MPhil Thesis (pdf) - Image Processing and Analysis Group

Level Set Implementations on
Unstructured Point Cloud
by
HO, Hon Pong
A Thesis Submitted to
The Hong Kong University of Science and Technology
in Partial Fulfillment of the Requirements for
the Degree of Master of Philosophy
in Electrical and Electronic Engineering
Copyright c○ by HO, Hon Pong (2004)
ALL RIGHTS RESERVED.
27 August, 2004

Authorization
I hereby declare that I am the sole author of this thesis.
I authorize the Hong Kong University of Science and Technology to lend this thesis
to other institutions or individuals for the purpose of scholarly research.
I further authorize the Hong Kong University of Science and Technology to reproduce
this thesis by photocopying or by other means, in total or in part, at the request of
other institutions or individuals for the purpose of scholarly research.
HO, Hon Pong
ii

Level Set Implementations on
Unstructured Point Cloud
by
HO, Hon Pong
This is to certify that I have examined the above MPhil thesis
and have found that it is complete and satisfactory in all respects,
and that any and all revisions required by
the thesis examination committee have been made.
Prof. Pengcheng SHI, Thesis Supervisor
Prof. Oscar C.L. AU, Thesis Examination Chairperson
Prof. Chi-Keung TANG, Thesis Examination Examiner
Prof. Khaled Ben LETAIEF, Head of the Department
Department of Electrical and Electronic Engineering
27 August, 2004
iii

Acknowledgements
I want to express my deep thanks to my supervisor Dr. Pengcheng Shi for his guidance
over the past years. I learnt great lessons through his kindness effort on every paper
we wrote and also his fruitful advices and comments from research-related to non-
academic issues. And thank again for his patience to my many mistakes. I also thank
Dr. Oscar C. Au and Dr. Chi-Keung Tang for being the thesis examination committee
members.
Special thanks go to Prof. Yunmei Chen of the University of Florida for her clear
analysis and precious suggestions about this thesis idea during her visit to our research
group.
I must thank my family and Janet for their love and care at anytime. Their support
are crucial and invaluable to me.
iv

Contents
Acknowledgements iv
List of Figures ix
List of Tables xi
1 Introduction 1
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 The need of computational grid . ............... 2
1.2.2 The need of grid refinement . . . ............... 3
1.2.3 A grid-less approach . . . ................... 4
1.3 Application to image segmentation ................... 4
1.4 Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.5 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 Review on Level Set Method and its domain representation 8
2.1 Level Set Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
v

2.2 Level set domain representation schemes . ............... 9
2.2.1 Sampling problem of a moving interface . ........... 9
2.2.2 Regular level set grid . . . ................... 10
2.2.3 Refined level set grid . . . ................... 12
2.2.4 Moving level set grid . . . ................... 12
2.2.5 Level set domain triangulation . . ............... 13
2.2.6 Adaptive level set domain triangulation . ........... 14
3 Level set implementations on unstructured point cloud 15
3.1 Level set initialization . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.1.1 Narrow band of level set nodes . . ............... 15
3.1.2 Signed distance function . ................... 18
3.2 Domain interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2.1 Influence domain . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2.2 Moving Least Square Approximation . . ........... 19
3.2.3 Generalized Finite Difference: An alternative to MLS . . . . . 22
3.3 Evolution process and level set re-initialization . ........... 24
3.3.1 Evolution process . . . . . . . . . . . . . . . . . . . . . . . . 24
3.3.2 Level set re-initialization . ................... 26
4 Application to image segmentation 30
4.1 Deformable model . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
vi

4.1.1 FEM contour mesh . . . . . . . . . . . . . . . . . . . . . . . 30
4.1.2 Irregular FEM contour mesh . . . ............... 31
4.2 Level set based Geometric Deformable Model (GDM) . ....... 32
4.2.1 GDM segmentation process . . . ............... 35
4.2.2 Level set velocity field for GDM segmentation . ....... 36
4.2.3 Gradient Vector Flow . . . ................... 38
4.3 Adaptive unstructured sampling of computational domain ....... 38
4.3.1 Sampling node distribution ................... 39
4.3.2 Feature-adaptive node distribution algorithm . . ....... 40
5 Experimental results and conclusion 43
5.1 General GDM segmentation results in 2D ............... 43
5.2 Segmentation in 3D . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.4 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
A Topology-preserving GDM through domain partitioning 55
A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
A.2 Problem definition . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
A.3 Domain representation . . . . . . . . . . . . . . . . . . . . . . . . . 58
A.4 Domain partitioning . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
A.5 Movable partitioning . . . . . . . . . . . . . . . . . . . . . . . . . . 60
vii

