed out e.g. in [120, 129]. Following these references
the saturation condition is as follows.
Saturation condition. An error indicator value S(x) for x ∈ N (E) is always greater than every
error indicator S(x C ) for x C ∈ N C (E). In this formula, N (E) are the nodes of the element E and
N C (E) are the new nodes due to refinement of the element E.
Figure 4.4: Refinement of elements leads to hanging nodes (circles) which are no degrees of freedom,
instead the values of the constraining nodes (squares) restrict them.
45

Chapter 4 Discretization of SPDEs
Figure 4.5: For an unsaturated error indicator, the appearance of hanging nodes constrained by hanging
nodes (due to level transitions of more than one between neighboring elements) is
possible (left). The saturation of the error indicator ensures that there are level one transitions
between neighboring elements only (right).
This saturation condition ensures that there is a level one transition between neighboring elements
only. Furthermore, we have to avoid the refinement of coarsened elements. Otherwise it is possible
to end up in a situation where an element is refined in step n, coarsened in step n + 1 and so on. A
slightly modified error indicator ˜S, which we define as the minimum of the actual error indicator and
the error indicator of the previous iteration, achieves this. Alternatively, the refinement of coarsened
elements can be avoided by using different thresholds for coarsening and refinement [34].
4.5.1 Combining GSD and Adaptive Grids
The combination of adaptive grids with the GSD method is straightforward. We assemble the
stochastic matrices in the same way as the deterministic matrices. After the solution of the system
is available, we interpolate the values on the hanging and inactive nodes. The only difficulty
arises for the generation of the equation system for the new stochastic basis element (equation (4.30)
respectively line 7 of the algorithm). There we have to compute the scalar product 〈U i ,AU i 〉 using the
adaptive matrix A. The product AU i has a different weight for the constraining nodes than the vector
U i , because the matrix has additional weights from the hanging nodes at the constraining nodes.
We propose to add these weighting factors to the vector U i .
Conclusion
We presented methods for the discretization of SPDEs. Based on sampling strategies we presented
Monte Carlo simulation and stochastic collocation with full or sparse grids constructed via Smolyak’s
algorithm. This thesis uses the sampling based approaches to verify the implementations of the intrusive
methods. Intrusive methods do not use a sampling strategy to solve the SPDEs. Instead, they
are based on a development of numerical schemes acting on random variables. Intrusive methods
are the key to the efficient numerical solution of SPDEs arising in image processing, because other
methods are orders of magnitude too slow or provide inaccurate results after an adequate period. We
presented the SFEM and the GSD method that tries to speed up the solution process on the SFEM
by constructing an optimal, problem dependent, subspace.
With this chapter, we have the fundamentals at hand to develop the concept of stochastic images
and to design image processing operators acting on these stochastic images.
46

Chapter 5
Stochastic Images
As described in Section 2.5, noise corrupts classical images. The repeated acquisition of the same
scene does not give identical images, because the noise typically is a stochastic quantity. Furthermore,
applying segmentation methods to two randomly chosen samples from the same scene yield
different results due to the noise. To model the noise of the acquisition process, we identify pixels
by random variables, i.e. identify images by random fields. Assuming that these stochastic images
fulfill mild regularity assumptions (H 1 -regularity in the spatial dimensions and L 2 -regularity in the
stochastic dimensions), they are elements of the tensor product space H 1 (D) ⊗ L 2 (Ω) introduced in
Section 3.1. We discretize this tensor product space using the polynomial chaos expansion introduced
in Section 3.3 and finite elements or finite differences for the spatial dimensions. This chapter
combines the methods presented so far to introduce the concept of stochastic images. Preusser et
al. [130] introduced stochastic images, but used a pointwise product space, which is a subspace of
the tensor product space H 1 (D) ⊗ L 2 (Ω). We compare both approaches at the end of this chapter.
5.1 Polynomial Chaos for Stochastic Images
It is popular in PDE based image processing to model an image f : D → IR on a domain D ⊂ IR d ,
d = 2,3 using a finite element space and a representation
f (x) = ∑ i∈I
f i P i (x) , (5.1)
where f i ∈ IR is the value of the ith pixel from the pixel set I and P i the shape function (e.g. tent
function) of the ith pixel (see e.g. [17]). In a stochastic image, a single pixel no longer has a fixed
value. Instead, it depends on a vector of random variables ξ (ω) = (ξ 1 (ω),...,ξ n (ω)) and on a
random event ω ∈ Ω. Note that it is possible to combine the concept of stochastic images with
other spatial discretizations, e.g. finite difference schemes. Then we have a pointwise representation
f (x i ,ξ ) and apply an interpolation rule for positions located between pixel positions.
Following [130], we obtain the representation of an image whose pixel values are random variables
from (5.1) by replacing the fixed f i by random variables f i (ξ ):
f (x,ξ ) = ∑ i∈I
f i (ξ )P i (x) . (5.2)
Fig. 5.1 shows a schematic sketch of this idea. Note that we omit denoting the dependence of ξ on
ω to simplify the notation. The polynomial chaos expansion (3.31) approximates any second order
random variable f i (ξ ) by a weighted sum of orthogonal multidimensional polynomials. This yields
f (x,ξ ) = ∑ i∈I ∑ N α=1 f i αΨ α (ξ )P i (x) (5.3)
as the representation of stochastic images, i.e. images whose pixels are random variables, discretized
using finite elements for the spatial dimensions. Using finite differences, the value at a pixel is
f (x i ,ξ ) = ∑ N α=1 f i αΨ α (ξ ) (5.4)
47

Chapter 5 Stochastic Images
❳❳
❳❳ ❳ supp ξ j , j = 1,...n
✘ ✘ ✘ ✘✘ ✘ x i
❳ ❳ ❳ ❳ ❳ supp Pi (x)
Figure 5.1: Sketch of the ingredients of a stochastic image. We discretize the spatial dimensions
using finite elements, but the coefficients of the FE basis functions are random variables.
Every random variable has a support, which spans over the complete image, thus pixels
depend on a random vector.
and an interpolation rule provides the values at positions between neighboring pixels. For fixed α
we call the coefficient f i α a stochastic mode of the pixel i. The set { f i α} i∈I collects the stochastic
modes of all pixels for fixed α. Thus, it is possible to visualize the set as a classical image.
From the polynomial chaos expansion of a stochastic image, we compute stochastic moments of
the image. With the use of the orthogonal set of basis functions we have E(Ψ 1 ) = 1, E(Ψ α ) = 0 for
α > 1 and E(Ψ α Ψ β ) = 0 if α ≠ β. The expected value and the variance of a stochastic pixel are
E( f (x i ,·)) = f i 1 ,
Var( f (x i ,·)) = ∑ N α=2
(
f
i
α
) 2
E
(
(Ψ α ) 2) .
(5.5)
We obtain higher stochastic moments in a similar way. Furthermore, it is possible to visualize the
complete stochastic information of every pixel, e.g. via a visualization of the PDFs of all pixels.
Note that the representation of stochastic images presented here differs from the one discussed by
Preusser et al. [130]. There, a space is used in which every pixel depends on one random variable
only. However, for most of the image acquisition processes and image processing methods the
assumption that the noise is independent for every pixel is not true. To represent these images in the
ansatz space we let every pixel depend on a random vector ξ . Section 5.3 compares both concepts.
The first step, required before the processing of the stochastic images starts, is the identification
of the random variables in the input data. We estimate these random variables from data samples
through the Karhunen-Loève expansion [41]. The Karhunen-Loève expansion, a stochastic version
of the principal component analysis (PCA) [74], determines the eigenvalues and eigenvectors of the
covariance matrix of the data samples and identifies the significant random variables in the data
samples with these eigenvectors and eigenvalues.
5.2 Generation of Stochastic Images from Samples
We obtain the polynomial chaos coefficients of the random variables X ∈ L 2 (Ω) by a maximum
likelihood estimation [41], leading to a representation of X ∈ L 2 (Ω) in the polynomial chaos by
X = ∑ N α=1 a αΨ α (ξ ) . (5.6)
To use the notion of stochastic images developed in the previous sections for image processing, we
need to obtain the coefficients of the representation (5.3) for the image undergoing the analysis. Let
48

5.2 Generation of Stochastic Images from Samples
u (1) ,...,u (M) , with u (k) ∈ IR r , r = |I |, denote sample images, e.g. images resulting from repeated
acquisitions. The goal is to identify these image samples as the samples of a vector of independent
random variables X. To this end, the empirical Karhunen-Loève decomposition [95] yields
u (k) = ū +∑ r √
j=1 s j U j X (k)
j , (5.7)
where ū is the mean of the input samples. The pairs (s j ,U j ) for j = 1,...,r are the eigenpairs sorted
in descending order of the r × r covariance matrix
C := 1
M − 1 ∑M k=1 (u(k) − ū) T (u (k) − ū) . (5.8)
Moreover, the
X (k)
j = 1 √ s j
U T j (u (k) − ū) (5.9)
are samples of the desired vector of random variables X = (X 1 ,...,X n ), where n < r.
The samples computed via the Karhunen-Loève expansion are samples of uncorrelated random
variables, but the random variables are not necessarily independent. Using Gaussian random variables,
we end up with independent random variables, because uncorrelated Gaussian random variables
are independent. Gaussian random variables have the drawback of the infinite support of the
density function, which causes problems in numerical schemes. Using other distributions, we assume
the independence as well, leading to a small additional error, because we neglect correlation
effects. Stefanou et al. [141] justified the assumption of independence by numerical experiments. In
addition, they developed numerical methods for uncorrelated random variables.
For a standard uniform random variable X it is possible to find a transformation g to an arbitrary
distributed random variable with finite variance Y : Y = g(X). Since X(ω) ∈ [0,1] we transform it
into a random variable Y with the desired distribution by applying the inverse cumulative distribution
function (CDF) FY −1 of Y :
Y = FY −1 (X) . (5.10)
This mapping is a standard result used in textbooks about probability, e.g. in [62]. It can also be
written as Y = FY
−1 (F X(ω)), because X has a standard uniform distribution and X(ω) = F X (ω).
Following [28], this result holds for arbitrary distributed random variables, too. In the next step, we
project the random variable Y on an element of the polynomial chaos basis by multiplying with the
basis element and taking the expected value.
The estimation of the coefficients of the polynomial chaos expansion (5.6) of the random vector
X from these samples is achieved by inverting the discrete empirical CDF F Xj , which is based on the
samples X (k)
j . This leads to a staircase-like approximation of the random variable X j . Following [141]
we get X j,α from the projection on Ψ α via
∫
X j,α = E(X j Ψ α ) = F −1 (
X Fξ j
(y) ) Ψ α (ξ (y))dΠ . (5.11)
Γ
Note that the assumption of independence allows to use basis functions, which depend on one random
variable only, i.e. Ψ α (ξ ) = Ψ α (ξ i ), i ∈ {1,...,n}. The empirical CDF and its empirical inverse are
F Xj (x) = 1 M
M
∑
k=1
{
FX −1
j
(y) = min x ∈
( )
I X (k)
j ≤ x
{
X (k)
j
} M
k=1
,
}
∣ FXj (x) ≥ y
,
(5.12)
49

Chapter 5 Stochastic Images
Figure 5.2: Decay of the sorted eigenvalues of the centered covariance matrix of 45 input samples
from an ultrasound device.
where I is the indicator function attaining value 1 for true arguments and 0 else. Note that the
random variables X j are related to the eigenpairs (s j ,U j ) of the Karhunen-Loève decomposition via
(5.9). With the expression for the inverse FX −1
j
and a numerical quadrature associated with the density
ρ(ξ ) we compute the polynomial chaos coefficients Xj
α independently from each other via
X j,α ≈ ∑ R k=1 w kF −1
X j
(
Fξ (y k ) ) Ψ α (y k ) , (5.13)
where we used the notation R for the number of nodes of the quadrature rule and w k for the quadrature
weights associated with the nodes.
We emphasize that the assumption of independence of the random variables X j is strong and in
general not true. However, following [141] in particular for a few input samples this assumption is
reasonable. When the assumption of independence is not true, it is possible to get the polynomial
chaos representation via methods presented by Stefanou et al. [141]. These methods require the
resolution of an optimization problem on a Stiefel manifold [72], which is time-consuming. Desceliers
[41] gives more details about the theoretical background of the presented method.
Remark 8. It is necessary to store a few leading eigenvalues and eigenvectors of the covariance
only to capture the significant stochastic effects in the input data. Fig. 5.2 shows the decay of the
eigenvalues of the covariance matrix computed from 45 samples from an ultrasound device. The
biggest eigenvalue is associated with the mean. The other two larger eigenvalues are most likely due
to motion of objects in the images during the acquisition. The stochastic effects take place on scales
that are orders lower than the expected value.
5.2.1 Efficient Eigenpair Computation of the Covariance Matrix
The computation of the covariance matrix of the input samples is a time-consuming and especially
memory-consuming process, because the covariance matrix is typically dense and the memory consumption
is the squared memory consumption of a single input sample. The storage of this matrix
limits the usability for high-resolution images. To avoid the generation of the complete covariance
matrix, we use the low rank approximation recently developed by Harbrecht et al. [63]. This approximation
is based on the pivoted Cholesky decomposition and an additional post-processing step to
generate a smaller matrix with the same leading eigenvalues.
50

5.2 Generation of Stochastic Images from Samples
Figure 5.3: Left picture group: The first mode (=expected value), second mode, third mode and
fourth mode of a stochastic CT image. Right: The sinogram, i.e. the raw data produced
by the CT imaging device for the head phantom [139].
Pivoted Cholesky Decomposition
The pivoted Cholesky decomposition is based on the Cholesky decomposition, a decomposition for
symmetric and nonsingular matrices [57]. The matrix A ∈ IR q×q is factorized into A = LL T , where L is
a lower triangular matrix. The computation of the complete factorization requires O(q 3 ) operations.
The pivoted Cholesky decomposition computes a rank m, m ≪ q, approximation of the matrix A,
where the trace norm measures the difference between matrix A and low rank approximation A m :
(√
)
‖A − A m ‖ tr = trace (A − A m ) T (A − A m ) . (5.14)
We achieve this by a modification of the Cholesky decomposition by introducing a pivot search.
This pivot search guarantees that the incomplete decomposition has the same leading eigenvalues as
the original matrix A. A rank m approximation of the matrix A is given by the product of the two
Cholesky factors L m and L T m, i.e.
A m = L m L T m , (5.15)
where the Cholesky factors are computed using Algorithm 1 from [63]. This algorithm needs access
to the diagonal of the matrix A and m rows of the matrix only. The storage requirement decreases
from q 2 to (m + 1)q and the number of operations from O(q 3 ) to O(m 3 ). However, this algorithm
computes the exact values for the leading eigenvalues, not an approximation. Harbrecht [63] provides
details about the theoretical background.
The eigenvalue computation of the eigenvalues of A m benefits from the fact that the eigenvalues
of A = L m L T m are the same as the eigenvalues of à = L T mL m . Thus, we transformed the computation
of the m leading eigenvalues from a IR q×q matrix into the computation of the m eigenvalues of a
IR m×m matrix, where m ≪ q. The eigenvectors of the initial matrix A are x = L m ˆx, where ˆx are the
eigenvalues of the small matrix L T mL m (see [63]).
5.2.2 Getting Stochastic Images from CT-data
The construction of stochastic images from image samples requires the acquisition of a huge number
of samples to get accurate results. For medical imaging techniques like US, or in other applications
51

Chapter 5 Stochastic Images
like quality control, the repeated acquisition is possible. However, we cannot apply this technique to
CT data, because the acquisition of CT data uses high-energy radiation [66]. Thus, the acquisition
of multiple samples is unethical for medical applications. Therefore, we present another possibility
for the generation of stochastic images from CT data based on the sinogram, the collection of rays
through the object under different angles and directions [66].
The approach is based on the hypothesis that the sinogram (see Fig. 5.3), the raw data of the
acquisition process (see [21] for details), is free of noise and that the noise and the artifacts in the
final CT images are due to the reconstruction step, which is necessary to transform the sinogram
into the final data set. We use multiple reconstruction techniques and parameter settings to generate
the input samples and use the technique described in the previous section to generate the stochastic
images. The reconstruction techniques range from Fourier based methods to iterative methods with
different settings for the data interpolation and the filter window for the low-pass filtering [154]. For
the computation of the reconstructions, we use CTSim [134], for which source code is available.
Thus, we combine the generation of input samples and the computation of the resulting stochastic
image in one program that runs without user interaction.
Another possibility to generate a stochastic image from the available CT image sample is to use a
noise model.
5.3 Comparison of the Space from [130] and the Space Used in this Thesis
Preusser et al. [130] made a first step for the application of SPDEs in the image processing context.
They proposed to use the space H h,p
still
:= V2 h ⊗ P p ⊂ H 1 (D) ⊗ L 2 (Γ) as ansatz space, where V2
h
is the classical finite element space spanned by multi-linear tent-functions P i and P p the space
spanned by one-dimensional polynomials H 1 ,...,H p . Then, the authors identified a stochastic image
f (x,ξ ) ∈ H h,p
still
with the polynomial chaos approximation:
f (x,ξ ) = ∑ i∈I ∑ p α=1 f i αH α (ξ i )P i (x) . (5.16)
In this representation, every pixel has its own random variable and the pixel is dependent on this
random variable only. Remember, the space used in this thesis uses a limited number of random
variables, but the support of these random variables ranges over the whole image.
An SPDE having stochastic images as input or solution is discretized using the SFEM. The authors
multiplied the equation by a test function of the form H β (ξ i )P i (x) ∈ H 1 (D) ⊗ L 2 (Γ), yielding to a
block system matrix for the unknown polynomial chaos coefficients of the solution.
The ansatz space and the discretization presented by Peusser et al. [130] have drawbacks in comparison
to the space used in this thesis. These drawbacks are listed below:
1. The authors used only test functions of the form H β (ξ i )P i (x), but functions of the form
H β (ξ k )P i (x), k ≠ i, are also elements of the product space H 1 (D) ⊗ L 2 (Γ). This leads to a
much too small system matrix of the SFEM method. Thus, the solution is computed in a
subspace of the tensor product space H 1 (D) ⊗ L 2 (Γ) only.
2. The dependence of pixels on independent random variables allows no propagation of stochastic
information between the pixels. This is a serious problem when dealing with diffusion equations
like in [130], because the diffusion transports stochastic information from a pixel into the
surrounding region. The ansatz space chosen in [130] cannot store this information, because
the neighboring pixels are independent of this specific random variable. Thus, the information
is lost. To be more precise, the solution of the diffusion process of the random variables has
to be projected on the ansatz space and the ansatz space is unable to store this information.
Especially for diffusion equations, stochastic information is lost due to this projection step,
leading to inaccurate results.
52