A.6 Separation enforcement . . . . . . . . . . . . . . . . . . . . . . . . . 61
A.7 TPLSS results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
A.8 Summary of the algorithm . . . . . . . . . . . . . . . . . . . . . . . 64
viii

List of Figures
2.1 Explicit representation for the moving front . . . ........... 9
2.2 Various types of level set domain representation schemes ....... 11
3.1 Initialization for the point-based level set evolution . . . ....... 17
3.2 Influence domain, cubic spline function and interpolated surface . . . 20
3.3 Flowchart of general level set evolution process . ........... 24
3.4 An illustration of a complete evolution process for embedded level set
front . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.1 Flowchart of the meshfree GDM segmentation process . ....... 34
4.2 Data inputs for GDM segmentation: intensity, gradient magnitude and
Gradient Vector Flow (GVF) . . . ................... 37
4.3 Process of recursive gradient-adaptive node distribution . ....... 42
5.1 Segmentation of the endocardium and epicardium from cardiac mag-
netic resonance image . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5.2 Segmentation of the brain ventricles from noisy BrainWeb image . . . 44
5.3 Examples of point-based GDM evolution ............... 45
ix

5.4 Comparison of segmentation results among different implementation
strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5.5 An illustration of the relationship between node distribution and the
precision of implicit contour . . . ................... 48
5.6 Meshfree GDM Segmentation of a synthetic cube ........... 49
5.7 Meshfree GDM Segmentation on BrainWeb image . . . ....... 50
5.8 Solution refinement using unstructured point cloud . . . ....... 51
5.9 Enlarged and overlay views of segmentations from BrainWeb image . 52
5.10 Narrow band nodes and adaptive influence domains . . . ....... 53
A.1 Brain image and the closeup view of the ventricles. . . . ....... 57
A.2 Illustration of the saddle map detecting the crossing of intensity derivative 60
A.3 Comparison between separation-unconstrained and separation-constrained
segmentation on synthetic and real images ............... 62
A.4 Comparison between separation-unconstrained and separation-constrained
GDM evolution processes . . . . . . . . . . . . . . . . . . . . . . . . 63
x

List of Tables
5.1 Error statistics comparison between point-based and regular grid level
set implementations . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
xi

Level Set Implementations on
Unstructured Point Cloud
HO, Hon Pong
Thesis for the Degree of Master of Philosophy
Department of Electrical and Electronic Engineering
The Hong Kong University of Science and Technology
ABSTRACT
We present a novel level set representation and Geometric Deformable Model
(GDM) evolution scheme where the analysis domain is sampled by unstructured cloud
of sampling points. The points are adaptively distributed according to both local im-
age information and level set geometry, hence allow extremely convenient enhance-
ment/reduction of local curve precision by simply putting more/fewer points on the
computation domain without grid refinement (as the cases in finite difference schemes)
or remeshing (typical in finite element schemes). The GDM evolution process is then
conducted on the point-sampled analysis domain, without the use of computational
grid or mesh, through the precise but expensive moving least squares (MLS) approxi-
mation of the continuous domain and calculations, or the faster yet coarser generalized
finite difference (GFD) calculations. Because of the adaptive nature of the sampling
point density, our strategy performs fast marching and level set local refinement con-
currently. The performance of our effort is evaluated and analyzed using synthetic and
real images.
xii

Chapter 1
Introduction
1.1 Background
Level Set Method [27, 28] is a numerical device to solve for interface motion under a
velocity field. Many things can be modelled as interface, such as water front, electrical
potential or physical object boundary. This method is very useful because it can suc-
cessfully capture and evaluate the motion of matters in wide range of both application
and research areas.
One key issue in level set implementation is the tradeoff between computational
efficiency and accuracy of solution (e.g. the interface being modelled), and several
algorithmic improvements such as the fast marching scheme [1] and the narrowband
search [24] have provided some reliefs. However, since most Level Set implementa-
tions rely on the finite difference computational schemes, the resolutions of the final
results are fundamentally dependent on the computational grids used to solve the dis-
creteized level set partial differential equations (PDEs). While a highly refined grid
allows high accuracy, the computational cost may be prohibitively expensive for prac-
tical applications, for example, image segmentation. On the other hand, coarse grid
allows fast computation at the expense of low fidelity to the original continuous solu-
tions.
1

Since the purpose of domain representation is to offer an appropriate embedding
for various calculations needed by the interface or front representation and evolution,
one is obviously not limited to the specific pre-structured rectangular sampling for
finite difference computations. Finite element mesh has emerged as an alternative
scheme for level set algorithms by offering a piecewise continuous approximation of
the domain where the level set computations are performed [2, 34]
1.2 Motivation
1.2.1 The need of computational grid
The need of more samples for representing high frequency signal is obvious under
sampling theory, and 2D curve sampling follows the same rule. In computer graphics,
lines in-between samples are created for curve visualization with low cost. Indeed, it
is just a first order interpolation of the curve signal. For 3D surface, we will have a
mesh instead. If shading is not required, we can simply use huge amount of samples
to replace the 2D lines or 3D mesh if resources available. In dynamic model of the
moving interface discussed here, tangents, normals and their higher derivatives are
needed for measuring geometric properties, like bending energy of the FEM-Snake
or |∇φ| values in the Level Set Method, these are similar to the use of normals in
many shading and illumination models. But normal or derivatives of a particular point
on the moving interface is varying over time. In 2D implementation, computing the
gradient of a 2D scalar function requires, for example, one sample on the right and
one sample above the point. Left, right, above or below are just positional relationship
among samples. Mesh or grid formation for the scalar function is actually a default
establishment of such a relationship so that local neighboring samples can be consulted
immediately upon the computation of derivatives and also integrations at a particular
2

point.
1.2.2 The need of grid refinement
Once the mesh or grid has been established, it also means that we divide our interface
model into lines or surface pieces, and assume the structure between vertices is well-
defined geometric such as straight lines, quadratic curves, flat plains or quadratic sur-
faces. Indeed, the idea of domain subdivision is critical for simplifying many dynamic
problems concerning complex rigid body, where a node on the mesh corresponds to a
particular sample of the body. But this assumption is valid only if the body is not de-
formed seriously during its motion. Unfortunately, our evolving interface is a non-rigid
body in nature and we do not know the final length nor shape in advance. Therefore,
many refinement strategies for the contour vertices have been derived to overcome the
limitation of using default mesh or grid for computation.
In Level Set Method, an n-D scalar function φ is used to embed one or more
(n − 1)D surface by the zero set {φ =0} of the φ function. In order to get higher
precision for certain locations, we prefer more samples over there. But we also need
to compute gradient of φ, i.e. ∇φ, to implement the PDE of the level set evolution.
In other word, we can notice that the refinement method of φ is intuitively constrained
by the arithmetics of various PDE formulations, which usually involve finite differ-
ence among values of grid-defined neighboring points. So refining the φ grid seems a
straight forward way to increase the sampling rate of φ at the same time maintaining
∇φ tractable.
3

1.2.3 A grid-less approach
It is clear to see that if the arithmetic of ∇φ is based on arbitrary neighboring points but
not grid-defined ones, then we can increase the sampling rate of φ freely without a pre-
defined pattern. This can be done by not using finite difference between φ samples, but
first approximating the continuous φ function from its samples by polynomials fitting
and then taking derivative of the approximated polynomial surface to obtain ∇φ. An
early developed mathematical tool known as Moving Least Squares [17, 18, 5] exactly
matches our needs for data fitting. By having a continuous φ approximation, we can
also extract the continuous contour {φ =0} on the 2D image by marching precisely
on the continuous φ surface so that the contour precision is solely dependent on the
marching step size, which is just up to our design.
1.3 Application to image segmentation
Segmentation has been nearly the most fundamental process for many computer vision
problems. Edges on an image could be very useful information for identifying object
in an image. On digital images, edges can be regarded as pixels with high gradient of
intensity, and gradient can be obtained through various basic image processing tech-
niques [11]. One generic description of object on images could be the outer boundary
of the object which could be viewed as an continuous closed contour on the image.
Looking for the correct closed contour is not trivial at all, as edges are likely to be
broken if noise exists or the object is party occluded. Despite of broken edges, it is still
possible to define some generic ways to connect the edges or high gradient areas up to
form a closed contour. Assume that we do not have particular knowledge towards the
shape of our target object. But we know that our final result should be a closed curve
completely around an object, and it should be located mainly on high gradient areas
4