5.4 Visualization of Stochastic Images
Figure 5.4: Second (left) and fifth (right) mode of a stochastic US image. The information encoded
in these images is hard to interpret, because there is no deterministic equivalent.
3. The ansatz space from [130] allows one basic random variable for the representation of arbitrary
random variables in the polynomial chaos only. This is a strong limitation, because
random variables reasonably representable in a polynomial chaos in one random variable can
be properly approximated only. Other random variables with more complicated density functions
have to be projected on this limited space, also leading to a loss of precision. This is due
to the double limit in the Cameron-Martin theorem [27]. They showed the approximation of
L 2 -random variables when the number of basic random variables ξ i ,i = 1,...,n and the degree
of the polynomials p goes to infinity.
The ansatz space from [130] is useful only when the solution is independent for every pixel and
the representation of the arbitrary random variable of a pixel through a polynomial in one random
variable is sufficient. These applications are rare, especially the diffusion equations used for demonstration
purposes in [130] and the segmentation methods presented in this thesis are critical.
5.4 Visualization of Stochastic Images
During the last years, many authors developed methods for the visualization of uncertainty, see [61,
125] and the references therein. The proposed visualization techniques are often limited to 1D or
2D data. For 1D data, it is possible to draw additional information in the graph of the function,
e.g. displaying the standard deviation and other stochastic quantities like kurtosis or skewness [125].
The stochastic images introduced in this chapter are two- or three-dimensional. Furthermore, due
to the polynomial chaos expansion, we have to visualize the additional stochastic dimensions.
A stochastic image is given by (5.3) and thus, the visualization techniques for classical images are
only partially feasible. One possibility for the visualization is via the images shown in Fig. 5.4. There
the set fα,i i ∈ I for fixed α is visualized as a single image. The complete stochastic image can be
visualized as N of such images, which is disappointing for images with high stochastic dimension.
Another possibility, shown in Fig. 5.5, is to calculate the variance for pixels. The variance image is
( ) (
f
i 2
α E (Ψ α (ξ )) 2) P i (x) . (5.17)
Var( f (x,ξ )) = ∑ i∈I ∑ N α=2
Visualizing expected value and variance allows for getting an impression about the pixels variability.
Another possibility for the visualization is to draw a set of samples from the computed output distribution,
visualized in Fig. 5.6. With this sampling, we look at classical, well-known, pictures, but
samples randomly drawn from the distribution highly influence the result. For a moderate number of
53

Chapter 5 Stochastic Images
Figure 5.5: Expected value (left) and variance (right) of a stochastic US-image. The expected value
looks like a deterministic image and in the variance, regions with a high gray value
uncertainty are visible as white dots.
Figure 5.6: Two samples drawn from a stochastic image. The images differ due to realizations of the
noise. In a printed version, these images look nearly the same.
random variables it is also possible to generate selected samples from stochastic images by prescribing
the values for every random variable. Then, we evaluate the basis functions from the polynomial
chaos at these points and generate the image as the sum of the deterministic images, one for every
basis function from the polynomial chaos. This can be automated by generating a dynamic image,
which automatically loops over all possible realizations of the stochastic image [61].
In the chapter dealing with stochastic level sets it is necessary to visualize stochastic contours,
i.e. contours whose position and shape are dependent on random variables. The easiest possibility
is to visualize realizations of the stochastic contour (see Fig. 5.7). Using this approach, we visually
detect regions with a high uncertainty of the contour position, i.e. regions where the distance between
realizations of the contour is greater than in other regions.
For 3D stochastic surfaces, the visualization is even harder, because a slicing through 2D-images
is cumbersome. Thus, a technique for the visualization of 3D stochastic surfaces is required. One
possibility is to visualize the expected value and to color-code them by the variance [125]. Fig. 5.8
shows such a visualization. The result is an image, which is comparable to the 2D result from
Fig. 5.5, but combines the information into one image. Furthermore, Djurcilov [43] presented ideas
for the volume rendering of stochastic images.
54

5.4 Visualization of Stochastic Images
Figure 5.7: Visualization of realizations of a stochastic 2D contour. Every yellow line corresponds
to a MC realization of the stochastic contour encoded in the stochastic image.
Conclusion
In this chapter, we presented the concept of stochastic images and introduced the polynomial chaos
approximation of stochastic images. With the projection method from Section 5.2, we are able to construct
stochastic images from samples. This is a crucial task, because without this projection method,
stochastic images are a theoretical construct only, but applications cannot use them. Furthermore,
we presented visualization techniques for stochastic images. The visualization is important to bring
stochastic images into applications. Without an intuitive visualization of the additional stochastic
content, it might be difficult to bring the concept of stochastic images into applications.
Having the concept of stochastic images at hand, we investigate in the next chapters how segmentation
methods can be extended to be able to accept stochastic images as input.
Figure 5.8: Visualization of a 3D contour encoded in a 3D stochastic image. The expected value of
the 3D stochastic contour is color-coded by the variance. Regions with a high variance
are red and regions with a low variance green.
55

Chapter 6
Segmentation of Stochastic Images
Using Elliptic SPDEs
The task of this chapter is to combine the notion of stochastic images with the concept of SPDEs
introduced in Chapter 3. SPDEs arise from variational formulations of image processing problems,
when we apply these variational methods on stochastic images. In this chapter, we investigate segmentation
methods based on elliptic SPDEs. Chapter 7 investigates parabolic SPDEs.
Based on elliptic SPDEs we develop two segmentation methods for stochastic images, random
walker segmentation and Ambrosio-Tortorelli segmentation of stochastic images. The segmentation
methods differ in reference to user interaction and the number of parameters. The extension of the
random walker segmentation is interactive. Thus, it is possible to improve the segmentation quality
by adding additional seed regions interactively. On the other hand, the extension of the Ambrosio-
Tortorelli segmentation is fully automatic. The user tunes the parameters only, but has no possibility
to improve the quality of the segmentation afterwards, except for choosing a new set of parameters
and trying to improve the quality this way.
6.1 Random Walker Segmentation on Stochastic Images
Section 2.2 summarized random walker segmentation [59]. A stochastic extension of the random
walker segmentation has to combine the notion of stochastic images developed in Chapter 5 with the
concept of SPDEs from Chapter 3 and the discretization of SPDEs from Chapter 4.
6.1.1 Deriving a Stochastic Random Walker Model
The extension of the random walker segmentation [59] to a stochastic segmentation method is
straightforward and follows the way for the generation of stochastic methods for image processing
described by Preusser et al. [130] and by the author [1,3]. Furthermore, the author published the
stochastic extension of the random walker method [5]. Stochastic images, described in Chapter 5,
replace the classical images and all further steps are performed on the stochastic images.
More precisely, we replace the classical image u : D → IR by a stochastic image v : D × Ω → IR
as defined in (5.3). Random walker segmentation needs no assumptions about the regularity of
the input images, because it transforms the problem into a partition problem of a graph. To proof
existence and uniqueness of the deduced SPDE related to the continuous formulation, we restrict the
method to images with a H 1 -regularity in the spatial dimensions. This is the typical regularity for
image processing tasks assumed for classical image processing [17]. To use the polynomial chaos
expansion, we assume that the images are L 2 -regular in the stochastic dimensions. Thus, we use the
tensor product space H 1 (D) ⊗ L 2 (Ω) introduced in Section 3.1. For the discretization we use the
spaces V h ⊂ H 1 (D) consisting of multi-linear tent-functions for every pixel of the input image and
S n,p ⊂ L 2 (Ω), a polynomial chaos expansion in n random variables with order p.
We start by building a graph for the spatial dimensions of the stochastic image. On this graph,
we define stochastic analogs of the edge weights and node degrees. The stochastic edge weight, the
57

Chapter 6 Segmentation of Stochastic Images Using Elliptic SPDEs
edge weight is a random variable, is given by the same expression as the classical edge weight, but
the quantities extracted from the image are random variables. Thus, the random variable describing
the edge weight of the edge between neighboring pixels i and j is, cf. (2.5)
(
w i j (ξ ) = exp −β (g i (ξ ) − g j (ξ )) 2) . (6.1)
Replacing the random variables by their polynomial chaos expansion, we have to compute
w i j (ξ ) = exp
( ( ) )
N
−β ∑ α=1 gi αΨ α (ξ ) −∑ N 2
α=1 g αΨ j α (ξ )
. (6.2)
Section 3.3 describes how to perform calculations for random variables represented in the polynomial
chaos. Note that we do not calculate the exponential of the polynomial chaos expansion explicitly.
Instead, we compute a Galerkin projection of the exponential in the polynomial chaos via (3.39).
From the definition of the stochastic edge weights, it is easy to generalize the node degrees to
stochastic node degrees represented in the polynomial chaos:
d i (ξ ) =
∑
{ j∈V :e i j ∈E}
w i j (ξ ) =
N
∑ ∑
{ j∈V :e i j ∈E} α=1
w i, j
α Ψ α (ξ ) . (6.3)
The normalization step, to ensure that the maximal difference between g i and g j is one, is not straightforward
because the quantities g i are random variables. A normalization of random variables is to
ensure that the expected value of the random variable is one. This is achieved by dividing the difference
of neighboring pixels by the maximal difference of the expected value of neighboring pixels:
(g i (ξ ) − g j (ξ )) 2 (u i (ξ ) − u j (ξ )) 2
=
. (6.4)
max k,l∈V,ek,l ∈E E
((u k (ξ ) − u l (ξ ))
2)
From the stochastic edge weights and the stochastic node degrees it is easy to build the stochastic
analog of the Laplacian matrix given by, cf. (2.9)
⎧
⎨ d i (ξ ) if i = j
L i j (ξ ) = −w i j (ξ ) if v i and v j are adjacent nodes
⎩
0 otherwise
(6.5)
= ∑ N α=1 Lα Ψ α (ξ ) .
The stochastic combinatorial Laplacian matrix has a representation in the polynomial chaos. The
coefficient L α in this polynomial chaos expansion is a matrix containing at position Li α j the αth
coefficient of the polynomial chaos expansion of either d i (ξ ) if i = j or of −w i j (ξ ) respectively zero.
To define the linear system of equations to solve the stochastic random walker problem, we start
with the stochastic analog of the weighted Dirichlet integral. It is given by taking the expected value
of the classical weighted Dirichlet integral R w and inserting the stochastic quantities there:
( ∫
)
1
E(R w [u(ξ )]) = E w|∇u(ξ )| 2 dx . (6.6)
2 D
As for the classical energy (cf. Section 2.2), a minimizer is a harmonic function satisfying
−∇ · (w(ξ )∇u(ξ )) = 0 in D × Ω
u = 1 on V O
u = 0 on V B .
(6.7)
58

6.1 Random Walker Segmentation on Stochastic Images
Remark 9. The methods from Section 3.2 ensure existence and uniqueness for the solution of (6.7).
The coefficient fulfills w ∈ F l (D), because we use a truncated polynomial chaos representation.
Using the polynomial chaos discretization of stochastic images from Chapter 5 and collecting all
pixels in a vector x we get the discrete version of the Dirichlet integral
) 1
E(R w (x)) = E(
2 xT Lx , (6.8)
where L is the stochastic combinatorial matrix from (6.5). This equation requires a special ordering of
the vector x and the matrix L. The vector x is organized by grouping the coefficients for polynomials
from the polynomial chaos together, x = (x1 1 h ,...,x|V |
1
,...,xN 1 h ,...,x|V |
N ). The matrix L is a N × |V h |
block matrix, with non-zero entries in the diagonal blocks only:
L = diag(L 1 ,...,L N ) . (6.9)
Reordering the vector x with respect to seeded and unseeded nodes (cf. Section 2.2) and using the
same stochastic ordering scheme for the new vectors x U and x M yields
( 1 [
E(R w [x U ])=E x
T
2 M xU] [ ][ ]) ( T L M B xM 1 (
B T
=E x
T
L U x U 2 M L M x M + 2xUB T T x M + xUL T ) )
U x U . (6.10)
A stochastic minimizer of the discretized stochastic Dirichlet problem is given by
L U (ξ )x U (ξ ) = −B(ξ ) T x M (ξ ) . (6.11)
This system of linear equations is solved using the GSD. Section 4.4 describes the combination
of the GSD with a discretization of the stochastic dimensions using the polynomial chaos. For
the stochastic random walker segmentation, all quantities are available in their polynomial chaos
approximation. The matrix is L = ∑ N α=1 Lα Ψ α (ξ ) and the vectors x U and x M are also available
in their polynomial chaos approximation. Thus, we apply the algorithm presented in Section 4.4
directly on these quantities. The next paragraph presents the results obtained from the GSD.
Remark 10. Due to the construction of the solution via a problem on a graph, we end up with a
much simpler stochastic problem in comparison to a direct solution of the SPDE (6.7). Using the
definition of the solution via the graph, the matrix L has a representation L = diag(L 1 ,...,L N ). If we
discretize the SPDE (6.7) via a SFEM approach, we end up with a matrix that has nonzero blocks
away from the diagonal. This difference is due to a projection step of the graph representation,
because the quantity w i j (ξ ) is projected back to the polynomial chaos at an early stage. When using
the SFEM this projection is at the end of the solution process.
6.1.2 Results
In the following, we demonstrate the benefits of the stochastic extension of the random walker segmentation
on three data sets. The first data set consists of M = 5 samples with a resolution of
100 × 100 pixels from the artificial “street sequence” [99]. Note that we do not consider the images
as a sequence, instead we treat them as five samples of the noisy and uncertain acquisition of the
same scene. The second data set consists of 45 samples with a resolution of 300 × 300 pixels from
an ultrasound device 1 . The third data set is a liver mask on a varying background with resolution
129 × 129. The whole image is corrupted by uniform noise and 25 samples with different noise
realizations are treated as input. We computed a stochastic image containing n = 5 random variables
1 Thanks to Dr. Darko Ojdanić for providing the data set.
59

Chapter 6 Segmentation of Stochastic Images Using Elliptic SPDEs
Figure 6.1: Expected value (top row) and variance (bottom row) of the street image (left) and the US
image (right). Color-coded are the seed regions for interior (yellow) and exterior (red).
for the ultrasound device, n = 3 random variables for the liver samples, and n = 2 random variables
for the street scene. The number of random variables is chosen in dependence on the decay of the
eigenvalues of the covariance matrix of the input samples (cf. Section 5.2). The eigenvalues of the
covariance matrix show an exponential decay. Thus, it is sufficient to store a few of them to capture
the important stochastic effects. For the three data sets, we used a polynomial degree p = 3 and
computed a stochastic image from these samples using the method presented in Section 5.2. The
polynomial degree of three for the polynomial chaos expansion is a good balance between accuracy
of the polynomial expansion and computational effort. The user defines the seed points for the segmentation
of the image on the expected value of the stochastic image. Fig. 6.1 shows the expected
value, the variance, and the seed points. With the stochastic image and the seed points as input, we
perform the stochastic random walker segmentation. The only free parameter β varies during the
experiments. Fig. 6.2 shows the expected value of the segmented object for different values of β.
Together with the expected value, we are able to show the variance of the segmentation result for
every pixel. The variance of the pixels gives information, how the gray value uncertainty in the
stochastic input image influences the segmentation results. Thus, the variance, respectively the polynomial
chaos coefficients of the result, contains the information how the gray value uncertainty in
the input image propagates through the segmentation process and influences the result. Regions with
a high variance indicate regions where the input gray value uncertainty influences the detection of
the object. It is obvious from the images that the uncertainty changes from the input to the output. In
the input data, the gray value uncertainty spreads over the whole image, whereas in the segmentation
result, the gray value uncertainty concentrates at the object boundary.
60

6.1 Random Walker Segmentation on Stochastic Images
Stochastic Images
Mean only
β = 3 β = 5 β = 5
E
Ultrasound Image
Var
n/a
E
Cartoon Image
Var
n/a
Figure 6.2: Mean and variance of the probabilities for pixels to belong to the object. Furthermore, we
show in red Monte Carlo realizations of the object boundary sampled from the stochastic
result. A high variance indicates pixels where the gray value uncertainty highly influences
the result. For comparison we added a classical random walker segmentation result
in the last column. There the variance image is not available, because the method acts on
a classical image.
61

Chapter 6 Segmentation of Stochastic Images Using Elliptic SPDEs
Figure 6.3: MC-realizations of the stochastic object boundary for the stochastic liver image segmented
with the stochastic random walker approach with β = 10. On the right we highlight
a region of the image, where the noise in the input image influences the result.
Remark 11. The classical random walker result can be interpreted as a probability map, i.e. the
result of random walker segmentation is a probability for every pixel to belong to the object or not.
When we apply the stochastic random walker method, the first interpretation of the result is that we
compute “probabilities of probabilities”, because we are computing the probability distribution of
the values at every pixel. Let us emphasize that the probability interpretation of the classical result is
one possible interpretation only. Mathematically, we are computing the result of a diffusion problem,
and the stochastic extension is, in this interpretation, a diffusion problem with stochastic diffusion
coefficient. The results can be interpreted as the stochastic solution of the stochastic diffusion problem.
The analog to the probability interpretation is that we computed the probabilities for belonging
to the object in dependence on the input gray value uncertainty.
The stochastic object boundary can be visualized by tracking the deterministic object boundary (the
value 0.5 in the result image) for realizations of the random variables. The work of Prassni et al. [126]
inspired this kind of visualization. The difference is that Prassni et al. [126] visualized the iso-lines
of different probabilities, whereas we visualize the same iso-line for realizations of the stochastic
image. Fig. 6.3 shows the result of such a visualization.
It is possible to compute and visualize other quantities extracted from the segmentation, e.g. the
volume of the segmented object. The obvious visualization of the stochastic volume is to draw the
PDF of the volume. The PDF of the segmented volume can be computed from the segmentation by
summing up the random variables at every pixel, because they specify the “probability” that the pixel
belongs to the object. Thus, the random variable v(ξ ) specifying the objects volume is
v(ξ ) =
N
∑
α=1
v α Ψ α (ξ ) := ∑ x i (ξ ) . (6.12)
i∈I
Having a look at the PDF, it is easy to decide whether the image noise influences the segmented
volume strongly or not. If the segmented volume is strongly influenced the PDF is broad, otherwise
the function is narrow. Fig. 6.4 shows the PDF of the segmented volume from the street image.
Moreover, the choice of the parameter β influences the profile of the PDF. A smaller β leads to a
diffuse object boundary and to a broader PDF.
62

6.1 Random Walker Segmentation on Stochastic Images
Figure 6.4: PDF of the area of the segmented person from the street image for β = 25 (black) and
β = 50 (gray). From the PDF we judge the reliability of the segmentation, a narrow PDF
indicates that the image noise influences the segmentation marginally.
Remark 12. Another possibility to calculate the object volume is to count pixels with a value above
0.5 only. Thus, we compute image samples from the stochastic result via a Monte Carlo approach,
threshold these samples, count the number of object pixels, and calculate the volume PDF. This
method is time-consuming. The proposed method has the advantage to include the partial volume
effect [21] at the boundary, because it considers pixels with a probability less than 0.5 partially.
6.1.3 Comparison with Monte Carlo Simulation and Stochastic Collocation
To verify the intrusive solution of the resulting SPDE via polynomial chaos, stochastic finite elements,
and the GSD method, we compared this solution with the solutions obtained via Monte Carlo
sampling and a stochastic collocation approach. Fig. 6.5 shows the comparison of the expected value
and the variance computed via GSD, stochastic collocation, and Monte Carlo sampling. The small
difference between the variances of the three solutions might be due to the projection of the Laplacian
matrix on the polynomial chaos. However, the great benefit of the GSD method is the significantly
better performance. We now investigate this in detail.
6.1.4 Performance Evaluation
Due to the availability of the implementation possibilities for the solution of SPDEs, we are able
to compare the execution times of the approaches. We did the detailed comparison for the random
walker segmentation in this thesis only, but the results generalize to the Ambrosio-Tortorelli
approach, because it uses the same methods.
Table 6.1 shows the comparison of the execution times of the GSD method, the Monte Carlo
method, and the stochastic collocation method with Smolyak and full grid. It is easy to see that
the GSD method outperforms the sampled based approaches. This supports the decision to prefer
the GSD method and the finite difference method for random variables throughout this thesis. The
stochastic collocation methods suffer from the “curse of dimension” [119], because the execution
times grow exponentially with the number of random variables in the stochastic images.
63

Chapter 6 Segmentation of Stochastic Images Using Elliptic SPDEs
GSD Monte Carlo Stochastic Collocation
E
Var
Figure 6.5: Comparison of the discretization methods for the computation of the stochastic random
walker result to verify the intrusive discretization. The small difference between the
intrusive discretization via the GSD method and the two other sampling based approaches
might be due to the projection of the Laplacian matrix on the polynomial chaos.
The great benefit of the Monte Carlo method is that the method is independent on the number of
random variables. Nevertheless, the 1000 samples used in this comparison are a lower bound for the
number of runs needed to get accurate results. Recall that the rate of convergence is O(( √ R −1 )) and
even with this number of runs, the Monte Carlo method is slower than the GSD method.
6.1.5 Segmentation and Volumetry of an Object
In many applications, the noise of every pixel in the image is independent of the noise of the neighboring
pixels. It is possible to model this kind of stochastic images with our approach, too. In this
case, we have to use one basic random variable for every pixel, i.e. we end up with n = |I | basic
random variables for the polynomial chaos.
Street (n = 2) Liver (n = 3) Ultrasound (n = 5)
Monte Carlo 76 113 1814
Stoch. Collocation (full grid) 16 390 ≈ 1 400 000
Stoch. Collocation (sparse grid) 6 18 634
GSD 9 15 437
Table 6.1: Comparison of the execution times (in sec) of the discretization methods.
64

6.1 Random Walker Segmentation on Stochastic Images
Figure 6.6: Input “doughnut” without noise (left) and noisy input image treated as expected value of
the stochastic image (right).
To demonstrate the possibility to model such images, we used an artificial test image, a “doughnut”
with an area of 60 pixels in front of a constant background with resolution 20 × 20 pixels. Fig. 6.6
shows the noise-free initial image. We corrupted the image by uniform noise (see Fig. 6.6) and treated
the noisy image as the expected value of our stochastic image. This modeling is close to the situation
in real applications. There, the real noise-free image is not available and thus, the sample at hand is
the best available estimate of the expected value. Due to the high number of random variables, we
restricted the polynomial chaos to a degree of one, i.e. we are able to capture the effects expressible
in
(
uniform random variables only. Using a polynomial degree of order one the polynomial chaos has
401
) (
1 = 401 coefficients, using a polynomial degree of two we would end up with 402
)
2 = 80601.
An up-to-date personal computer cannot store such a high number of stochastic modes. A solution
could be the sparse polynomial chaos introduced by Blatman and Sudret [22].
After initialization of the expected value with the noisy image, we have to prescribe values for the
remaining polynomial chaos coefficients of the input image. Since we assume that the noise at every
pixel is independent, we have to prescribe a value for the coefficient corresponding to the random
variable of the pixel. We set this coefficient to 0.5/ √ 3, modeling a uniform distributed random
variable with support [w − 0.5,w + 0.5] around the expected value w given by the noisy input image.
The result of the random walker on this stochastic image is a stochastic image with the same
number of random variables. Since the random walker method requires the solution of a stochastic
diffusion equation, stochastic information is transported between the pixels. Thus, a pixel in
the result image depends on all basic random variables of the input image. The visualization of
polynomial chaos coefficients of the solution is unintuitive and cumbersome, because we have 401
coefficients per pixel. Consequently, we use the visualization techniques from Section 5.4. Fig. 6.7
shows realizations of the stochastic object boundary and the seed points for the segmentation.
In applications, features of the segmented object are of interest, e.g. in medical applications the
volume of the object is of interest to get information about the growth or shrinkage of the segmented
lesion. The volume of the segmented object in the stochastic image is a stochastic quantity, because
it depends on the particular noise realization. Thus, it is possible to visualize the PDF of the object
volume. We investigated two possibilities to compute the volume PDF from the stochastic segmentation
result. Section 6.1.2 introduced the first method. There the polynomial chaos expansions of
all pixels are added, and the PDF of the resulting random variable is computed via Monte Carlo
sampling from this random variable. This method is comparable with methods that consider partial
volume effects, because there is no binary decision whether the pixel belongs to the object or not. In
fact, we add all the stochastic possibilities of the pixels to belong to the object.
65

Chapter 6 Segmentation of Stochastic Images Using Elliptic SPDEs
Figure 6.7: Left: The object seed points (yellow) and background seed points (red) used as initialization
of the stochastic random walker method. Right: The MC-realizations of the
stochastic segmentation result differ significantly for different noise realizations.
The other possibility to compute the volume PDF from the stochastic result is inspired by the
classical method to compute the random walker result. We generate samples from the stochastic
segmentation result via Monte Carlo sampling and estimate the volume of the object given by pixels
with value above 0.5 on every sample. Fig. 6.8 compares the two approaches for the computation of
the object’s volume. Having in mind that the “real” object volume is 60 pixels, both methods slightly
overestimate the object’s volume, but the real object volume is close to the expected value (60.39 for
the summation of the random variables and 60.83 for the object thresholding) of both PDFs.
Figure 6.8: The PDF for both possibilities of the volume computation, the summation of the random
variables (gray) and the thresholding (black). The true volume is 60 pixels.
66

6.2 Ambrosio-Tortorelli Segmentation on Stochastic Images
6.2 Ambrosio-Tortorelli Segmentation on Stochastic Images
In the following, we focus on the combination of the notion of stochastic images with the segmentation
approach in the spirit of Ambrosio and Tortorelli [14]. Section 2.3 introduced this approach.
The author published the stochastic Ambrosio-Tortorelli extension in [3].
For the segmentation of stochastic images by the phase field approach of Ambrosio and Tortorelli,
we replace the deterministic u and φ by their stochastic analogs. The stochastic energy components
are then defined as the expectations of the classical energy components (cf. Section 2.3), i.e.
∫
Efid s (u) := E(E fid) =
E s reg(u,φ) := E(E reg ) =
E s phase (φ) := E(E phase) =
∫
Γ D
∫ ∫
Γ D
∫ ∫
Γ
(u(x,ξ ) − u 0 (x,ξ )) 2 dxdΠ
and we define the stochastic energy as the sum of these, i.e.
D
µ
(φ (x,ξ ) 2 + k ε
)
|∇u(x,ξ )| 2 dxdΠ
ν ε |∇φ(x,ξ )| 2 + ν 4ε (1 − φ (x,ξ ))2 dxdΠ
(6.13)
EAT(u,φ) s = Efid s (u) + Es reg(u,φ) + Ephase s (φ) . (6.14)
The Euler-Lagrange equations of the stochastic Ambrosio-Tortorelli energy are obtained from the
first variation of (6.13). Since the stochastic energies (6.13) are the expectations of the classical
energies (2.16), the computations are analog. For example, we get for a test function θ : D × Γ → IR
d
∣
dt Es fid (u +t θ) ∣∣t=0
= d ∫ ∫ (
) 2 ∣
∣∣t=0
u(x,ξ ) +tθ(x,ξ ) − u 0 (x,ξ ) dx dΠ
dt
Γ D
∫ ∫ (
)
= 2 u(x,ξ ) − u 0 (x,ξ ) θ(x,ξ ) dx dΠ .
Γ
D
(6.15)
With analog computations for the remaining energy contributions, we arrive at the following system
of SPDEs: We seek for u,φ : D × Γ → IR as the weak solutions of
−∇ · (µ(φ(x,ξ
) 2 + k ε )∇u(x,ξ ) ) + u(x,ξ ) = u 0 (x,ξ )
( 1
−ε∆φ(x,ξ ) +
4ε + µ )
2ν |∇u(x,ξ )|2 φ(x,ξ ) = 1
4ε . (6.16)
This system is analog to the classical system (2.18) in which stochastic images replace the classical
images. The equations are SPDEs, because the coefficients φ(x,ξ ) 2 and |∇u(x,ξ )| 2 are random
fields. Moreover, the right hand side of the first equation, u 0 (x,ξ ), is a random field. We use random
fields from the tensor product space H 1 (D) ⊗ L 2 (Ω). This space enables us to use finite elements for
the discretization of the spatial part and the polynomial chaos expansion for the stochastic part.
Remark 13. Recently, Krajsek et al. [86] developed an extension of the Ambrosio-Tortorelli model
based on Bayesian estimation theory [77]. This concept is related to the approach presented here,
but limited to Gaussian random variables, whereas the approach presented here deals with arbitrary
distributions with finite variance. The investigation of a link between the approaches is future work.
67

Chapter 6 Segmentation of Stochastic Images Using Elliptic SPDEs
6.2.1 Γ-Convergence of the Stochastic Ambrosio-Tortorelli Model
Ambrosio and Tortorelli [14] showed the Γ-convergence of their model towards the Mumford-Shah
model. It is possible to extend this result to show the Γ-convergence of the stochastic extension of
the Ambrosio-Tortorelli model towards a stochastic Mumford-Shah model. For the formulation of
the result, we use the stochastic analog of the space D h,n from [14], the space D h,n ⊗ L 2 (Ω), which
contains admissible functions for the energies. In the notation of [14], n is the space dimension and
h = 1/ √ ε. Thus, letting the scale of the phase field ε tend to zero is equivalent to letting h → ∞.
Theorem 6.1. The stochastic Ambrosio-Tortorelli model E(E AT ) Γ-converges to the stochastic
Mumford-Shah model E(E MS ) as ε → 0. More precisely let (u h ,φ h ) ∈ D h,n ⊗ L 2 (Ω) be a sequence
that converges to (u,φ) in D h,n ⊗ L 2 (Ω). Then we have
∫
∫
E MS (u(ω),K(ω))dω ≤ liminf E AT (u h (ω),φ h (ω))dω (6.17)
h→∞
Ω
and for every (u,φ) there exists a sequence (u h ,φ h ) ∈ D h,n converging to (u,φ) such that
∫
∫
E MS (u(ω),K(ω))dω ≥ limsup E AT (u h (ω),φ h (ω))dω . (6.18)
Ω
h→∞ Ω
In both inequalities, the edge set K is defined accordingly as the discontinuity set of u.
Proof. We begin the proof by citing a famous theorem for the interchange of a limit process and
integration, Fatou’s lemma (see [32]):
Theorem 6.2 (Fatou’s lemma). For a sequence of nonnegative measurable functions f n ,
∫
∫
liminf f n ≤ liminf f n . (6.19)
We have to show that we can interchange the limit process and the integration. Let us assume that
this interchange is possible (all requirements of Fatou’s lemma are satisfied). Then, we have
∫
∫
∫
liminf E AT (u h ,φ h )dω ≥ liminf E AT (u h (ω),φ h (ω))dω = E MS (u(ω),K(ω))dω (6.20)
h→∞ Ω
Ω h→∞ Ω
and by using the reverse of Fatou’s lemma we get
∫
∫
∫
limsup
h→∞
Ω
E AT (u h ,φ h )dω ≤
Ω
limsupE AT (u h (ω),φ h (ω))dω =
h→∞
Ω
Ω
E MS (u(ω),K(ω))dω , (6.21)
because for every realization ω ∈ Ω we have the Γ-convergence of the Ambrosio-Tortorelli model to
the Mumford-Shah model initially proved by Ambrosio and Tortorelli [14]. Thus, we have to show
that the interchange of the limit process and the integration is possible.
The existence of the deterministic series ensures the existence of a series for which the limit
superior is less than the Γ-limit. For every ω ∈ Ω we choose the deterministic series constructed by
Ambrosio and Tortorelli [14]. The inequality is ensured because Fatou’s lemma yields
∫
∫
∫
limsup
h→∞
Ω
E AT (u h ,φ h )dω ≤
Ω
limsupE AT (u h (ω),φ h (ω))dω =
h→∞
Ω
E MS (u(ω),K(ω))dω . (6.22)
We justify the applicability of Fatou’s lemma in the following.
To use Fatou’s lemma we have to show that E AT is nonnegative and measurable. The first condition
is trivially ensured, because E AT is the sum of integrals of positive (squared) functions and thus
nonnegative. The second condition is also ensured, because of the following theorem from [142]:
Theorem 6.3. Any lower semicontinuous function f is measurable.
Following [14], the functional E AT is semicontinuous when we use the space D h,n . Thus, the
Ambrosio-Tortorelli functional is nonnegative and measurable and Fatou’s lemma can be applied.
68

6.2 Ambrosio-Tortorelli Segmentation on Stochastic Images
✛
✛
j
✻
i
❄
✲
α
✲
✻ + L α,β
✟ ✟✟✟✟✟Mα,β β
❄
Figure 6.9: Structure of the block system of an SPDE. Every block has the sparsity structure of a
classical finite element matrix and the block structure of the matrix is sparse, meaning
that some of the blocks are zero. The sparsity structure on the block level depends on the
number of random variables and the polynomial chaos degree used in the discretization.
6.2.2 Weak Formulation and Discretization
The system (6.16) contains two elliptic SPDEs, which are supposed to be interpreted in the weak
sense. To this end, we multiply the equations by a test function θ : H 1 (D) × L 2 (Γ) → IR, integrate
over Γ with respect to the corresponding probability measure, and integrate by parts over the physical
domain D. For the first equation in (6.16) this leads us to
∫
Γ
∫
D
)
∫
µ
(φ (x,ξ ) 2 + k ε ∇u(x,ξ ) · ∇θ(x,ξ ) + u(x,ξ )θ(x,ξ )dxdΠ =
Γ
∫
D
u 0 (x,ξ )θ(x,ξ )dxdΠ
(6.23)
and to an analog expression for the second part of (6.16). Here we assume homogeneous Neumann
boundary conditions for u and φ such that no boundary terms appear in the weak form. For the existence
of solutions of this SPDE, the constant k ε is supposed to ensure the positivity of the diffusion
coefficient µ(φ 2 + k ε ). In fact, there must exist c min ,c max ∈ (0,∞) and I = [c min ,c max ] such that
(
P ω ∈ Ω ∣ )
µ
(φ (x,ξ (ω)) 2 + k ε ∈ I
)
∀x ∈ D = 1 , (6.24)
i.e. the coefficient is bounded almost sure by c min and c max .
The Doob-Dynkin lemma (see Section 3.2.3) ensures that the solutions of the SPDEs have a representation
in the same basis as the input, allowing us to use the same polynomial chaos approximation
for the input and the solution of the SPDEs. This is due to the continuity and measurability of the
stochastic partial differential operators.
The weak system (6.23) is discretized by a substitution of the polynomial chaos expansion (5.3) of
the image and the phase field. As test functions, products P j (x)Ψ β (ξ ) of spatial basis functions and
stochastic basis functions are used. Denoting the vectors of coefficients by U α = (u i α) i∈I ∈ IR |I |
and similarly for the phase field φ and the initial image u 0 we get the fully discrete systems
N (
∑
) M α,β + L α,β U α =
α=1
(
N
∑
α=1
εS α,β + T α,β ) Φ α =
N
∑
α=1
N
∑
α=1
M α,β (U 0 ) α
A α (6.25)
69

Chapter 6 Segmentation of Stochastic Images Using Elliptic SPDEs
Figure 6.10: Nonzero pattern of the SFEM matrix for the smoothed stochastic image using n = 5
random variables and a polynomial degree p = 3. A black dot denotes a block that has
a nonzero stochastic part, thus having the sparsity structure of a classical FEM matrix.
for all β ∈ {1,...,N}, where M α,β ,L α,β ,S α,β and T α,β are blocks of the system matrix, defined as
(M α,β ) i, j = E (Ψ α Ψ β ) ∫
(S α,β ) i, j = E (Ψ α Ψ β ) ∫
D
D
P i P j dx ,
∇P i · ∇P j dx
(6.26)
and (
(
L α,β ) i, j = ∑
k
T α,β ) i, j = ∑
k
(
∑E
Ψ α Ψ β Ψ γ) ∫
(˜φ 2 ) k γ
γ
∫
∑E
γ
(
Ψ α Ψ β Ψ γ) u k γ
D
D
∇P i · ∇P j P k dx ,
P i P j P k dx .
(6.27)
Here, (˜φ 2 ) k γ denotes a coefficient of the polynomial chaos expansion of the Galerkin projection of φ 2
onto the image space (cf. Section 3.3.3). The right hand side vector of the phase field equation is
∫
(A α ) i =
Γ
∫
Ψ α dΠ
D
⎧ ∫
1
⎪⎨
4ε P i dx =
D ⎪⎩
1
4ε P i dx if α = 1 ,
0 else .
(6.28)
Note that the expectations of the products of stochastic basis functions involved above are again
the components of the lookup table introduced in Section 3.3.5. The deterministic integrals can be
precomputed, because they are needed several times during the assembling of the system matrix.
Analog to the classical finite element method the systems of linear equations can be treated by an
iterative solver like the method of conjugate gradients [67].
The general block structure of an SFEM matrix is depicted in Fig. 6.9 and the sparsity structure on
the block level for five random variables and a polynomial degree of three is depicted in Fig. 6.10.
In addition, the matrix generation can be accelerated using lookup tables. The memory consumption
is enormous, because the matrix has N 2 -times the storage requirement of the deterministic
matrix, where N is the dimension of the polynomial chaos. Thus, we use the GSD method for the
solution to avoid the generation of the SFEM matrix.
70