with inconsistency.
Active deformable models have been among the most successful strategies in object
representation and segmentation [16, 24]. While in general the parametric deformable
models (PDMs) have explicit representations as parameterized curves in a Lagrangian
formulation [10, 16, 25, 35], the geometric deformable models (GDMs) are implicitly
represented as level sets of higher-dimensional, scalar level set functions and evolve
in an Eulerian fashion [7, 24, 28]. The geometric nature of the GDMs gives them
several often desirable properties, i.e. independence from curve parametrization, easy
computation of curvatures, and natural compatibility with curve topological changes.
1.4 Contribution
In the Lagrangian formulation of PDMs and object motion tracking, meshfree particle
methods have been introduced as more efficient and possibly more effective object
representation and computation alternatives because of trivial h − p adaptivity [21].
Following the same spirit, it is easy to see that if level set computations can be based on
meshfree particle representation of the analysis domain, we can almost freely increase
or decrease the sampling rate of the domain without many extra efforts. This can be
achieved through point-wise continuous approximation of the domain and the level set
functions through polynomials fitting at the local point cloud.
Hence, we introduce a novel level set representation and evolution scheme where
the analysis domain is sampled by adaptively distributed cloud of points. The curve
evolution process is performed on the point-sampled analysis domain through either
Moving Least Squares (MLS) approximation or generalized finite difference (GFD)
schemes. Because of the h − p adaptive nature of the sampling point, our strategy nat-
urally possesses multiscale property and performs fast marching and local refinement
5

concurrently.
1.5 Outline
The focus of this thesis is about enhancing general level set implementation with higher
accuracy and flexibility. There are many level set motion governing equations which
involve different utilizations of available data or information. Most of their implemen-
tations are built upon grid-based finite difference scheme. We are not going to add
something on top of those ideas, but we offer a different computation environment
behind their implementations. This new environment can be applied to most of the
currently available level set formulation with minor modification. So we will explain
and illustrate the key components for this new environment in first two chapters and
show how this new strategy can be used to implement the level set based GDM for
segmentation.
In chapter 2, the Level Set Method for modelling general front propagation is in-
troduced as it is the foundation of our numerical strategy. Particularly in chapter 2.2,
different types of level set representation schemes for the moving interface/front will
be reviewed. Representation scheme has huge impact on the speed and accuracy of the
interface evolution process. This review is to give a broad discussion about the pros
and cons of, from simple to sophisticated, representation schemes in terms of speed
and accuracy. More importantly, it shows the trend of improvement on representations
which enlightened our thought to move towards a grid-less / meshfree representation.
We will then move on to the details of the new environment in chapter 3. Chap-
ter 3.2 discusses how to obtain derivatives on unstructured samples using moving least
squares (MLS) approximation or generalized finite difference (GFD) scheme. We state
the changes related to level set re-initialization in chapter 3.3. Common level set liter-
6

atures usually disregard the re-initialization procedures because they assume the zero
set of a sampled function can be exacted easily [23]. Although it is true for grid-based
implementation, our grid-less approach requires more effort on this issue.
To demonstrate the feasibility of the new computational environment, chapter 4
systematically combines all procedures needed for GDM segmentation. The corre-
sponding GDM formulations, especially the force term in that formulation, can be
found in chapter 4.2.2. And chapter 4.3 is about how we adaptively re-sample an im-
age according to selected image features.
Experimental segmentation results on synthetic, real, 2D and 3D images are at
chapter 5. Conclusion and future work will also appear at the last chapter of this
thesis.
Appendix A is an example of how this new computational environment gives a
simplified solution to an old problem. The problem is related to the side effect of
GDM topological changeability [14] and our simple solution takes the advantage of
the infinite resolution capability using grid-less approach.
7

Chapter 2
Review on Level Set Method and its
domain representation
2.1 Level Set Method
The basic idea of the Level Set Method is to implicitly represent the moving front Γ,
propagating along its normal direction, by the zero level set of a higher dimensional
hypersurface φ (the level set function) [1, 24, 30, 26]:
Γ={x|φ(x,t)=0} (2.1)
Differentiate the above equation with respect to t, and define the scalar velocity field
F as F = n · x ′ (t) with n = ▽φ the normal direction, we have
∂φ
∂t
where |▽φ| represents the norm of the gradient of φ.
= F |▽φ| (2.2)
The front evolution is now driven by Equation (2.2) and we are going to solve for
convergence of that dynamic equation by time-domain finite difference:
φ t+∆t = φ t +∆tF|∇φ| (2.3)
Equation (2.3) is the basic level set formulation for modelling front propagation.
8