6.2 Ambrosio-Tortorelli Segmentation on Stochastic Images
Stochastic Generalization of the Edge Linking Step
The edge linking step from Section 2.3 can be applied on the stochastic Ambrosio-Tortorelli model,
too. We introduce an additional coefficient c for the image equation. This coefficient is a random
field, i.e. c ∈ H 1 (D) × L 2 (Ω). The modified image equation in the stochastic context reads
−∇ · (µc(x,ξ
)(φ(x,ξ ) 2 + k ε )∇u(x,ξ ) ) + u(x,ξ ) = u 0 (x,ξ ) . (6.29)
The random field c is composed of the stochastic generalizations of the edge continuity and the edge
consistency step. Thus, c is
c(x,ξ ) = c dc (x,ξ ) · c h (x,ξ ) , (6.30)
whereas these quantities are
(
(c dc (ξ )) i = ζ (ξ ) dc) + 1 − ( ζ (ξ ) dc) i
i φ(ξ ) i
( (
))
(
ζ (ξ ) dc) = exp ε dc 1
i |η s | ∑ ∇u i (ξ ) · ∇u j (ξ ) − 1
j∈η s
1
(c h (ξ )) i =
1 + α ( )
φ(ξ ) i − φ(ξ ) 2 .
i
(6.31)
To calculate c dc and c h , it is necessary to use the calculations for random variables approximated in
the polynomial chaos presented in Section 3.3.3. The only quantity for which a stochastic generalization
is not obvious is the orthogonal edge direction ∇u ⊥ i . This direction is needed, because we
have to sum up over pixels perpendicular to the image gradient in the second equation of (6.31). This
perpendicular direction is also a stochastic quantity, but it is not possible to sum up in a stochastic direction.
To overcome this, we use the direction E ( (∇u) ⊥) and neglect the error due to the inaccurate
direction. This is similar to the upwinding problem for stochastic equations [92].
Remark 14. Erdem et al. [49] proposed additional feedback measures for textures and the local
scale. These measures are not included here, but can be generalized in a similar fashion.
6.2.3 Results
In the following, we demonstrate the performance and advantages of the stochastic extension of
the Ambrosio-Tortorelli segmentation approach. We use three data sets which cover a broad range
of possible input data. Furthermore, we compare the results of the stochastic Ambrosio-Tortorelli
model with the adaptive extension for the spatial dimensions and the stochastic version of the edge
linking step. Thus, the organization of this section is the following: First, we demonstrate the method
on three data sets. Then we show the results of the combination of the stochastic method with an
adaptive grid approach for the spatial dimensions. Finally, we demonstrate that the stochastic method
benefits from the idea of edge linking [49].
The first input image data set consists of M = 5 samples from the artificial “street sequence” [99].
The second data set consists of M = 45 image samples from ultrasound (US) imaging of a structure in
the forearm, acquired within two seconds. The third data set contains ten images of a scene acquired
with a digital camera 2 . Note that we do not consider the street sequence as an image sequence
here. Instead, we use five frames as samples of the noisy and uncertain acquisition of the same
object. From the samples, we compute the polynomial chaos representation using n = 5 (digital
camera), n = 10 (US), respectively n = 4 (street scene) random variables with the method described
2 Thanks to PD Dr. Christoph S. Garbe for providing the data set.
71

Chapter 6 Segmentation of Stochastic Images Using Elliptic SPDEs
Figure 6.11: Mean value of the three data sets used to demonstrate the stochastic Ambrosio-Tortorelli
method. For the second data set, we denoted image regions the text refers to.
in Section 5.2. The images have a resolution of 100 × 100 pixels for the street sequence, 129 × 129
pixels for the US data set, and 513 × 513 pixels for the digital camera data set. We use a polynomial
degree of p = 3 for the street scene and the US data and p = 2 for the digital camera sequence. This
leads to a polynomial chaos dimension of N = 56 (digital camera), N = 286 (US), and N = 35 (street
scene), respectively. For the reduction of the complexity by the GSD, we set K = 6. Furthermore,
we use ν = 0.00075 and k ε = 2.0h in all computations, where h is the grid spacing. To show the
influence of the random variables, we used the US data using the expected value only (n = 0).
Before we proceed with the presentation and interpretation of the results, let us remember the
power of the method. For the stochastic image and stochastic phase field it is possible to visualize
the PDF of every pixel (see Fig. 6.12), because we compute with this method the coefficients which
describe these random variables in a basis spanned by orthogonal polynomials in random variables.
Street Image Data Set
We use five samples of the street sequence to compute the stochastic image. Fig. 6.14 shows the
expected value and the variance of the stochastic input image computed using the method presented
Figure 6.12: PDF of a pixel from the phase field computed from the polynomial chaos expansion of
the pixel via a sampling approach. Although we use uniform basic random variables for
the polynomial chaos, the resulting random variables have skewed and Gaussian like
distributions due to the use of higher order polynomials in the basic random variables.
72

6.2 Ambrosio-Tortorelli Segmentation on Stochastic Images
Samples E(φ) Var(φ)
Monte Carlo
GSD
Figure 6.13: Segmentation result of the street scene. On the left we show the five samples the stochastic
input image is computed from. On the right we compare the results computed via
the GSD method and a Monte Carlo sampling.
in Section 5.2. It is visible from the pictures that the gray value uncertainty is high close to the edges
of moving objects. Thus, we expect the highest phase field variance in these regions. The results
depicted in Fig. 6.13 match with these expectations. Indeed, in the region around the wheels of the
car and around the right shoulder of the person, the edge detection is most influenced by the moving
camera, respectively the varying gray values between the samples at the edges. Also around the
edges in the background, the variance increases due to the moving camera. However, the stochastic
method can detect the edges in the image properly. The result of the stochastic method contains
much more information than the deterministic method. The expected value of the stochastic method
is comparable to the result of the classical method, the stochastic information like chaos coefficients,
variance, etc. are the real benefit of the method. Thus, we use the variance, indicating the robustness
of the detected edges to get information, which is not available in the classical model.
To verify the intrusive GSD method, we compared the results of the GSD implementation with
a simple Monte Carlo method with 10000 sample computations. Fig. 6.13 shows the results and
Figure 6.14: Expected value and variance of the stochastic input image of the street scene.
73

Chapter 6 Segmentation of Stochastic Images Using Elliptic SPDEs
10 random variables mean only
ε = 0.2h, µ = 1/300 ε = 0.4h, µ = 1/300 ε = 0.4h, µ = 1/400 ε = 0.4h, µ = 1/400
E
Image
Var
n/a
E
Phase Field
Var
n/a
Figure 6.15: Mean and variance of the image and phase field for varying ε and µ using the US data.
For comparison, we added the result from the deterministic method applied on the mean.
reveals that both approaches lead to similar results, but again, the GSD implementation is 100 times
faster than performing Monte Carlo simulation with a suitable number of samples.
Ultrasound Samples
The conversion of the input samples into the polynomial chaos as described in Section 5.2 leads to
the representation of the stochastic US image with 286 coefficients per pixel. Thus, a visualization of
this stochastic image via stochastic moments like expected value and variance is necessary. Fig. 6.15
shows the expected value and the variance of the phase field φ and the smoothed image u for settings
of the smoothing coefficient µ and the phase field width ε. The algorithm needs about 100 iterations,
i.e. alternating solutions of (6.16) for u and φ, to compute a solution. However, in the first steps, the
convergence is fast and after about 10 iterations, there is no visible difference in u and φ.
From the variance image of the phase field, the identification of regions where the input distribution
has a strong influence on the segmentation result (areas with high variance) is straightforward.
A benefit of the new stochastic edge detection via the phase field φ is that it allows for an identifi-
74

6.2 Ambrosio-Tortorelli Segmentation on Stochastic Images
without edge linking edge linking edge linking + adaptive
E
Image
Var
E
Phase Field
Var
Figure 6.16: Comparison of the stochastic Ambrosio-Tortorelli model (left column) with the extended
model using the edge linking procedure described in Section 2.3.3 (middle column)
and a combination of the edge linking and adaptive grid approach (right column).
Note that these results are computed with the same parameter set. The differences in
the results are due to the additional edge linking parameter c only.
75

Chapter 6 Segmentation of Stochastic Images Using Elliptic SPDEs
cation of edges in a way that is robust with respect to parameter changes. In particular, within the
four regions marked in Fig. 6.11 the expectation of the phase field is highly influenced by the choice
of µ and ν as visible in Fig. 6.15. The blurred edge at position 1 can be seen in the expectation of
the phase field only when we use a narrow phase field. In region 2, we have a different situation in
which the edge can be identified using a widish phase field. In addition, the edges at positions 3 and
4 can be identified using adjusted parameters. However, note that one of these edges is not seen in
the expectation of φ because of the particular choice of parameters; a high variance of φ indicates
the possible existence of an edge. In particular, this is true for the regions 1 and 2.
Moreover, the algorithm estimates the reliability of detected edges: A low expected value of the
phase field and a low variance indicate that the edge is robust and not influenced by the noise and
uncertainty of the acquisition process. This is true for the upper edges of the structure. In contrast, a
high variance in regions with a high or low expected value of the phase field, e.g. the labeled regions
1–4, indicates regions where the detected edge is sensitive to the noise.
In addition, we can easily extract the distribution of the gray values for any pixel location inside
the image and the phase field from the polynomial chaos expansion obtained via the GSD method.
Fig. 6.12 shows the PDF of a pixel from the phase field computed via the GSD.
Adaptive Grids
With a combination of the stochastic method and the adaptive grid approach described in Section 4.5,
we decrease the memory requirements and increase the performance further. Fig. 6.17 shows the
results of the adaptive method and compares them with the results from the uniform grid. For the
computations, we used a threshold for the error indicator of ι = 0.005. Thus, elements where the
error indicator S(x) is smaller than ι at every node are not further refined. The choice of a suitable
value for the error indicator threshold is important, because too small values lead to unnecessary
fine grids, whereas too high values for the threshold lead to coarse grids even close to edges. This
causes over-shootings in the numerical computation of the phase field, i.e. the phase field has no
longer values between zero and one. The results shown in Fig. 6.17 use a maximal level of 7, leading
to a uniform grid of size 129 × 129 at the beginning. The final adaptive grid depicted on the right
of Fig. 6.17 has about 70% fewer degrees of freedom, but yields nearly the same results. Fig. 6.17
shows no visible difference between the uniform grid and the adaptive grid solution.
From Fig. 6.17 it is visible that the saturation condition, required to build admissible grids, leads
to an area of increasing element size around detected edges. In flat regions, i.e. where the image
gradient magnitude |∇u| is small, the elements are coarser compared to regions close to edges.
Edge Continuity and Edge Consistency
The combination of the stochastic Ambrosio-Tortorelli segmentation with an edge linking step is a
great advance on the way to a detection and volumetry of objects in stochastic images. Fig. 6.16
shows the results of the stochastic extension of the edge linking step. We used α = 10, which turns
out to be a good weighting between smoothing out unwanted edges and sharpening regions to get
closed contours, respectively linked edges. We used s = 4, i.e. we used the two neighboring pixels in
the directions perpendicular to the image gradient to compute the feedback measure ζ dc (ξ ).
The figures indicate that the use of the modified diffusivity in the image equation of (6.16) leads
to a better detection of closed contours in the stochastic images. These closed contours lead to
cartoon-like smoothed images, because they avoid the smoothing over undetected edges.
It is possible to combine the edge linking step with the adaptive grid approach, leading to a fast and
accurate extension of the initial stochastic Ambrosio-Tortorelli model. The last column of Fig. 6.16
shows the result of such a combination.
76

6.2 Ambrosio-Tortorelli Segmentation on Stochastic Images
full grid adaptive grid full vs. adaptive grid
E
Image
Var
E
Phase Field
Var
Figure 6.17: Comparison of the full grid and adaptive grid solution. The full grid and adaptive grid
solution are visually identical, but the computation of the adaptive grid solution needs
significantly less DOFs. Thus, it can be applied on high-resolution images.
Conclusion
We presented extensions of the random walker and the Ambrosio-Tortorelli model to stochastic images
and applied the methods on different data sets. Especially, the intuitive visualization of the
stochastic random walker method via the visualization of contour realizations and the objects volume
PDF can be useful to convince the image processing community of stochastic modeling.
Furthermore, we presented a detailed theoretical foundation of the stochastic Ambrosio-Tortorelli
extension. The availability of the theoretical foundation along with the intuitive visualization of the
results is the key to a widely accepted method in image processing. The acceleration of the algorithm
via an adaptive grid approach and the integration of the edge linking step shows the potential of the
proposed methods to be extended to the users’ needs easily.
77

Chapter 7
Stochastic Level Sets
Level sets are widely used in applications ranging from computer vision [148] over material science
to computer-aided design [138] for the tracking and representation of moving interfaces arising
e.g. in the simulation of radiofrequency ablation [13]. Dervieux and Thomasset [40] and Sethian
and Osher [121, 138] introduced level sets in the form used today. The main idea is to embed the
moving interface as the zero level set of a higher-dimensional function φ. The moving boundary is
then equivalent to a propagation of the level sets of the function φ over time. The actual position of
the boundary at time t is reconstructed from the function φ by tracking the zero level set at time t.
Level sets are used for the segmentation of images as well. They are more flexible in comparison
to a parametrization of the boundary used e.g. for snakes [76]. In addition, advanced segmentation
methods like geodesic active contours [30, 82], an energy minimization method, are based on
level sets.
When we try to combine a level set based segmentation approach with stochastic images, we end
up with a stochastic velocity for the level set propagation, i.e. we have to solve a hyperbolic SPDE.
The development of numerical methods for hyperbolic SPDEs is an active research field. To the
best of the authors knowledge, there is no method available in the literature that can be applied to
the stochastic level set equation. The use of classical methods, like upwinding schemes [138], is
not possible, because they are based on the sign of the propagation speed, which is in the stochastic
context a random variable, too. Thus, we use a parabolic approximation of the level set equation.
This enables us to use the methods developed in the previous chapters.
Due to the importance of the level set equation in other applications besides the segmentation of
images, this chapter is split into two parts. First, we present the derivation of the parabolic approximation
of the stochastic level set equation along with the numerical discretization. Furthermore,
we present numerical tests showing the applicability of the discretization. The second part of this
chapter deals with the application of the stochastic level set equation for image segmentation. We
introduce stochastic extensions of three widely used segmentation methods based on the level set
equation: gradient-based segmentation, geodesic active contours, and Chan-Vese segmentation.
7.1 Derivation of a Stochastic Level Set Equation
The discretization of the classical level set equation is based on techniques for the discretization of
hyperbolic conservation laws. The discretization of hyperbolic SPDEs is still a challenging task. To
the best of the authors knowledge, there are two possibilities, which are less accurate [92] or timeconsuming
[147]. Thus, we focus on a parabolic approximation of the level set equation to avoid
the numerical problems related to the hyperbolic level set version. The parabolic stochastic level set
equation is based on the work of Sun and Beckermann [143] for the classical level set equation. The
stochastic level set equation is derived from the equation
φ(y(t,ω),t,ω) = 0 almost sure in Ω , (7.1)
79

Chapter 7 Stochastic Level Sets
where t is the time, ω a stochastic event, and y(t,ω) the path of a particle on the interface. Using the
chain rule, we get the stochastic version of the advection equation
φ t (t,x,ω) + v(t,x,ω) · ∇φ(t,x,ω) = 0 , (7.2)
where v = ∂y(t,ω)
∂t
is the speed of the level set propagation. The speed decomposes in a component in
the normal direction N and in the tangential directions T of the interface:
where v N and v T are
v(t,x,ω) = v N (t,x,ω) + v T (t,x,ω) , (7.3)
v N (t,x,ω) = (v(t,x,ω) · N(t,x,ω))N(t,x,ω) resp. v T (t,x,ω) = v(t,x,ω) − v N (t,x,ω) . (7.4)
Note that the decomposition is dependent on the stochastic event ω, because for every realization
ω ∈ Ω of the level set φ we get a different normal N(t,x,ω) and a different decomposition of the
stochastic quantity v(t,x,ω). Substituting (7.3) and (7.4) into (7.2) and using the relations
v T (t,x,ω) · ∇φ(t,x,ω) = 0 and v N (t,x,ω) · ∇φ(t,x,ω) = v n (t,x,ω)|∇φ(t,x,ω)| , (7.5)
where v n is the speed in the normal direction, yields the stochastic extension of the level set equation:
φ t (t,x,ω) + v n (t,x,ω)|∇φ(t,x,ω)| = 0 . (7.6)
As already mentioned, the discretization of this deterministic equation uses methods for hyperbolic
conservation laws, e.g. upwinding schemes. To the best of the authors knowledge, there is no accurate
and fast upwinding scheme for SPDEs available. To avoid the use of a numerical upwinding
scheme for hyperbolic SPDEs, we modify the stochastic level set equation in the spirit of Sun and
Beckermann [143]. We start with a decomposition of the speed v n into a component independent and
a component dependent on the interface curvature κ:
v n (t,x,ω) = a(t,x,ω) − b(x,t,ω)κ(t,x,ω) . (7.7)
The curvature κ is expressed using the level set φ, this is a standard approach for deterministic level
sets [138], and rewritten using the quotient rule:
( )
∇φ(t,x,ω)
κ(t,x,ω) = ∇ · N(t,x,ω) = ∇ ·
|∇φ(t,x,ω)|
(
) (7.8)
1
(∇φ(t,x,ω)) · ∇(|∇φ(t,x,ω)|)
=
∆φ(t,x,ω) − .
|∇φ(t,x,ω)|
|∇φ(t,x,ω)|
The previous modeling is valid for sufficiently smooth level set functions. If we prescribe a special
behavior of the level set in the normal direction of the level set, quantities like the gradient or the
curvature can be computed easily. For the special choice of the level set function
( ) n(t,x,ω)
φ(t,x,ω) = −tanh √ , (7.9)
2W
where n is the distance to the interface in the normal direction and W ∈ IR an additional parameter
controlling the width of the tangential profile, we get for the norm of the gradient:
|∇φ(t,x,ω)| = − ∂φ(t,x,ω)
∂n
This is because the derivative of the hyperbolic tangent is
= 1 − φ(t,x,ω)2 √
2W
. (7.10)
(tanhx) ′ = 1 − tanh 2 x . (7.11)
80

7.1 Derivation of a Stochastic Level Set Equation
Remark 15. Prescribing a special behavior of the level set function, the hyperbolic tangent profile,
is a standard technique in the level set context. A typical choice for classical level sets is the signed
distance function, i.e. to ensure that |∇φ| = 1 (see [138]).
The last term in (7.8) is the second derivative of φ in the normal direction, i.e.
(∇φ(t,x,ω)) · ∇(|∇φ(t,x,ω)|)
|∇φ(t,x,ω)|
= ∂ 2 φ(t,x,ω)
∂n 2 , (7.12)
we use the special profile of the level set from (7.9), the derivation rule for the hyperbolic tangent
and the chain rule to simplify the expression:
∂ 2 φ(t,x,ω)
∂n 2 = − φ(t,x,ω)( 1 − φ(t,x,ω) 2)
W 2 . (7.13)
Substituting this into (7.8), we get
κ(t,x,ω) =
1
(∆φ(t,x,ω) + φ(t,x,ω)( 1 − φ(t,x,ω) 2) )
|∇φ(t,x,ω)|
W 2
. (7.14)
Now we are able to substitute the findings from (7.3) to (7.14) into the level set equation (7.6). First,
we substitute the decomposition of the speed from (7.3) and the expressions for the level set gradient
from (7.10) and the curvature from (7.14) into the level set equation (7.6):
φ t (t,x,ω) + a 1 − φ(t,x,ω)2 √
2W
= b
(∆φ(t,x,ω) + φ(t,x,ω)( 1 − φ(t,x,ω) 2) )
W 2
. (7.15)
This equation is parabolic for b > 0 and the hyperbolic term a|∇φ| is converted into a nonlinear term
in φ. Sun and Beckermann [143] derived the deterministic equivalent of (7.15). They showed that
it is possible to implement (7.15), but the resulting phase field has the prescribed tangential profile
across the interface. Hence, it is not a signed distance function, which is preferred in applications.
A possibility to sustain the parabolic term in the absence of a curvature dependent speed, i.e. if
b = 0, is to subtract the curvature from (7.8) from the reformulated level set equation (7.15):
φ t + a 1 − φ 2 (
√ = b ∆φ + φ(1 − φ 2 )
2W W 2 − |∇φ|∇ · ∇φ )
|∇φ|
. (7.16)
This subtraction is based on the idea of the counter term approach developed by Folch et al. [51] and
should not be confounded with setting b = 0, because the first term on the right hand side conserves
the tangential profile of the level set. If we set b = 0, the equation moves an arbitrary shaped level set,
instead of producing the tangential shaped level set. Note that in (7.16) and the following equations
we use the notation φ for the phase field instead of writing φ(t,x,ω) to ease notation. Of course,
the phase field stays dependent on time t, spatial position x and stochastic event ω. Furthermore, we
omit denoting the dependence of other quantities from t, x, and ω, when this is obvious.
To summarize the modifications of the level set equation: We have a stochastic parabolic level
set equation. When the speed is curvature dependent, we use (7.15). Otherwise, we use (7.16).
Furthermore, the hyperbolic term a|∇φ| becomes a term nonlinear in φ.
In a last step, we use the nonlinear preconditioning technique from [56]. With the substitution
(
φ = −tanh ψ/( √ )
2W)
, (7.17)
81

Chapter 7 Stochastic Level Sets
Figure 7.1: Stochastic level sets do not have a fixed position where φ(x) = 0. Instead, there is a band
with positive probability that the level set is equal to zero, i.e. the position of the zero
level set is random and it is possible to estimate the PDF of the interface location in the
normal direction of the expected value of the interface (lower right corner).
which ensures that ψ is a signed distance function to the interface because of φ = −tanh √
2W
, we
get the final version of the stochastic level set equation
( (
1 − |∇ψ|
2 )√ 2
ψ t + a|∇ψ| = b ∆ψ +
tanh
ψ
( ) ) ∇ψ
√ − |∇ψ|∇ ·
, (7.18)
W
2W |∇ψ|
where we omitted the dependence of the function ψ on time t, spatial position x, and random event ω.
Following [143], where the deterministic equivalent of this equation was derived, the right hand side
of this function serves as an integrated reinitialization scheme for the level set ψ. Thus, further
reinitialization is not required for deterministic level sets.
7.1.1 Interpretation of Stochastic Level Sets
Having (7.18) at hand, we have to interpret the result of the level set motion with random speed.
Due to the random variable/field that controls the speed of the level set motion the position of the
zero (and all other) level sets is a random quantity, too. A possibility to estimate the influence of the
random speed component on the level set motion is to calculate the probability that the zero level set
is at a specific position. Furthermore, we can calculate the whole band with positive probability that
the zero level set is located there, i.e. where
P(φ(x) = 0) > 0 (7.19)
holds. In the normal direction of the expected value E(φ) = 0 of the zero level set location, we can
estimate the PDF of the interface position (see Fig. 7.1).
Remark 16. Using Gaussian random variables in combination with stochastic level sets, we end up
with a nonzero probability for the interface location in the whole domain. This is due to the infinite
support of Gaussian random variables. Thus, we limit the following investigations to a polynomial
chaos in uniform random variables. We denote a random variable X uniformly distributed in the
interval [a,b] by X ∼ U [a,b]. Uniform random variables have a compact support, leading to a band
with finite thickness for the potential interface location.
n
82

7.2 Discretization of the Stochastic Level Set Equation
Stochastic Signed Distance Functions
It is desirable to use signed distance functions as level sets in the stochastic context, too. A stochastic
signed distance function has to be a classical signed distance function for every realization ω ∈ Ω.
Theorem 7.1. A stochastic signed distance function fulfills E(|∇φ|) = 1 and Var(|∇φ|) = 0.
Proof. The first property is ensured by
the second property by
∫
∫
E(|∇φ(x)|) = |∇φ(x,ω)|dω = 1 dω = 1 , (7.20)
Ω
Ω
∫
∫
Var(|∇φ(x)|) = (|∇φ(x,ω)| − 1) 2 dω = 0 dω = 0 . (7.21)
Ω
Ω
7.2 Discretization of the Stochastic Level Set Equation
For the numerical tests, we discretize (7.18) using the explicit Euler scheme for the discretization of
the time derivative. The time discrete version of (7.18) is
(
( √
2
ψ t+τ = ψ t +τ −a|∇ψ t | + b ∆ψ t +
W
(
1 − |∇ψ t | 2) ( ) ))
tanh √ ψt
∇ψ
− |∇ψ t t
|∇ ·
2W |∇ψ t , (7.22)
|
where ψ t is the phase field at time t. The spatial discretization is done via a uniform grid like in
Section 5.1. The stochastic part is discretized using the polynomial chaos. Thus, we have to build
numerical schemes for the gradient norm, the curvature, and the hyperbolic tangent that deal with
polynomial chaos expansions.
Gradient Norm and Laplacian
The gradient norm is computed using finite difference schemes. The directional derivatives are computed
using central differences in the interior of the domain and forward resp. backward differences
at the domain boundary. The necessary computations of the square and the square root are performed
using the methods from Section 3.3.
The Laplacian is computed as the sum of the second directional derivatives, which we compute
using central differences in the interior and forward resp. backward differences at the boundary.
Hyperbolic Tangent
We compute the hyperbolic tangent using tanhx = 1 − 2(exp(2x) + 1) −1 . Thus, we use the computation
of the exponential function in the polynomial chaos from Section 3.3 and other methods from
there, i.e. we compute the Galerkin projection of the hyperbolic tangent on the polynomial chaos.
Curvature
The computation of the curvature is the most critical process for the computation of the update,
because the update is done in the whole domain, not in a narrow band around the zero level set. This
makes it necessary to compute a stable curvature even in regions with a high curvature. These regions
arise in simple settings, e.g. when the level set is initialized as a circle. The curvature in the midpoint
of the circle goes to infinity. We use a method for the stable curvature computation proposed by Sun
83

Chapter 7 Stochastic Level Sets
and Beckermann [143] based on an idea by Echebarria et al. [46]. It is given by
( ) ∇ψ
∇ ·
|∇ψ|
⎛
≈ 1 ⎝
h
ψ i+1, j − ψ i, j
√
(ψ i+1, j − ψ i, j ) 2 + (ψ i+1, j+1 + ψ i, j+1 − ψ i+1, j−1 − ψ i, j−1 ) 2 /16
ψ i, j − ψ i−1, j
− √
(ψ i, j − ψ i−1, j ) 2 + (ψ i−1, j+1 + ψ i, j+1 − ψ i−1, j−1 − ψ i, j−1 ) 2 /16
+ √
ψ i, j+1 − ψ i, j
(ψ i, j+1 − ψ i, j ) 2 + (ψ i+1, j+1 + ψ i+1, j − ψ i−1, j+1 − ψ i−1, j ) 2 /16
⎞
−√
ψ i, j − ψ i, j−1
⎠ .
(ψ i, j − ψ i, j−1 ) 2 + (ψ i+1, j−1 + ψ i+1, j − ψ i−1, j−1 − ψ i−1, j ) 2 /16
(7.23)
Due to the independence of the update for the spatial positions, this finite difference scheme can be
parallelized on multiple processor cores easily.
Remark 17. Due to the hyperbolic tangent profile of the level sets across the interface, we have
to respect a condition on the maximal curvature of the represented object. For a high curvature,
the hyperbolic tangent profiles overlap for points on the interface. This leads to instabilities of the
numerical schemes for the discretization.
7.3 Reinitialization of Stochastic Level Sets
The right hand side of (7.18) is an integrated reinitialization of the level set function. Following
[143], this reinitialization is sufficient to get accurate results for deterministic level sets. When
using a stochastic velocity, we have to reinitialize all polynomial chaos coefficients, which are on different
scales. Typically, the first coefficient, the expected value, is orders of magnitude bigger than
the remaining coefficients. Furthermore, the coefficients of polynomials in uncoupled random variables
are close to zero. During the numerical experiments, we observed that the reinitialization via
(7.18) is not sufficient. Thus, we need an additional reinitialization to get accurate stochastic results.
The classical reinitialization methods for level sets are not applicable in the stochastic context.
The Fast Marching method [138] is based on an upwinding scheme. As discussed in Section 7.1,
a stochastic upwinding scheme is not available. Iterative reinitialization via φ t = sign(φ)(1 − |∇φ|)
is not possible because the signature of a stochastic quantity is not well-defined. The equation for
energy minimization [90], i.e. α|E(|∇φ| − 1) 2 | + β|Var(|∇φ|)| → min, is unstable if φ converges to
the stochastic signed distance function.
To get a working reinitialization scheme for stochastic level sets, we use a modification of the
stochastic level set equation (7.18). As already mentioned, the right hand side of this function is
an integrated reinitialization. We use this equation, set the speed to zero, i.e. a = 0, and solve the
equation for artificial time T . Doing this, the reinitialization equation is
( (
1 − |∇ψ|
2 )√ 2
ψ t = b ∆ψ +
tanh
ψ
( ) ) ∇ψ
√ − |∇ψ|∇ ·
. (7.24)
W
2W |∇ψ|
In all numerical experiments, we apply this reinitialization equation. We use for every time step of
(7.18) ten reinitialization time steps of (7.24) with a time step size for the reinitialization of 0.5τ,
where τ is the time step size of the original problem.
84

7.4 Numerical Experiments
cosine inward
cosine outward
E Var E Var
PC
SC
MC
MCL
Figure 7.2: Comparison of expected value and variance of the resulting phase field for the cosine
test of (7.18) using the polynomial chaos (PC), stochastic collocation (SC), Monte Carlo
simulation (MC), and Monte Carlo simulation of the original level set equation (MCL).
7.4 Numerical Experiments
In this section, we present numerical experiments for the verification of the proposed algorithm and
for the implementation of the algorithm. To validate the intrusive implementation in the polynomial
chaos, we verify the results with Monte Carlo experiments and a stochastic collocation approach.
To show that the phase field equation is comparable with the native level set equation, we added a
Monte Carlo experiment based on the original level set equation
φ t + a|∇φ| = 0 . (7.25)
We are able to compare four implementations of the stochastic level set evolution: The intrusive
implementation of the preconditioned phase field in the polynomial chaos (PC), a stochastic collocation
approach based on the preconditioned phase field (SC), a Monte Carlo simulation of the
85

Chapter 7 Stochastic Level Sets
rarefaction fan
shock
E Var E Var
PC
SC
MC
MCL
Figure 7.3: Comparison of the expected value and variance of the resulting phase field for the rarefaction
fan and the shock, two classical tests for level set propagation. The figure shows
the comparison of the four discretizations of the stochastic phase field equation.
preconditioned phase field (MC), and a Monte Carlo simulation of the original level set implementation
(MCL). The comparison is performed on two typical tests for level set evolution, the evolution
of a cosine curve in the inward and outward direction and the evolution of an edge of a square in the
inward and outward direction. Furthermore, we demonstrate the extension of the proposed method
to three spatial dimensions on the Stanford bunny data set [149]. In contrast to other publications
dealing with mean curvature motion [138], we use the Stanford bunny and apply the preconditioned
phase field equation with stochastic speed on it. In all numerical experiments, we set W = 2.5h,
where h is the grid spacing and in the absence of a curvature dependent speed, we set b = 1.25h.
For the evolution of the cosine (see Fig. 7.2), the challenge is the development of a shock [138],
when the curve moves inward. Due to the stochastic velocity, we used a uniformly distributed speed a
with E(a) = 1.0 and Var(a) = 0.04, i.e. a ∼ U [1 − 0.2 √ 3,1 + 0.2 √ 3], the position of the shock is
uncertain, and the discretization has to be adequate in a vicinity of the possible shock positions.
For the numerical experiments, we use a spatial resolution of 129 × 129, a polynomial chaos in one
86

7.4 Numerical Experiments
Figure 7.4: Expected value color-coded by the variance for the Stanford bunny after shrinkage under
an uncertain speed in the normal direction. Red indicates regions with a high variance
and green regions with low variance. In addition, we show one slice of the variance.
random variable with order two and apply 30 steps with step size 0.1h. Furthermore, we computed
20 time steps of the reinitialization equation (7.24) with step size 0.2h after every time step. The
polynomial chaos coefficients of the speed are set to a 1 = 1, a 2 = 0.2, and a 3 = 0, such that the
expansion fulfills E(a) = 1 and Var(a) = 0.04.
It is visible from Fig. 7.2 that the methods based on the stochastic preconditioned phase field
formulation lead to the same results. Only the discretization of the level set method leads to deviating
results. This is due to the reinitialization of the level set via Fast Marching [138]. The Fast Marching
method assumes that the level set values at a grid point near the interface, the trial nodes (see [138])
is the signed distance to the interface. This is not true in the presence of a shock due to the crossing
of the zero level set. For deterministic level sets this error can be neglected, even for stochastic level
sets the expected value is accurate. However, for the stochastic part of the solution (the variance in
Fig. 7.2), which is orders of magnitude smaller than the expected value, this error becomes relevant.
Thus, it is more precise to use the reinitialization via (7.24) in the presence of a shock.
When the interface moves outward, we have a rarefaction fan (see [138]), because one point on
the zero level set is the closest point for multiple points away from the interface. The same problem
as for the shock arises and again, the reinitialization via (7.24) is the better method.
The second test is the evolution of one edge of a square in the inward and outward direction
depicted in Fig. 7.3. Again, we have the development of a shock when the curve moves inward and
of a rarefaction fan when the curve moves outward.
The last test is the contraction of the Stanford bunny under an uncertain velocity. Again, we used
the speed E(a) = 1.0 and Var(a) = 0.04 from the previous tests. Fig. 7.4 shows the results. For the
Stanford bunny, we have the evolution of a 3D object. We use a method for the visualization of 3D
stochastic images from Section 5.4 by visualizing the expected value color-coded by the variance.
As expected, we see a high variance of the contour in regions with high curvature. This is due to the
development of shocks, when the contour moves inside these regions.
87

Chapter 7 Stochastic Level Sets
Figure 7.5: Mean of the CT data set (left) and the liver data set (right) for the segmentation test.
7.5 Segmentation of Stochastic Images Using Stochastic Level Sets
Level set evolution with an uncertain speed can be useful in applications, e.g. in physical applications
where level sets track the interface between materials. Often, the material parameters are not known
exactly and the interface speed can be modeled dependent on the random material parameters.
The focus of this thesis is on the application of segmentation methods. This is why we use this new
concept for stochastic level sets at the moment for segmentation only. The author used level sets for
the modeling of physical effects, the evaporation of water during radiofrequency ablation [10,11,13].
It is possible to use stochastic level sets in this context, because the material parameters can be
modeled as random variables [87, 128] which leads to an uncertain interface speed.
For segmentation, we investigate three segmentation methods based on level sets for stochastic
extensions: gradient-based segmentation, geodesic active contours, and Chan-Vese segmentation.
Other segmentation methods based on level sets can also be suitable for stochastic extensions.
7.5.1 Gradient-Based Segmentation of Stochastic Images
Gradient-based segmentation of an image u : D → IR is introduced in Section 2.4.2 and given by
φ t + v(1 − bκ)|∇φ| = 0 , (7.26)
where the function v is called stopping function, because this function controls the stopping of the
level set evolution at the desired boundaries. Often, the function v is given by
v = 1/(1 + |∇u|) , (7.27)
where u is the image to segment. Typically, the level set is initialized as the signed distance function
of a small circle inside the object to segment. There is no theoretical justification of this method
besides the observation that the level set speed is close to zero at sharp edges due to the reciprocal
dependence between image gradient and speed. We replace the classical image u(x) by a stochastic
image u(x,ω). The equation for stochastic gradient-based segmentation is
and the speed is a stochastic quantity, too:
φ t (t,x,ω) + v(t,x,ω)(1 − bκ(t,x,ω))|∇φ(t,x,ω)| = 0 (7.28)
v(t,x,ω) =
1
1 + |∇u(t,x,ω)|
. (7.29)
88

7.5 Segmentation of Stochastic Images Using Stochastic Level Sets
Figure 7.6: Left: Mean contour during the evolution of the stochastic level set. The iso-contours
are drawn on the variance image of the final, magenta contour. The contour detection is
influenced by the image noise on the bottom and the right of the object (high variance).
Right: Contour realizations of the stochastic gradient-based segmentation of the CT data.
This method can be implemented by using the stochastic preconditioned phase field implementation
introduced in the last section by rearranging the equation to
where the stochastic speed ṽ is
φ t (t,x,ω) + ṽ(t,x,ω)|∇φ(t,x,ω)| = 0 , (7.30)
ṽ(t,x,ω) = 1 − bκ(t,x,ω)
1 + |∇φ(t,x,ω)|
. (7.31)
Using the decomposition into curvature dependent and independent parts, we end up with
Numerical Results
φ t +
1
1 + |∇φ| |∇φ| − b
κ|∇φ| = 0 . (7.32)
1 + |∇φ|
For the presentation of the results of the gradient-based segmentation of stochastic images we use
two data sets. The first data set consists of 289 reconstructions of a CT data set with 100 × 100
pixels. Section 5.2.2 gives details about the generation of the reconstructions. These reconstructions
are treated as independent realizations of a stochastic image and the polynomial chaos expansion of
the stochastic image is calculated using the methods from Section 5.2. The second data set consists
of a liver mask embedded into a 129 × 129 pixel image with a varying gradient strength to the
background. This image is corrupted by uniform noise and 25 samples, i.e. noise realizations, of this
image are treated as input for the generation of the stochastic image. In both data sets, the generated
stochastic image contains two random variables, and we use a polynomial degree of two, i.e. n = 2
and p = 2. Fig. 7.5 shows the expected value of the stochastic image of both data sets. The parameter
b is set to the grid spacing h and the level set is initialized as a circle centered in the center of the
image with a radius of 0.15 of the image width.
Fig. 7.6 shows the expected value during the level set evolution (the colored contours) and the
variance after 280 iterations of the gradient-based segmentation with time step τ = 0.2h of the second
data set. It shows the typical behavior of a rapid propagation of the level set towards the object
boundary and the influence of the stopping function that tries to stop the evolution at the boundary.
89

Chapter 7 Stochastic Level Sets
MC SC PC
E
Var
Figure 7.7: Resulting image with the expected value of the contour (red) of the segmented object
and the phase field variance with the expected value of the contour for gradient-based
segmentation of a stochastic CT image. The variance is constant in the normal direction
of the expected value of zero level set.
In Fig. 7.6, we depicted realizations of the stochastic contour encoded in the stochastic result of
the segmentation of the first data set. The figure shows that the noise in the input image influences
the segmentation in regions with a low gradient, i.e. in the bone regions of the head phantom. In
regions where the level set has not entered the bone or in regions where the evolution reached the
outer bone boundary, the segmentation is more stable with respect to noise. This is visible from the
realizations of the contour lines, which are close together in these regions.
A comparison of the intrusive implementation using the polynomial chaos expansion and the sampling
based implementations using Monte Carlo simulation and stochastic collocation is depicted in
Fig. 7.7 and shows a good consistency of the implementations. The expected value is visually at the
same position for the three methods, and the variance is similar, too.
7.5.2 Stochastic Geodesic Active Contours
Geodesic active contours try to minimize the energy of a curve (cf. Section 2.4.3). For a stochastic
curve C(q,ω) : [0,1] × Ω → IR 2 and a stochastic edge indicator g(x,ω) : IR × Ω → IR the expected
value of the geodesic curve energy is
∫ ∫ 1
∫ ∫ 1
E(B(C)) = βg u (|∇u(C(q,ω))|)dqdω + α|C ′ (q,ω)|dqdω . (7.33)
Ω 0
Ω 0
This energy tries to minimize the expected value of the curve length ∫ Ω
∫ 1
0 |C′ (q,ω)|dqdω weighted
by the edge indicator g, i.e. the functional is minimal when we found a short path along an edge
90