In fact, there are many motion governing formulations for level set implicit front Γ.
Different application-dependent properties can be achieved by having specific F term
in Equation (2.3), for example, if F =1all the time and initial Γ is a circle, then
subsequent results of the Γ will be a sequence of expanding concentric circles. Apart
from the F term, when modelling real interface in 3D, gradient of the φ can be obtained
through:
∇φ = [ ∂φ ∂φ ∂φ
, , ]
∂x ∂y ∂z
(2.4)
|∇φ| = ( ∂φ
∂x )
2
+( ∂φ
∂y )
2
+( ∂φ
∂z )
2
(2.5)
Our concern here is not to get a tailor-made F term for particular application,
but we want to address the underlying sampling issue for all existing level set based
solutions and suggest a sample way to improve their quality with minor efforts.
2.2 Level set domain representation schemes
2.2.1 Sampling problem of a moving interface
Continuous
front
Discrete
vertex
Front
Mesh
Figure 2.1: Explicit representation of the moving front. Left: regular sampling rate
(vertices) along the continuous front. Right: irregular sampling rate with more samples
at high curvature locations.
When implementing any of the moving interfaces in computers, we also need to
have a proper way to sample the interfaces and have a data structure to represent their
status during motion. Before the proposal of using implicit representation, an intuitive
9

way to represent the interfaces under analysis is to use finite number of samples (ver-
tices in Figure 2.1) taking exactly on the ideal continuous front. Relations between
neighboring vertices are also kept in the form of a mesh.
This configuration enables us to approximate the behavior of the actual front by
using discrete values which can be processed by computers. From a mathematical
perspective, this is just a parametrization of a continuous 2D curve: C(x, y) =0into
two scalar functions: {Sx(s),Sy(s)}, where s ∈ [0, 1]. In discrete case, the parameter
s is further restricted to a finite set of values.
Well, one reason for level set representation being more flexible is that we can have
multiple interfaces embedded in one φ function without increasing the complexity of
φ. But, without s and vertices mentioned before, the representable shape variety is
still limited by another factor ; how accurate the φ function can be? It should not
be a problem if we can have a continuous φ function. But in practice, very often we
would use a sampled φ and zero-crossings of φ as front vertices, see Fig.2.2(a). So the
sampling rate and sampling pattern for the φ function take command of the accuracy
of contour representation. The sampling rate actually means how many samples are
used and the sampling pattern is related to where the samples are taken.
2.2.2 Regular level set grid
If we are using the Level Set Method for image processing, a trivial way to perform
sampling in 2D is that we make a sample for the φ function at every image pixel point.
This is equivalent to say that the sampling is based on a rectangular grid which is the
same as the source image. But image grid and level set grid (φ grid) do not necessarily
be the same. Making them the same just implies a full coverage of possible contour
positions on the image, i.e. the solution domain. The immediate drawback is that
10

(a) Regular Level Set Grid (b) Refined Level Set Grid (c) Moving Level Set Grid
(d) Level Set Domain
Triangulation
(e) Refined Level Set
Domain Triangulation
Supporting
Nodes
(f) Adaptive Level Set
Domain Triangulation
(g) Meshfree
Representation
Figure 2.2: Various types of level set domain representation schemes, see text for
details
11

we need more computations for obtaining the derivatives of φ. As we are moving a
implicit curve, speed-up can be done by limiting the φ function available only around
the moving curve. The area of valid φ is defined to be the narrow band of the level set
function, and it moves together with the curve. Note that the image grid and level set
grid are still the same within the narrow band.
2.2.3 Refined level set grid
Recall that we are moving interfaces or curves under a known velocity field (see be-
ginning of this chapter), we could expect those interfaces will have more complicated
shapes if the spatial complexity of the field is higher. So further speed-up can be
achieved by starting with a coarser level set grid, and refining the level set grid if
the interfaces need to have higher precisions [31]. This is known as adaptive grid
refinement, see Fig.2.2(b). Unfortunately, grid refinement, which is very similar to
re-sampling the explicit front mesh (Figure 2.1) as mention before, carries most of the
disadvantages of re-meshing since the sampling grid is also a mesh in nature. At this
point, changing the sampling rate may not be good, but there is room for improvement
on the sampling pattern.
2.2.4 Moving level set grid
Without changing the grid density, a recent proposed strategy [12] to move grid lines
towards feature-likely locations can locally increase the sampling rate in those regions
while maintaining the overall sampling rate of the φ function constant, see Fig.2.2(c).
The overhead is an additional mapping between the deformed level set grid and the
original input grid. It is because the velocity field F is partly derived from input data,
but spatial grid points correspondence between two grids are distorted. Also, the phys-
12