7.5 Segmentation of Stochastic Images Using Stochastic Level Sets
MC SC PC
E
Var
Figure 7.8: Mean and variance of the stochastic geodesic active contour segmentation of the stochastic
CT data set. The variance is constant in the normal direction of the zero level set.
inside the image. Typically, the edge indicator is
g u =
1
(1 − εκ) , (7.34)
1 + |∇G σ ∗ u| p
where G σ is a Gaussian smoothing kernel with width σ and p ∈ {1,2}. Computing the stochastic
Euler-Lagrange equation as necessary condition for a minimum of the function is done in the same
fashion as in [30, 82], but we have to respect the outer integration over Ω. We end up with the
stochastic Euler-Lagrange equation
φ t (t,x,ω) = g u (t,x,ω)β|∇φ(t,x,ω)| − α∇g u (t,x,ω)∇φ(t,x,ω) + εκ|∇φ(t,x,ω)| , (7.35)
which is analog to the deterministic one. The parameters α,β, and γ can be freely chosen to optimize
the segmentation result. The meaning of the parameters is the following:
• α: The parameter α controls the attraction of the minima of the edge indicator g u when it is
positive. Otherwise, the level set is pushed away from the minima.
• β: The parameter β controls the shrinkage or expansion of the level set. A negative value of β
leads to a shrinkage of the level set and positive β to an expansion of the level set. Thus, this
parameter controls whether the initial level set is inside or outside of the desired contour.
• ε: The parameter ε acts as a weighting term for the curvature smoothing.
The stochastic geodesic active contour level set equation is discretized using the methods presented
in Section 3.3 and by using the stochastic preconditioned phase field equation.
91

Chapter 7 Stochastic Level Sets
Figure 7.9: Left: Evolution of the expected value contour of the stochastic geodesic active contour
method. The shown variance corresponds to the contour after 240 iterations (the magenta
contour). Right: Mean value of the stochastic image to be segmented and the contours at
time points of the level set evolution. The final contour matches the object boundary.
Numerical Results
The application of the stochastic geodesic active contour method is demonstrated on the same data
sets as the gradient-based segmentation on stochastic images, namely the stochastic CT image and
the liver mask with two random variables and a polynomial degree of two.
A comparison of the intrusive implementation using the polynomial chaos expansion and the sampling
based implementations using Monte Carlo simulation and stochastic collocation is depicted in
Fig. 7.8 for the CT data and shows a good consistency of the implementations.
Fig. 7.9 shows the expected value contour for time points during the level set evolution together
with the variance of the final level set of the segmentation of the second data set. Again, the regions
with a high variance are the bottom and the upper-right side of the object. This is consistent with the
results from the gradient-based segmentation (cf. Fig. 7.6). Furthermore, the right picture of Fig. 7.9
shows the evolution of the expected value contour on the expected value image. The expected value
contour after 240 iterations with a time step of 0.2h is aligned to the object boundary. The variance
corresponding to this contour is the same as the variance in the left picture of the same figure.
The advantage of the stochastic geodesic active contour approach over the stochastic gradientbased
segmentation is that a running over the edges is mostly avoided (cf. Fig. 7.11).
7.5.3 Stochastic Chan-Vese segmentation
We derive the stochastic Chan-Vese model from classical Chan-Vese model by replacing all quantities
by their stochastic counterparts:
( ( ) )
∇φ
φ t = δ ε (φ) µ∇ · − ν − λ 1 (u 0 − c 1 ) 2 + λ 2 (u 0 − c 2 ) 2
|∇φ|
, (7.36)
where the phase field φ, the initial image u 0 , the mean values c 1 and c 2 , and the smooth delta
approximation δ ε are stochastic quantities, i.e. they are dependent on the random event ω ∈ Ω. The
function δ ε is the derivative of the stochastic smooth Heaviside approximation
H ε (z(ω)) = 1 2
(
1 + 2 ( )) z(ω)
π arctan ε
. (7.37)
92

7.5 Segmentation of Stochastic Images Using Stochastic Level Sets
Figure 7.10: Mean (left) of the stochastic CT and the variance (right) of the stochastic Chan-Vese
solution. Additionally, we show the expected value contour at different time steps.
The regularized stochastic δ-function δ ε is
1
δ ε (z(ω)) =
πε + π . (7.38)
ε
z(ω)2
The mean value of the object and the background is a stochastic quantity because we have to average
over a collection of random variables. The mean values are
∫
D
c 1 (φ) =
u ∫
0(x)H ε (φ(x))dx
∫
D
D H and c 2 (φ) =
u 0(x)(1 − H ε (φ(x)))dx
∫
ε(φ(x))dx
D (1 − H . (7.39)
ε(φ(x)))dx
Note that we average over the spatial dimensions, i.e. over the deterministic image domain only.
Thus, c 1 and c 2 are random variables. In (7.39) we have to evaluate the Heaviside approximation,
which involves the computation of the inverse tangent of a stochastic quantity. To avoid the necessity
to develop a numerical scheme for the stochastic inverse tangent, we use a well-known approximation,
see e.g. [131]:
{
x
arctan(x) ≈
1+0.28x 2 if |x| ≤ 1
else
π
2 − x
x 2 +0.28
. (7.40)
This is not a real drawback of the stochastic discretization, because it can be interpreted as an alternative
approximation of the Heaviside function and is not as bad as an approximation of an approximation
as it might look. The remaining part of the Chan-Vese model is generalized to stochastic
quantities by using Debusschere’s methods for the computation with polynomial chaos quantities
(see Section 3.3 and [38]). The main driving force of the stochastic Chan-Vese model is the difference
between the mean value of the separated region and the actual gray value. The mean value
of the image regions is computed via an averaging of a collection of random variables. Thus, the
stochastic information cancels out of the stochastic Chan-Vese model, because we are approximating
the “real”, noise-free, mean value when we average over a huge number of random variables.
The Variance as Homogenization Criterion for Stochastic Chan-Vese Segmentation
Up to now, we have used the (spatial) mean value of the stochastic image as homogenization criterion
only. Thus, we ignore stochastic information, e.g. the variance, of the stochastic image. Homogenizing
the variance of the segmented object and background can improve the segmentation result
further. For example, in medical images different organs or tissue components can have different
93

Chapter 7 Stochastic Level Sets
Figure 7.11: Left: MC-realizations of the stochastic contour from the stochastic Chan-Vese segmentation
applied on the CT data set. Right: Realizations of the stochastic contour from the
stochastic geodesic active contour approach applied on the CT data set.
noise levels. Thus, they can be separated by homogenizing the variance. To include the homogenization
of the variance we add additional terms to the stochastic Chan-Vese model that are inspired by
the terms for the homogenization of the mean value. The inclusion of stochastic moments in functionals
has been investigated e.g. by Tiesler et al. [146]. To be more precise, we add two additional
components to the Chan-Vese energy leading to
( ( )
∇φ
φ t = δ ε (φ) µ∇·
)−ν −λ 1 (u 0 − c 1 ) 2 +λ 2 (u 0 − c 2 ) 2 −ρ 1 (Var(u 0 ) − v 1 ) 2 +ρ 2 (Var(u 0 )−v 2 ) 2 .
|∇φ|
(7.41)
In (7.41) we added two parameters ρ 1 and ρ 2 to weight the additional components. Furthermore, the
new components v 1 and v 2 are defined as
∫
D
v 1 (φ) =
Var(u 0(x))H ε (φ(x))dx
∫
D H ε(φ(x))dx
∫
and v 2 (φ) =
D Var(u 0(x))(1 − H ε (φ(x)))dx
∫
D (1 − H ε(φ(x)))dx
. (7.42)
Remember, the variance can be computed from the polynomial chaos expansion of the stochastic image
easily. Moreover, it is possible to homogenize every polynomial chaos coefficient independently,
leading to various additional constraints.
Numerical Results
We apply the stochastic Chan-Vese model on the same data sets as the other methods: the stochastic
CT and the stochastic liver mask. Fig. 7.10 shows the expected value of the liver data set along
with the expected value contour at stages of the evolution. The stochastic Chan-Vese model slightly
overestimates the object, because the final (green) contour is not perfectly aligned with the boundary.
This is due to the homogenization criterion the stochastic Chan-Vese model tries to fulfill. Fig. 7.10
shows the variance of the final level set along with the contours at stages of the evolution. The
variance indicated that the segmentation is uncertain in the critical areas at the object’s bottom and
top. Furthermore, the variance identifies two critical regions on the right and the left of the object.
Fig. 7.11 shows realizations of the final contour via a MC-sampling from the stochastic result.
For the liver data set, we show the level set evolution on the expected value of the initial image and
on the variance of the final level set in Fig. 7.12. The data set is constructed by adding artificial noise
to a noise-free image. This noise nearly cancels out due to the averaging process for the computa-
94

7.5 Segmentation of Stochastic Images Using Stochastic Level Sets
Figure 7.12: Mean (left) of the stochastic liver image and the variance of the stochastic Chan-Vese
solution. In addition, we show the expected value contour at different time steps.
Figure 7.13: Variance of the stochastic image to segment (left), the expected value is not depicted,
because the expected value is an image with the same gray value at every pixel. On
the right, the segmentation result is depicted on one realization (one sample) of the
stochastic image to segment.
tion of the random variable for the mean inside the regions. The realizations drawn from the final
stochastic level set fit to each other and to the final contour of the level set evolution from Fig. 7.12.
The extension of the stochastic Chan-Vese approach that tries to homogenize the variance of the
object and the background allows to segment objects in images with constant mean, i.e. it allows to
segment objects from constant images where the classical method fails. Fig. 7.13 shows the result
of the segmentation of an image with constant mean, but non-constant variance. Drawing samples
of this image (cf. Fig. 7.13) the object is visible on samples through the different variance levels, but
again, the classical Chan-Vese approach cannot segment the object in the image due to the constant
mean value, whereas the variance extension of the stochastic Chan-Vese approach yields the correct
result. In fact, the Chan-Vese approach without variance homogenization would not move the initial
contour because the driving force is zero due to the constant mean value.
Conclusion
We presented an extension of the level set approach to use random variables or random fields as
propagation speed. The use of this uncertain speed leadss to an uncertain interface position char-
95

Chapter 7 Stochastic Level Sets
acterizing the influence of the uncertain propagation speed. The resulting stochastic level set equation,
a hyperbolic SPDE, is transformed into a parabolic SPDE to end up with an equation that can
be discretized with intrusive numerical methods for SPDEs. The extension of the classical level
set equation is important in many applications, because the modeling of imprecise known material
parameters, boundaries, or source terms through random fields [53, 84, 110, 128, 161] is a rapidly
growing field of research. Furthermore, we presented a method for the reinitialization of stochastic
level sets and showed that the commonly used classical reinitialization methods like fast marching
cannot be applied in the stochastic context.
Based on the stochastic level set equation, we extended three segmentation methods. Using a
stochastic image as input for these methods, we end up with an uncertain speed as driving term for
the segmentation that depends on information extracted from the stochastic image. Using gradient information
only, as in the first presented method, we end up with a method that uses local information
only. Thus, this method is highly sensitive to the noise characterized by the stochastic components of
the stochastic image. Using additional global stochastic information, as in the stochastic Chan-Vese
approach, weakens the influence of the input uncertainty on the segmentation result.
Additional to the stochastic extensions of the classical segmentation methods, we presented an
extension of the Chan-Vese approach that tries to homogenize the variance of the segmented object.
Thus, this method is not an extension in the spirit of the other method extensions, where we replaced
classical images by their stochastic counterparts. Instead, this extension allows to use stochastic
information as driving force of the segmentation. This enables us to segment images that cannot be
segmented with the classical methods. For example, we are able to segment objects in an image with
constant mean, when they have different noise properties, i.e. a different variance.
96

Chapter 8
Segmentation of Classical Images Using
Stochastic Parameters
In the previous chapters, we presented methods for the segmentation of stochastic images. All these
methods are based on the solution of SPDEs and we get a stochastic segmentation result characterizing
the influence of the gray value uncertainty on the segmentation result. This chapter uses a
different approach that leads also to SPDEs for the segmentation of images.
In applications, the user tweaks the parameters of the segmentation methods to get satisfying
results. Often, the user performs this tweaking for every single data set. Thus, the segmentation
result is not dependent on the input image and the selected segmentation methods only, but also on
the particular choice of the parameters by the user. This yields the problem that the segmentation
result is not reproducible among users. The influence of the user parameters on the segmentation
results is important e.g. in medical applications, when different users segment base and follow-up
scans using different segmentation parameters. It is difficult to decide whether the segmentation
result is different due to a growth of the tumor or due to the different segmentation parameters.
In cancer therapy the further treatment of the patient is based on the segmentation results using
RECIST [48, 145]. Thus, information about the stability of the segmentation with respect to the
parameters might be useful to come to an informed decision.
In this chapter, we try to investigate the influence of the segmentation parameters on the segmentation
result. This task is known as sensitivity analysis [26, 135]. The main idea is to replace the
deterministic segmentation parameters by random variables and apply the segmentation methods on
deterministic images. The stochastic segmentation result is comparable to the result of the segmentation
of stochastic images. The difference is that the components are stochastic due to the stochastic
parameters instead of the stochastic image. We visualize the results using the same techniques as for
stochastic images, showing the influence of the segmentation parameters on the segmentation result.
With this approach, we detect regions in the image highly influenced by the choice of the segmentation
parameters and regions, where the segmentation is robust with respect to parameter changes.
In addition, we investigate which segmentation parameters have a strong influence on the segmentation
result. For geodesic active contours, the influence of the smoothing term should be nearly the
same on the whole image, whereas the weight related to the edge detector is important on the edges
in the image. This approach needs few random variables only, typically one for every segmentation
parameter. Hence, this approach is suitable for a discretization via the methods presented in this
thesis without the need to reduce the number of random variables necessary for stochastic images
via the Karhunen-Loève decomposition.
In the following, we investigate the use of stochastic segmentation parameters for random walker
segmentation, Ambrosio-Tortorelli segmentation, gradient-based segmentation, and geodesic active
contours. We discretize the methods using slightly adopted versions of the stochastic segmentation
methods for the segmentation of stochastic images presented in Chapter 6 and Chapter 7.
97

Chapter 8 Segmentation of Classical Images Using Stochastic Parameters
8.1 Random Walker Segmentation with Stochastic Parameter
The random walker segmentation has one free parameter that the user has to choose during the
segmentation process. This parameter, denoted by β, controls the influence of the image gradient on
the matrix entries because the edge weights for random walker segmentation (cf. Section 2.2) are
(
w i j = exp −β (g i − g j ) 2) . (8.1)
Making the parameter β a random variable and approximating this random variable in the polynomial
chaos (cf. Section 3.3), the stochastic edge weights for the sensitivity analysis are
w i j (ξ ) = exp
(
−
( N∑
β α Ψ α (ξ )
α=1
)(g i − g j ) 2 )
. (8.2)
Note that the parameter β is not restricted, but can be, the standard random variable for the construction
of the polynomial chaos expansion, e.g. a uniform random variable. In fact, we use the power of
the polynomial chaos approximation by making the parameter β dependent on a couple of standard
random variables with an adequate polynomial degree in the polynomial chaos expansion.
Using the stochastic edge weights from (8.2), we define the node degree analog to Section 6.1:
d i (ξ ) =
∑
{ j∈V :e i j ∈E}
w i j (ξ ) =
∑
N
∑
{ j∈V :e i j ∈E} α=1
(w i j ) α Ψ α (ξ ) . (8.3)
Note that for the sensitivity analysis we use the exact normalization of the image gradient given
by (2.6), because the pixel values are deterministic values in this setting. From the stochastic edge
weights (8.2) and the stochastic node degrees (8.3), we construct the stochastic Laplacian matrix via
⎧
⎨
L i j (ξ ) =
⎩
=
N
∑
α=1
d i (ξ ) if i = j
−w i j (ξ ) if v i and v j are adjacent nodes
0 otherwise
L α Ψ α (ξ ) .
Finally, we end up with the same stochastic equation system as in Section 6.1, but the stochastic
components are due to the stochastic parameter instead of stochastic pixels inside the image:
(8.4)
L U (ξ )x U (ξ ) = −B(ξ ) T x M (ξ ) . (8.5)
We have to use stochastic images to store the stochastic solution. The stochastic images have to
contain the same random variables the parameter depends on.
Remark 18. The discretization of the random walker segmentation with a stochastic parameter uses
the generalized spectral decomposition. The only small variation in the implementation is that we
have to use a polynomial chaos approximation of the parameter β for the calculation of the edge
weights. The edge weights themselves are already random quantities in the stochastic random walker
implementation of Section 6.1.
98

8.1 Random Walker Segmentation with Stochastic Parameter
Figure 8.1: Left: Realizations of the stochastic contour obtained from the random walker segmentation
with stochastic parameter. Right: Mean and variance of the stochastic contour
obtained from the random walker segmentation with stochastic parameter.
Results
We perform random walker segmentation on the well-known data sets from the last chapters. Because
all stochasticity is due to the parameters, we use the expected value image of the stochastic
input data set only, i.e. we use a deterministic input image. Thus, this method is, in contrast to all
other methods presented so far, usable for classical images. To be able to capture the stochasticity
introduced by the stochastic parameters, we identify the deterministic input image with a stochastic
image containing one random variable (the random variable the stochastic parameter depends on)
and use a maximal polynomial degree of four. The stochastic components of the input are set to zero.
The only parameter in the random walker method is the parameter β for the estimation of the
graph weights. In the following, we model this parameter as a uniformly distributed random variable
with expected value ten, i.e. the first coefficient of the polynomial chaos expansion is β 1 = 10. For
the experiments we used a variance of the uniform random variable of three, resulting in a random
variable that is uniformly distributed between 7 and 13, i.e. β ∼ U [7,13]. For the polynomial chaos
expansion of the parameter β, setting β 2 = √ 3 and β i = 0,i > 2 models this behavior.
Fig. 8.1 shows the image to segment, realizations of the stochastic contour, the expected value
contour and the variance for the US data set. Note that there is no direct relation between regions
with a high variance and regions where the contour realizations are far from each other as one might
suggest. In fact, the distance between the contour realizations depends on the gradient of the underlying
probability map (the expected value of the result) and the variance. In regions where the
expected value is around 0.5 and has a low gradient, a small variance can influence the contour position
significantly, whereas in regions with a high gradient, even a high variance cannot influence
the contour position visually. The upper right corner of Fig. 8.1, where a low variance corresponds
to varying contour positions, shows this effect. Furthermore, this is visible in Fig. 8.2, where the
random walker segmentation of the liver data set is shown. There, the highest uncertainty in the
contour position is in the quadrant with the lowest gradient between object and background and the
highest uncertainty in the expected value of the probability map is at the bottom of the object.
Furthermore, the result depicted in Fig. 8.2 shows two problems of the random walker segmentation
method. First, the method needs a strong gradient between object and background. Otherwise,
99