ical distance among grid points on the deformed grid is needed for calculating the
derivatives of φ using finite difference. An efficiency gain can be justified by having
the overall sampling rate much lower than the original input grid with local rate sim-
ilar to the input grid for high gradient areas. But the local sampling rate for moving
grid strategy is still upper bounded by the number of initial grid lines we are using,
which occurs at an extreme situation when all grid lines moved towards one point on
the image.
2.2.5 Level set domain triangulation
Go back to the very beginning reason for constructing the level set grid, we just want
a proper coverage of the problem domain (e.g image) so that various solutions (e.g.
contours) can be found from the φ samples on the grid. As it is only a matter of
domain coverage, we are not limited to use ”squares” to tile up the problem domain.
For 2D problem domain, observe that φ can also be viewed as a surface in 3D and
the simplest form of surface element in 3D is triangle, so triangular mesh can also
properly represent the 3D surface rising up from the problem domain [9]. Similar to
the square-mesh surface, implicit fronts embedded in the triangulated surface can also
be moved by changing the z-coordinates of any (x, y, z) vertex of the triangulation .
Normals and z coordinates of any point on the triangulation can be interpolated from
the vertices of particular triangles where the point is located. Therefore, a 1D mesh
of φ can be recovered by marching along the zero-crossings of consecutive triangles,
see Fig. 2.2(d). Similar to the case in Fig. 2.2(b), we can re-fine the triangulation
to enhance the front precision, see Fig. 2.2(e). But refinement must follow some
predefined patterns in order to maintain the consistency of the triangulation, otherwise
we need to re-triangulate the surface as a whole [32, 34]. Then the precision gain is
13

estricted by the refining patterns and the speed of getting higher precision is restricted
by the speed of re-triangulation.
2.2.6 Adaptive level set domain triangulation
By using some prior knowledge associated with the velocity field, like divergence, we
know in advance that smaller triangles are preferred for certain areas [32, 22]. So adap-
tive pre-triangulation of the level set grid can avoid re-meshing the analysis domain at
the same time having higher front precision, see Fig.2.2(f). This idea of using prior
knowledge is pretty nice and narrow band speed-up is also applicable for triangulated
domain, but it could have problem for the evolving fronts when the velocity field causes
the fronts to have topological changes. That is, when a self-closed interface need to be
separated into two at location having big triangles, it is very difficult to represent two
fronts in one triangle as the zero-crossings along the three edges of that triangle may be
completely lost. In fact, this is just an aliasing problem in sampling theory since there
are not enough samples (triangles) to reconstruct the signal (fronts). In this thesis, we
introduce an even more accurate and flexible implementation strategy that local resolu-
tion is adaptive to selected measurements or features (like front curvature and gradient
of input data) but without upper bound or re-meshing problems, see Fig.2.2(g).
14

Chapter 3
Level set implementations on
unstructured point cloud
3.1 Level set initialization
3.1.1 Narrow band of level set nodes
Suppose we have a set of unstructured points x as sampling nodes for the level set
function φ on the problem domain. Computing φ t+∆t (xi) in Equation (2.3) for all the
nodes could be expensive and also unnecessary as our focus is the zero level set, i.e.
{φ =0}. Therefore, narrow band concept [1, 24, 30] was introduced for speeding
up purpose. In ordinary grid-based level set implementation, the narrow band can be
defined as a fixed number (3-4) of pixels around all zero crossings of φ. So the narrow
band in that case had a uniform width. However, we do not assume a fixed node
density in our point-based method, hence the nodes included in the narrow band may
have non-uniform distribution and therefore our narrow band could have a non-uniform
width. Nevertheless, the narrow band width w is not our primary interest, the key
concern is that the narrow band has included enough nodes for MLS to appropriately
interpolate the φ values around the zero level set so that the moving front(s) is properly
represented. The later description of our point-based narrow band will base on two
definitions below:
15

1. NodeSet refers to the constant set of level set sampling nodes distributed ac-
cording to the Algorithm 2 before starting the front evolution using Equation
(2.3).
2. NBSet refers to a dynamic set of nodes for the φ function around the moving
fronts. NBSet comprises of nodes mainly from NodeSet and also additional
nodes around high curvature location of the fronts.
To effectively implement the dynamic NBSet, we will first switch on / activate a
subset of nodes in the NodeSet which are near to the current front and then compute
the evolution of the front as usual but using only that subset of nodes. To further en-
hance the stability and precision of the evolving front, we would also like to include
additional non-NodeSet nodes into the Narrow Band subset (NBSet) near high cur-
vature locations. The additional node amount is set to be proportional to the curvature
of φ, again the proportional constant is relatively flexible as the analysis domain has
already been covered by original NodeSet.
Since our narrow band has non-uniform width, the word near is not trivial. To
avoid traversing the whole NodeSet all the time and at the same time guarantee nodes
sufficiency, we would like to use k-nearest nodes of all extracted contour points in the
most previous re-initialization plus additional nodes for high curvature as the set of
narrow band nodes (NBSet), where k is the number of nodes for surface interpolation.
When the front moves, the NBSet set is also modified by replacing a portion of nodes
in previous NBSet by their neighboring nodes at frontend. The time-saving trick is
that node neighborhood in NodeSet is constant during front evolution so that it can
be pre-computed and stored in a lookup table.
16

◮
Figure 3.1: Initialization for the point-based level set evolution. Top left: featureadaptive
node distribution. Bottom left: initial circular on the problem domain. Button
right: the signed distance function from the initial contour with negative values inside
the contour.
17

3.1.2 Signed distance function
Initial level set can be placed manually or using a statistical average position from
training data. In Figure 3.1, an initial circular contour is placed manually at the center
of the problem domain (image with two black dots). Multiple contours are also possi-
ble. Let xc be the manually selected center position with radius r, NodeSet is the set
of sampling nodes distributed on the problem domain beforehand, the signed distance
function φ can be constructed by:
φ(x) =|x − xc|−r ∀ x ∈ NodeSet (3.1)
For multiple initial contours with centers xc1, xc2, ..., xcn and radii r1,r2, ..., rn,
φ(x) =min n i (|x − xci|−ri) ∀ x ∈ NodeSet (3.2)
Since the front can be represented by using only nearby nodes, we can just work on a
subset of node during the evolution process. We call this subset to be a narrow band of
nodes (NBSet) which is defined as,
NBSet = x | φ(x)|

grid-based samples of φ(x, y) in 2D or φ(x, y, z) in 3D. But now we have changed our
data structure to a randomly distributed set of points instead of grid-based, exact near
neighbor along a particular x-ory-direction is almost unavailable.
3.2.1 Influence domain
In order to obtain those φ’s derivatives on unstructured point cloud, we further assume
the φ function at any location x is a polynomial surface within an area around x.
We then use the neighboring nodes xi in that area to recover the coefficients of φ by
performing a least square fitting, see Figure 3.2. Therefore, all nodes inside that area
will have certain amount of effect on the recovered coefficients, which depends on
how the least square is performed. That area is usually called the influence domain or
support region. In general data fitting problem, the size of such an area and the order
of polynomial basis control the complexity of the constructed surface.
In our practical problem, sampling nodes distribution can be feature-adaptive rather
than purely random, by fixing the polynomial order and choosing a fixed number of
nearest neighboring nodes for any φ(x) has implicitly the same meaning as determin-
ing a feature-adaptive size for the influence domain. Here, we define the Area that
the k neighboring nodes xi are found to interpolate the φ value at x to be the influence
domain of x. We will show how to do feature-adaptive node distribution and determine
the size of influence domain in chapter 4.3.
3.2.2 Moving Least Square Approximation
Originally developed for data fitting and surface construction [17, 18], the Moving
Least Square (MLS) approximation has found new applications in emerging mesh-
free particle methods [3, 20, 21]. MLS approximation is chosen because it is com-
19