Chapter 8 Segmentation of Classical Images Using Stochastic Parameters
Figure 8.2: Left: Realizations of the stochastic contour obtained from the random walker segmentation
with stochastic parameter. Right: Mean and variance of the stochastic contour
obtained from the random walker segmentation with stochastic parameter.
the segmentation fails, like in the upper left part of the segmentation result in Fig. 8.2. The second
problem is due to sharp corners of the object, like the corner in the middle of Fig. 8.2. The random
walker method tries to identify smooth objects, because internally it solves a diffusion equation that
prefers smooth solutions. Both problems can be reduced by defining additional seed points for the
segmentation close to the problematic regions.
Another observation from the stochastic segmentation result is that the PDF of the segmented volume
(cf. Section 6.1.2) is not uniformly distributed, even though the input (the stochastic parameter)
has a uniform distribution. Fig. 8.3 shows the PDF of the segmented areas for both test examples.
For the US data set, the resulting PDF is close to a uniform distribution (the left picture of Fig. 8.3),
for the liver data set the area PDF is concentrated around a peak (right picture of Fig. 8.3). Both
PDF are computed using the method described in Section 6.1.5 given by summing up the random
variables of all pixels, cf. (6.12).
Figure 8.3: Volume of the stochastic contour obtained from the random walker segmentation with
stochastic parameter. The left curve shows the PDF of the object in the US image, the
right curve the PDF of the liver in the liver data set. The PDFs are obtained using (6.12).
100

8.2 Ambrosio-Tortorelli Segmentation with Stochastic Parameters
8.2 Ambrosio-Tortorelli Segmentation with Stochastic Parameters
Ambrosio-Tortorelli segmentation on stochastic images requires the solution of a system of two
coupled SPDEs and involves four parameters the user has to choose:
−∇ · (µ(φ(x,ξ
) 2 + k ε )∇u(x,ξ ) ) + u(x,ξ ) = u 0 (x,ξ )
( 1
−ε∆φ(x,ξ ) +
4ε + µ )
2ν |∇u(x,ξ )|2 φ(x,ξ ) = 1
4ε . (8.6)
The parameter µ controls the influence of the phase field value on the image smoothing process and
ε controls the width of the phase field. The influence of the image gradient on the phase field is
controled by ν and k ε is an additional regularization parameter that ensures ellipticity of the first
equation. By making all four parameters random variables, it is possible to investigate which parameter
has the strongest influence on the segmentation result. The adoption of (8.6) is straightforward.
We replace the classical parameters µ,ν,k ε and ε by their stochastic counterparts µ(ξ ),ν(ξ ),k ε (ξ )
and ε(ξ ) and approximate these random variables in the polynomial chaos. We end up with
−∇ · (µ(ξ
)(φ(x,ξ ) 2 + k ε (ξ ))∇u(x,ξ ) ) + u(x,ξ ) = u 0 (x,ξ )
( 1
−∇ · (ε(ξ )∇φ(x,ξ )) +
4ε(ξ ) + µ(ξ ) )
2ν(ξ ) |∇u(x,ξ )|2 φ(x,ξ ) = 1
4ε(ξ ) . (8.7)
The discretization of this SPDE system is analog to the discretization of the SPDE system for stochastic
images. The differences are that the coefficient of the Laplacian in the second equation is a
stochastic quantity and that the right hand side of the second equation is a stochastic quantity, too.
∫D 1
4ε(ω)
Thus, we integrate ∫ Ω dxdω to compute the right hand side by an integration rule, and we
have to use an assembling method for the inhomogeneous stiffness matrix to discretize ∇ε(ω)∇φ.
The discretization of (8.7) using finite elements for the deterministic dimensions and the polynomial
chaos for the stochastic dimensions is
N
∑
α=1
(
) M α,β + L α,β U α =
N
∑
α=1
M α,β (U 0 ) α ,
N (
∑
α=1
S α,β + T α,β ) Φ α =
N
∑
α=1
A α (8.8)
for all β ∈ {1,...,N}, where M α,β ,L α,β ,S α,β and T α,β are blocks of the system matrix, defined as
(M ) (Ψ ) ∫
α,β = E α Ψ β P i P j dx
i, j D
( ) (
S α,β = ∑∑E
Ψ α Ψ β Ψ γ) ∫
(˜ε) k (8.9)
γ ∇P i · ∇P j P k dx ,
i, j
k γ
and (
(
L α,β ) i, j = ∑
k
T α,β ) i, j = ∑
k
(
∑E
γ
∑E
γ
Ψ α Ψ β Ψ γ) (˜φ 2 ) k γ
(
Ψ α Ψ β Ψ γ) u k γ
D
∫
D
∫
D
∇P i · ∇P j P k dx,
P i P j P k dx .
(8.10)
Here, (˜φ 2 ) k γ and u k γ denote the coefficients of the polynomial chaos expansion of the Galerkin projection
of µ(ξ )(φ(ξ ) 2 1
+ k ε (ξ )) respectively
4ε(ξ ) + µ(ξ )
onto the image space (cf. [38]).
2ν(ξ )|∇u(ξ )| 2
Finally, the right hand side vector of the phase field equation is
∫ ∫
(A α 1
) i =
4ε P i(x)dxΨ α (ξ )dΠ . (8.11)
Γ
D
We solve the SPDE system for the sensitivity analysis with the same methods as the SPDE system
for stochastic images from Section 6.2, i.e. it is possible to use the GSD for the resolution process.
101

Chapter 8 Segmentation of Classical Images Using Stochastic Parameters
Figure 8.4: Ambrosio-Tortorelli model applied on the expected value of the liver data set using a
stochastic parameter µ. The upper row shows the expected value (left) and the variance
(right) of the smoothed image, the lower row the expected value (left) and the variance
(right) of the phase field.
Results
We applied the Ambrosio-Tortorelli segmentation with stochastic parameters on the liver data set.
Again, we use the expected value of the stochastic data set as deterministic input and construct a
stochastic input image that contains one random variable and a maximal polynomial chaos degree
of four. As in the random walker tests, the remaining stochastic dimensions are filled up with zeros.
To separate the influence of the stochasticity of the parameters in the Ambrosio-Tortorelli model, we
use one stochastic parameter for the first tests and keep the other parameters deterministic.
Fig. 8.4 shows the result for a uniformly distributed parameter µ. To be precise, µ is uniformly
distributed between 200 and 600, i.e. µ ∼ U [200,600]. The parameter µ controls the influence of the
smoothing term in the image equation. For large µ we get sharper images with sharp edges. Thus,
a stochastic parameter µ influences the smoothing of the image. This is visible from the variance of
102

8.2 Ambrosio-Tortorelli Segmentation with Stochastic Parameters
Figure 8.5: Ambrosio-Tortorelli model applied on the expected value of the liver data set using a
stochastic parameter ε. The upper row shows the expected value (left) and the variance
(right) of the smoothed image and the lower row the expected value (left) and the variance
(right) of the phase field.
the smoothed image in Fig. 8.4, where a smoothing across the object boundaries leads to a variance
that looks similar to the original image. This is due to the cartoon-like initial image. Once energy is
transported across the edge, it is equally distributed in the whole region due to the smoothing term.
The smooth image resulting from the image equation influences the phase field, because it leads to
diffuse boundaries and to a wide phase field that is visible in the phase field variance in Fig. 8.4.
In Fig. 8.5 we used a stochastic parameter ε uniformly distributed between 0.0015 and 0.0035,
i.e. ε ∼ U [0.0015,0.0035]. The parameter ε influences the width of the phase field, but has no
influence on the smoothing parts of the equations. We observe changes in the variance around the
edges in Fig. 8.5. Directly, the parameter ε influences the width of the phase field and due to the
wider phase field, the image is smoothed differently close to edges.
103

Chapter 8 Segmentation of Classical Images Using Stochastic Parameters
8.3 Gradient-Based Segmentation with Stochastic Parameter
Gradient-based segmentation via a level set formulation contains one parameter b that controls the
influence of the curvature κ. Making this parameter a random variable, we end up with
φ t (t,x,ω) + v(1 − b(ω)κ(t,x,ω))|∇φ(t,x,ω)| = 0 . (8.12)
The stopping function v is v = 1
1+|∇u|
. Additionally to the necessary Galerkin projection in the
numerical scheme for the solution of the gradient-based segmentation, we have to project bκ back to
the polynomial chaos. For this, we use the standard methods presented in Section 3.3. The remaining
part of the discretization is analog to the discretization of the stochastic gradient-based segmentation
with stochastic image.
Results
We present the gradient-based segmentation with stochastic parameter using the CT data set and the
liver data set. As usual, we used the expected value as input and used one random variable and a
polynomial degree of four. For the experiment, we used a stochastic parameter b that is uniformly
distributed between 0.75 and 1.25, i.e. b ∼ U [0.75,1.25]. The parameter b controls the influence of
the curvature smoothing. A higher parameter b leads to smoother contours. This is shown in Fig. 8.6
where the contour realizations vary with respect to the curvature.
Figure 8.6: Result of the gradient-based segmentation with stochastic parameter b, i.e. with a stochastic
curvature smoothing. The upper row shows the results for the CT data set, expected
value of the image and contour realizations (left) and variance of the level set with contour
realizations (right). The lower row shows the same results for the liver data set. In
all figures we add Monte Carlo realizations of the stochastic object boundary. The red
contour corresponds to a b = 0.75, yellow to b = 1.0 and blue to b = 1.25.
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8.4 Geodesic Active Contours with Stochastic Parameters
8.4 Geodesic Active Contours with Stochastic Parameters
The sensitivity analysis for the geodesic active contour approach follows the procedure for the sensitivity
analysis of the other segmentation methods. Geodesic active contours are given by
φ t (t,x) = γg(t,x)κ(t,x)|∇φ(t,x)| + α∇g(t,x)∇φ(t,x) − βg|∇φ| . (8.13)
The parameters α,β, and γ can be chosen to optimize the segmentation result. The parameter α
controls the attraction of the minima of the speed function v. The parameter β controls the shrinkage
(negative β) or expansion (positive β) of the level set and the parameter γ acts as a weighting term
for the curvature smoothing.
Making the segmentation parameters random variables, we end up with
φ t = γ(ω)g(t,x,ω)κ(t,x,ω)|∇φ(t,x,ω)| + α(ω)∇g(t,x,ω)∇φ(t,x,ω) − β(ω)g|∇φ| . (8.14)
This equation is nearly identical to the stochastic geodesic active contour equation, but requires an
additional projection step during the discretization to projects the products γg,α∇g, and βg back to
the polynomial chaos. Besides this additional projection step, we use the same numerical methods
as for the discretization of the stochastic geodesic active contour equation in Section 7.5.2, i.e. we
use an explicit time step discretization via the Euler method and a uniform spatial grid.
Results
The geodesic active contour method with stochastic parameters is performed on the same data sets as
in the previous sections. Due to the smooth objects that we try to segment in the images, we ignore
the smoothing term by setting γ = 0. The parabolic approximation and the attraction term ∇g∇φ
ensure that we get smooth results in this setting, too. The parameters α and β are chosen by setting
α 1 = 0.08, α 2 = 0.002, β 1 = 1.0, and β 2 = 0.02. Thus, we use two stochastic parameters at the same
time and make them both dependent on the same random variable. Since we set the expected value
and the first coefficient to a nonzero value, we end up with uniformly distributed parameters.
Fig. 8.7 shows the result for the CT data set. The image is easy to segment due to the homogeneous
gradient between the inner parts of the head phantom and the bone. The problematic parts are the
Figure 8.7: Result of the geodesic active contour segmentation with stochastic parameters for the CT
data set. On the left the expected value of the image and contour realizations and on the
right the variance of the level set with contour realizations are shown.
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Chapter 8 Segmentation of Classical Images Using Stochastic Parameters
Figure 8.8: Result of the geodesic active contour segmentation with stochastic parameters for the
liver data set. On the left the expected value of a detail of the image and contour realizations
and on the right the variance of the level set with contour realizations are shown.
regions, where the object to segment does not have the “elliptic” contour behavior. In these regions,
the gradient differs from the remaining parts of the image. The geodesic active contour method has
different attractors depending on the particular value of α and β in these regions. This is visible
from Fig. 8.7, because the contour realizations are far from each other, and the variance is high in a
region in the upper part of the object boundary.
Remark 19. Note that for level sets there is, in contrast to the random walker method, a one-to-one
correspondence between the distance between the contour realizations and the variance, because we
use a stochastic equivalent of the signed distance function. Thus, deviations in the level set position
are related to the variance directly.
For the liver data set, Fig. 8.8 shows the results with the same stochastic parameters. Again, the
results are close together for parameter realizations, because we have one attractor for the level set
only (the object boundary). The differences in the lower part of the object are due to the weak
attraction of the liver boundary for some realizations of the parameter α.
Conclusion
The presented sensitivity analysis is a natural extension of the stochastic image processing framework
presented in this thesis. With the sensitivity analysis, we investigate the robustness of the
classical image segmentation methods with respect to parameter changes. This additional stochastic
information is available for the costs of a few Monte Carlo runs. However, we do not use the Monte
Carlo method, but have to solve a couple of deterministic problems when using the GSD method.
A possible application of this kind of sensitivity analysis is to warn the user, when the segmentation
result is sensitive to parameter changes. This can be done via background calculations, i.e. the system
computes the stochastic solution while the user examines the deterministic result. When necessary,
the system informs the user about the stochastic result and makes additional information like the
variance or contour realizations available.
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Chapter 9
Summary, Discussion, and Conclusion
In this thesis, we presented extensions of PDE-based segmentation methods to stochastic images,
i.e. images whose pixels are random variables. The characterization of such stochastic images is
based on the recently developed generalized polynomial chaos expansion. With this expansion, we
developed extensions of the well-known finite element and finite difference schemes for the discretization
of the PDE to the stochastic dimensions, leading to stochastic PDEs. To demonstrate the
power of using stochastic images, we extended the well-known segmentation methods proposed by
Mumford-Shah and the related approximation by Ambrosio-Tortorelli as well as the random walker
method and three methods based on a level set formulation. The input for the stochastic segmentation
is constructed via computing the leading random variables via a principal component analysis
of samples of the input scene and a projection on the polynomial chaos basis. Furthermore, we
used the stochastic images and model extensions to perform a sensitivity analysis of the methods by
identifying the parameters with random variables.
9.1 Discussion
The work presented in this thesis is a complete framework for the important task of error propagation
in mathematical image processing [36, 106]. For every step of the mathematical image processing
pipeline (data acquisition, data representation, operator modeling, discretization, solution strategies
and visualization) methods for the solution of the particular problems are presented. Besides the
development of the framework, theoretical justifications of the methods are presented as well. In
particular, these are the extensions of the Γ-convergence proof for the stochastic Ambrosio-Tortorelli
model and the proof of the existence of SPDE solutions used in this thesis.
This thesis applies the error propagation framework to mathematical operators for image segmentation,
but the thesis can also be seen as a case study to demonstrate the applicability of the methods
in image processing. Other image processing operators based on a PDE formulation can be extended
by the presented methods easily, because the framework and the implementation of all steps around
the operator extension are available. The only step that remains is the stochastic operator extension.
Furthermore, the stochastic parameter study presented in this thesis sensitizes users to be skeptic
about the segmentation results if these are not robust with respect to parameter changes.
9.2 Conclusion
We presented methods for all tasks along the stochastic image processing pipeline, but some of the
methods presented in this thesis can be improved to get more stable and more accurate results. For
example, the projection step for the estimation of the input distribution (cf. Section 5.2) is based
on a Monte Carlo sampling (it is based on the uncorrelated image samples) and the method has
the poor convergence speed O(1/ √ N) of the Monte Carlo method. Stefanou et al. [141] presented
two methods based on an optimization problem. These methods are computationally much more
expensive, but lead to a better convergence speed. Furthermore, the complete stochastic pipeline is
restricted to a few basic random variables, which might be problematic, because image noise has
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Chapter 9 Summary, Discussion, and Conclusion
Figure 9.1: Visualization of the PDF of the object boundary in the case of the segmentation of the
stochastic US image.
a short correlation length in many applications. Thus, multiple random variables are required to
characterize the input adequately. A possibility to deal with huge numbers of random variables,
and therefore a high dimensional polynomial chaos, is to use adaptive methods for the stochastic
dimensions. A starting point are the methods presented in [22, 80, 152]. In addition, the use of the
parabolic approximation of the level set equation might be problematic in applications where sharp
corners and shocks are important. A direct implementation of the stochastic level set equation could
be based on the work for hyperbolic SPDEs in [92, 147], but these methods are computationally too
expensive and not accurate enough for this task, but the discretization of hyperbolic SPDEs is still
an active field of research.
Another important task is the development of intuitive visualization techniques for the high dimensional
stochastic output. This thesis presented ideas for the visualization of the stochastic results, but
in cooperation with visualization experts, these techniques can be improved. The availability of such
visualization techniques is helpful to convince the image processing community of this kind of error
propagation and to use the error propagation in applications. The user, e.g. a physician, needs
an intuitive access to the stochastic data. A starting point might be the visualization of stochastic
boundaries depicted in Fig. 9.1.
9.3 Outlook and Future Work
Besides the improvement of the methods presented in this thesis, there are possibilities for future
research in the field of image processing with SPDEs. For example, advanced segmentation methods
based on level set formulations can be investigated for stochastic extensions. Furthermore, it is
planned to investigate registration methods [103] for stochastic extensions. Moreover, the stochastic
extensions presented in [130] have to be adapted to the new ansatz space.
Further work directions are the development of efficient methods for stochastic finite difference
schemes, especially for nonlinear operations, because the calculation of the square root of a polynomial
chaos expansion is a bottleneck for most algorithms in this thesis. In addition, the investigation
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9.3 Outlook and Future Work
of level set schemes, which do not need a reinitialization step, is important for efficient stochastic
methods, because at the moment 80% of the computation time is spent for the reinitialization.
In addition, the emerging field of tensor-structured methods [19, 60, 81] is important for the efficient
solution of the presented SPDEs. Tensor-structured methods represent the data and the operators
in a compressed form with a storage requirement linear in the number of dimensions, instead of
the exponential dependence when storing the uncompressed data. Up to now, there are first numerical
examples available in the literature [19, 81] and the methods are not applied on problems arising
in applications like image processing.
A big challenge for the future is to bring this error-aware image processing pipeline into applications.
To be able to achieve this, it is necessary to use problem-dependent basic random variables for
the polynomial chaos. For example, for the modeling of magnetic resonance images it is advantageous
to use Rice distributed basic random variables, because the noise of gradient magnitude images
is Rice distributed. To use a compatible basis leads to more accurate results with fewer basic random
variables. Other input data require different basic random variables. Therefore, it might be a good
idea to construct the basis on the fly if the input data is available based on the method from [157].
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List of Figures
1.1 Left: CT image of a lung lesion (the small roundish structure in the middle of the
image). Right: The segmentation mask computed via region growing [127]. . . . . . 1
1.2 Noisy images from an ultrasound device (left) showing a structure in the forearm and
a computed tomography (right) of a vertebra in a human spine. . . . . . . . . . . . . 2
1.3 This thesis combines findings from image processing with findings about SPDEs to
yield segmentation algorithms acting on stochastic images. . . . . . . . . . . . . . . 3
2.1 Sketch of the ingredients of a digital image. At every intersection of the regular grid
lines a pixel is located and for every pixel the corresponding FE basis function has
its support in the elements around this pixel. . . . . . . . . . . . . . . . . . . . . . . 8
2.2 The graph generated from a 3 × 3 image contains 9 nodes and 12 edges. The edges
e mn connect the nodes (the black dots) v l . Every edge e mn has a weight w mn describing
the costs for traveling along this edge. . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Left: Definition of the seed regions for the object (yellow) and the background (red).
Middle: The probability that a random walker reaches an object seed. Black denotes
probability zero, white probability one. Right: Random walker segmentation result
of the ultrasound image. As input we used the seed regions from the left image and
β = 200. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.4 From left to right: Three steps of the interactive random walker segmentation. We
show the seeds and the image to segment in the upper row and the segmentation
corresponding to this particular choice of the seeds in the lower row. The addition
of seed regions for the object and the background yield an iterative refinement of the
segmentation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.5 Left: The initial (noisy) US image treated as input for the Ambrosio-Tortorelli approach.
Middle: The smooth Ambrosio-Tortorelli approximation of the initial image.
Right: The corresponding phase field, i.e. the approximation of the edge set of the
smoothed image. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.6 Comparison of the Ambrosio-Tortorelli model (left) and the extended model using
the edge linking procedure (right). Data set provided by PD Dr. Christoph S. Garbe. . 17
2.7 Segmentation of a medical image based on a level set propagation with gradientbased
speed function. The time increases from left to right and the zero level set (red
line) approximates the boundary of the object (a liver mask) at the end. . . . . . . . 19
2.8 Segmentation using geodesic active contours. Left: The initial image. Right: Solution
of the geodesic active contour method initialized with small circles inside the
object. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.9 Segmentation of an object without sharp edges using the Chan-Vese approach. In red,
we show the steady-state solution of the Chan-Vese segmentation method initialized
with a small circle inside the object. . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.10 A test pattern corrupted by uniform (left), Gaussian (middle), and speckle noise (right). 22
3.1 Relation between the stochastic spaces. We avoid the integration over Ω with respect
to the measure Π. Instead, we transform the integral into integration over a subset of
IR (the space Γ i ) with respect to the known PDF ρ of the basic random variables ξ i . . 29
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List of Figures
3.2 Sparsity structure of the stochastic lookup table for n = 5 random variables and a
polynomial degree p = 3. The gray dots indicate positions in the three-dimensional
lookup table C αβγ that contain nonzero entries. . . . . . . . . . . . . . . . . . . . . 35
3.3 PDFs of initial uniformly distributed input intervals (gray) and the PDFs of the results
of the polynomial chaos computation (black) for squaring an interval (left) and
dividing an interval by itself (right). . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.1 Comparison between a sparse grid (left) constructed via Smolyak’s algorithm and a
full tensor grid (right). The sparse grid contains significantly less nodes than the full
tensor grid whose number of nodes growth exponentially with the dimension, but
has nearly the same approximation order. . . . . . . . . . . . . . . . . . . . . . . . 38
4.2 Comparison of discretization methods with respect to implementational effort and
speed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.3 Refinement of a rectangular element of a finite element mesh. A single element on a
coarser level splits up into four elements on the next finer level. . . . . . . . . . . . . 45
4.4 Refinement of elements leads to hanging nodes (circles) which are no degrees of
freedom, instead the values of the constraining nodes (squares) restrict them. . . . . 45
4.5 For an unsaturated error indicator, the appearance of hanging nodes constrained by
hanging nodes (due to level transitions of more than one between neighboring elements)
is possible (left). The saturation of the error indicator ensures that there are
level one transitions between neighboring elements only (right). . . . . . . . . . . . 46
5.1 Sketch of the ingredients of a stochastic image. We discretize the spatial dimensions
using finite elements, but the coefficients of the FE basis functions are random variables.
Every random variable has a support, which spans over the complete image,
thus pixels depend on a random vector. . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.2 Decay of the sorted eigenvalues of the centered covariance matrix of 45 input samples
from an ultrasound device. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5.3 Left picture group: The first mode (=expected value), second mode, third mode and
fourth mode of a stochastic CT image. Right: The sinogram, i.e. the raw data produced
by the CT imaging device for the head phantom [139]. . . . . . . . . . . . . . 51
5.4 Second (left) and fifth (right) mode of a stochastic US image. The information encoded
in these images is hard to interpret, because there is no deterministic equivalent. 53
5.5 Expected value (left) and variance (right) of a stochastic US-image. The expected
value looks like a deterministic image and in the variance, regions with a high gray
value uncertainty are visible as white dots. . . . . . . . . . . . . . . . . . . . . . . . 54
5.6 Two samples drawn from a stochastic image. The images differ due to realizations
of the noise. In a printed version, these images look nearly the same. . . . . . . . . . 54
5.7 Visualization of realizations of a stochastic 2D contour. Every yellow line corresponds
to a MC realization of the stochastic contour encoded in the stochastic image. 55
5.8 Visualization of a 3D contour encoded in a 3D stochastic image. The expected value
of the 3D stochastic contour is color-coded by the variance. Regions with a high
variance are red and regions with a low variance green. . . . . . . . . . . . . . . . . 55
6.1 Expected value (top row) and variance (bottom row) of the street image (left) and the
US image (right). Color-coded are the seed regions for interior (yellow) and exterior
(red). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
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List of Figures
6.2 Mean and variance of the probabilities for pixels to belong to the object. Furthermore,
we show in red Monte Carlo realizations of the object boundary sampled from
the stochastic result. A high variance indicates pixels where the gray value uncertainty
highly influences the result. For comparison we added a classical random
walker segmentation result in the last column. There the variance image is not available,
because the method acts on a classical image. . . . . . . . . . . . . . . . . . . 61
6.3 MC-realizations of the stochastic object boundary for the stochastic liver image segmented
with the stochastic random walker approach with β = 10. On the right we
highlight a region of the image, where the noise in the input image influences the
result. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
6.4 PDF of the area of the segmented person from the street image for β = 25 (black)
and β = 50 (gray). From the PDF we judge the reliability of the segmentation, a
narrow PDF indicates that the image noise influences the segmentation marginally. . 63
6.5 Comparison of the discretization methods for the computation of the stochastic random
walker result to verify the intrusive discretization. The small difference between
the intrusive discretization via the GSD method and the two other sampling based approaches
might be due to the projection of the Laplacian matrix on the polynomial
chaos. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
6.6 Input “doughnut” without noise (left) and noisy input image treated as expected value
of the stochastic image (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
6.7 Left: The object seed points (yellow) and background seed points (red) used as initialization
of the stochastic random walker method. Right: The MC-realizations of
the stochastic segmentation result differ significantly for different noise realizations. 66
6.8 The PDF for both possibilities of the volume computation, the summation of the
random variables (gray) and the thresholding (black). The true volume is 60 pixels. . 66
6.9 Structure of the block system of an SPDE. Every block has the sparsity structure
of a classical finite element matrix and the block structure of the matrix is sparse,
meaning that some of the blocks are zero. The sparsity structure on the block level
depends on the number of random variables and the polynomial chaos degree used
in the discretization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
6.10 Nonzero pattern of the SFEM matrix for the smoothed stochastic image using n = 5
random variables and a polynomial degree p = 3. A black dot denotes a block that
has a nonzero stochastic part, thus having the sparsity structure of a classical FEM
matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
6.11 Mean value of the three data sets used to demonstrate the stochastic Ambrosio-
Tortorelli method. For the second data set, we denoted image regions the text refers
to. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
6.12 PDF of a pixel from the phase field computed from the polynomial chaos expansion
of the pixel via a sampling approach. Although we use uniform basic random
variables for the polynomial chaos, the resulting random variables have skewed and
Gaussian like distributions due to the use of higher order polynomials in the basic
random variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
6.13 Segmentation result of the street scene. On the left we show the five samples the
stochastic input image is computed from. On the right we compare the results computed
via the GSD method and a Monte Carlo sampling. . . . . . . . . . . . . . . . 73
6.14 Expected value and variance of the stochastic input image of the street scene. . . . . 73
6.15 Mean and variance of the image and phase field for varying ε and µ using the US
data. For comparison, we added the result from the deterministic method applied on
the mean. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
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List of Figures
6.16 Comparison of the stochastic Ambrosio-Tortorelli model (left column) with the extended
model using the edge linking procedure described in Section 2.3.3 (middle
column) and a combination of the edge linking and adaptive grid approach (right
column). Note that these results are computed with the same parameter set. The
differences in the results are due to the additional edge linking parameter c only. . . . 75
6.17 Comparison of the full grid and adaptive grid solution. The full grid and adaptive
grid solution are visually identical, but the computation of the adaptive grid solution
needs significantly less DOFs. Thus, it can be applied on high-resolution images. . . 77
7.1 Stochastic level sets do not have a fixed position where φ(x) = 0. Instead, there is
a band with positive probability that the level set is equal to zero, i.e. the position of
the zero level set is random and it is possible to estimate the PDF of the interface
location in the normal direction of the expected value of the interface (lower right
corner). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
7.2 Comparison of expected value and variance of the resulting phase field for the cosine
test of (7.18) using the polynomial chaos (PC), stochastic collocation (SC), Monte
Carlo simulation (MC), and Monte Carlo simulation of the original level set equation
(MCL). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
7.3 Comparison of the expected value and variance of the resulting phase field for the
rarefaction fan and the shock, two classical tests for level set propagation. The figure
shows the comparison of the four discretizations of the stochastic phase field equation. 86
7.4 Expected value color-coded by the variance for the Stanford bunny after shrinkage
under an uncertain speed in the normal direction. Red indicates regions with a high
variance and green regions with low variance. In addition, we show one slice of the
variance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
7.5 Mean of the CT data set (left) and the liver data set (right) for the segmentation test. . 88
7.6 Left: Mean contour during the evolution of the stochastic level set. The iso-contours
are drawn on the variance image of the final, magenta contour. The contour detection
is influenced by the image noise on the bottom and the right of the object (high
variance). Right: Contour realizations of the stochastic gradient-based segmentation
of the CT data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
7.7 Resulting image with the expected value of the contour (red) of the segmented object
and the phase field variance with the expected value of the contour for gradientbased
segmentation of a stochastic CT image. The variance is constant in the normal
direction of the expected value of zero level set. . . . . . . . . . . . . . . . . . . . . 90
7.8 Mean and variance of the stochastic geodesic active contour segmentation of the
stochastic CT data set. The variance is constant in the normal direction of the zero
level set. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
7.9 Left: Evolution of the expected value contour of the stochastic geodesic active contour
method. The shown variance corresponds to the contour after 240 iterations (the
magenta contour). Right: Mean value of the stochastic image to be segmented and
the contours at time points of the level set evolution. The final contour matches the
object boundary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
7.10 Mean (left) of the stochastic CT and the variance (right) of the stochastic Chan-Vese
solution. Additionally, we show the expected value contour at different time steps. . . 93
7.11 Left: MC-realizations of the stochastic contour from the stochastic Chan-Vese segmentation
applied on the CT data set. Right: Realizations of the stochastic contour
from the stochastic geodesic active contour approach applied on the CT data set. . . . 94
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List of Figures
7.12 Mean (left) of the stochastic liver image and the variance of the stochastic Chan-Vese
solution. In addition, we show the expected value contour at different time steps. . . 95
7.13 Variance of the stochastic image to segment (left), the expected value is not depicted,
because the expected value is an image with the same gray value at every pixel. On
the right, the segmentation result is depicted on one realization (one sample) of the
stochastic image to segment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
8.1 Left: Realizations of the stochastic contour obtained from the random walker segmentation
with stochastic parameter. Right: Mean and variance of the stochastic
contour obtained from the random walker segmentation with stochastic parameter. . . 99
8.2 Left: Realizations of the stochastic contour obtained from the random walker segmentation
with stochastic parameter. Right: Mean and variance of the stochastic
contour obtained from the random walker segmentation with stochastic parameter. . . 100
8.3 Volume of the stochastic contour obtained from the random walker segmentation
with stochastic parameter. The left curve shows the PDF of the object in the US
image, the right curve the PDF of the liver in the liver data set. The PDFs are obtained
using (6.12). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
8.4 Ambrosio-Tortorelli model applied on the expected value of the liver data set using
a stochastic parameter µ. The upper row shows the expected value (left) and the
variance (right) of the smoothed image, the lower row the expected value (left) and
the variance (right) of the phase field. . . . . . . . . . . . . . . . . . . . . . . . . . 102
8.5 Ambrosio-Tortorelli model applied on the expected value of the liver data set using
a stochastic parameter ε. The upper row shows the expected value (left) and the
variance (right) of the smoothed image and the lower row the expected value (left)
and the variance (right) of the phase field. . . . . . . . . . . . . . . . . . . . . . . . 103
8.6 Result of the gradient-based segmentation with stochastic parameter b, i.e. with a
stochastic curvature smoothing. The upper row shows the results for the CT data set,
expected value of the image and contour realizations (left) and variance of the level
set with contour realizations (right). The lower row shows the same results for the
liver data set. In all figures we add Monte Carlo realizations of the stochastic object
boundary. The red contour corresponds to a b = 0.75, yellow to b = 1.0 and blue to
b = 1.25. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
8.7 Result of the geodesic active contour segmentation with stochastic parameters for
the CT data set. On the left the expected value of the image and contour realizations
and on the right the variance of the level set with contour realizations are shown. . . 105
8.8 Result of the geodesic active contour segmentation with stochastic parameters for
the liver data set. On the left the expected value of a detail of the image and contour
realizations and on the right the variance of the level set with contour realizations are
shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
9.1 Visualization of the PDF of the object boundary in the case of the segmentation of
the stochastic US image. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
115

List of Tables
3.1 The first ten one-dimensional Legendre-polynomials. The multi-dimensional polynomials
up to degree nine are based on these polynomials and (3.40). . . . . . . . . . 32
3.2 Important distributions and the corresponding polynomials for the expansion. . . . . 32
3.3 The first ten one-dimensional Hermite-polynomials. The construction of the multidimensional
polynomials up to degree 9 is based on these polynomials and (3.40). . . 33
6.1 Comparison of the execution times (in sec) of the discretization methods. . . . . . . 64
117

Appendix A
Publications Written During the Course
of the Thesis
Parts of the results of this thesis are already published or submitted for publication. Besides the
publications related to this thesis, the author published results about the simulation of radio frequency
(RF) ablation. We give a short introduction into RF ablation before we list the papers.
A.1 Publications Related to Stochastic Images
[1] T. Pätz, R. M. Kirby, and T. Preusser. Ambrosio-Tortorelli segmentation of stochastic images:
Model extensions, theoretical investigations and numerical methods. Submitted to International
Journal of Computer Vision, 2011.
[2] T. Pätz, R. M. Kirby, and T. Preusser. Segmentation of stochastic images using stochastic extensions
of the Ambrosio-Tortorelli and the random walker model. PAMM, 11(1):859–860, 2011.
[3] T. Pätz and T. Preusser. Ambrosio-Tortorelli segmentation of stochastic images. In K. Daniilidis,
P. Maragos, and N. Paragios, editors, Computer Vision - ECCV 2010, volume 6315 of Lecture
Notes in Computer Science, pages 254–267. Springer Berlin / Heidelberg, 2010. (This paper
received the ECCV 2010 Best Student Paper Award.).
[4] T. Pätz and T. Preusser. Segmentation of stochastic images using level set propagation with
uncertain speed. In preparation, 2011.
[5] T. Pätz and T. Preusser. Segmentation of stochastic images with a stochastic random walker
method. Submitted to IEEE Transactions on Image Processing, 2011.
[6] T. Pätz and T. Preusser. Variational image segmentation using stochastic parameters. In preparation,
2011.
A.2 Publications Related to Radiofrequency Ablation
RF ablation is a minimally invasive technique for a local ablation of abnormal tissue, like primary or
metastatic cancer. During the last years, RF ablation has become an alternative to the surgical resection
of the tumor. At the beginning of the treatment, an internally cooled RF probe is percutaneously
placed inside the tissue and connected to an RF generator. The generator delivers an electric current
in the radio-frequency range (typically 500 kHz) with a power between 25W and 200W. Due to the
electric impedance, the tissue close to the probe is heated and above 60 ◦ C it is destroyed.
The modeling and simulation of RF ablation is a multiple investigated research topic (see [20] for
a review). Many scientists presented simulations with varying detail, because multiple biophysical
effects take place during the ablations. Another challenge is the modeling of the physical parameters
influencing the ablation outcome, because these parameters are (nonlinearly) influenced by biophysical
effects. For example, the electric conductivity is nonlinearly dependent on the temperature, the
119

vaporization state, and the coagulation state of the tissue. The simulation of RF ablation typically
uses a coupled system of PDEs for the electric potential and the heat transfer.
[7] I. Altrogge, T. Pätz, T. Kröger, H.-O. Peitgen, and T. Preusser. Optimization and fast estimation
of vessel cooling for RF ablation. In World Congress on Medical Physics and Biomedical
Engineering, September 2009, Munich, Germany, volume 25/4 of IFMBE Proceedings, pages
1202–1205. Springer, 2010.
[8] I. Altrogge, T. Preusser, T. Kröger, S. Haase, T. Pätz, and R. M. Kirby. Sensitivity analysis for
the optimization of radiofrequency ablation in the presence of material parameter uncertainty.
Submitted to International Journal for Uncertainty Quantification, 2011.
[9] T. Kröger, T. Pätz, I. Altrogge, A. Schenk, K. S. Lehmann, B. B. Frericks, J.-P. Ritz, H.-O.
Peitgen, and T. Preusser. Fast estimation of the vascular cooling in RFA based on numerical
simulation. Open Biomed Eng J, 4:16–26, 2010.
[10] T. Pätz, T. Kröger, and T. Preusser. Simulation of radiofrequency ablation including water evaporation.
In World Congress on Medical Physics and Biomedical Engineering, September 2009,
Munich, Germany, volume 25/4 of IFMBE Proceedings, pages 1287–1290. Springer, 2010.
[11] T. Pätz and T. Preusser. Simulation of water evaporation during radiofrequency ablation using
composite finite elements. In Proceedings of the 1st Conference on Multiphysics Simulation –
Advanced Methods for Industrial Engineering, 2010.
[12] T. Pätz and T. Preusser. Composite finite elements for a phase-change model. Submitted to
SIAM Journal on Scientific Computing, 2011.
[13] T. Pätz and T. Preusser. Simulation of water evaporation during radiofrequency ablation using
composite finite elements. The International Journal of Multiphysics, Special Edition: Multiphysics
Simulations – Advanced Methods for Industrial Engineering, pages 145–156, 2011.
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