Figure 3.2: Three steps for interpolating ˜ φ surface at an arbitrary node from the Narrow
Band Set. Left: Determine the influence domain size Rx of node x by searching knearest
neighbors. Middle: The cubic spline weight function for the node and its
neighbor. Right: The portion of interpolated surface.
pact which means the continuity of the approximation within overlapping influence
domains is preserved. In the following sections, we use ˜ φ to denote the MLS approxi-
mated φ for better description. Using MLS, ˜ φ surface is defined to be in the form:
˜φ(x) =
m
pj(x)aj(x) ≡ p T (x)a(x) (3.4)
j
where p is a vector of polynomial basis, m is the order of polynomial, and a is a
vector of coefficients for the ˜ φ function at location x using the polynomial basis p.
For example, if p T (x) =[1,x,y], then a T (x) =[a1(x),a2(x),a3(x)] and they are
arbitrary function of x. To get ˜ φ(x) at arbitrary location x, we determine the coeffi-
cients a through the k-nearest neighboring nodes xi and their known φ(xi) values. The
weighted approximation error is defined as :
J =
k |x − xi|
W (
i
Rx
)[p T (xi)a(x) − φ(xi)] 2
W ( ¯ ⎧
2
⎨ 3
d)=
⎩
− 4 ¯ d2 +4¯ d3 for ¯ d ≤ 1
2
4
3 − 4 ¯ d +4¯ d2 − 4
3 ¯ d3 for 1
2 ≤ ¯ d ≤ 1
0 for ¯ d>1
(3.5)
(3.6)
where W is a cubic spline decaying weight function, |x − xi| is the spatial distance
between x and xi, and T denotes vector transpose. Solve for a(x) through minimizing
20

J we can get
˜φ(x) =N(x)Us
Us =[φ(x1),φ(x2), ··· ,φ(xk)] T
(3.7)
(3.8)
N(x) =p T (x)A(x) −1 B(x) (3.9)
k
A(x) = W ( |x−xi|
Rx )p(xi)p T (xi) (3.10)
i
B(x) =W ( |x−xi|
Rx )[p(x1), p(x2), ··· , p(xk)] (3.11)
The solution of a(x) has already embedded in N(x) of Equation (3.9) for simpli-
fication. The N(x) is called the shape function that interpolates the known vector Us,
which is a collection of known φ(xi) values, to get the approximated ˜ φ(x) value.
To get the first order derivatives of ˜ φ, let
A(x)Υ(x) =p(x) (3.12)
Υx = A −1 (px − AxΥ) (3.13)
˜φx = ΥxB + ΥBx
where ˜ φx is the shorthanded notation for ∂ ˜ φ(x)
(3.14)
∂x , and ˜ φy can be obtained in similar fashion.
Finally, |∇ ˜
φ| = ˜φ 2
x + ˜ φ2 y for 2D case or use Equation (2.5) for 3D case.
We may also need second order terms φxx,φyy and φxy:
Υxx = A −1 (pxx − 2 Ax Υx − Axx Υ)
Υxy = A −1 (pxy − (Ax Υy + Ay Υx + Axy Υ))
˜φxx = Υxx B +2Υx Bx + ΥBxx
˜φxy = Υxy B + Υx By + Υy Bx + ΥBxy
(3.15)
(3.16)
Moreover, ˜ φyy can be obtained in similar way as ˜ φxx. With approximated ˜ φxx, ˜ φyy and
21

˜φxy, we can get the curvature measurement κ by Equation (4.4) in later chapter.
3.2.3 Generalized Finite Difference: An alternative to MLS
MLS interpolation can give us a nice surface of φ, see the right column of Figure 3.4.
But it requires quite a lot of computation to get the interpolation function in Equation
(3.9). This may not be good for some time-critical applications, therefore we also
provide a faster but less accurate interpolation method to get φ(x) and its derivatives
[4, 15].
If we define our surface ˜ φ to be simple polynomial based, with the form
φ(x) =φ(x, y) =a1 + a2x + a3y + a4xy (3.17)
Then let x1, x2, ··· , x¯ k be ¯ k nearest neighboring nodes of x with known φ(xi)∀ i =
1... ¯ k. We can find the unknown coefficients ai∀ i =1...4 by solving for the following
simultaneous equations:
φ(x1) =φ(x1,y1) =a1 + a2x1 + a3y1 + a4x1y1
... = φ(x2,y2) =a1 + a2x2 + a3y2 + a4x2y2
=
.
φ(x¯ k)=φ(x¯ k,y¯ k)=a1 + a2x¯ k + a3y¯ k + a4x¯ ky¯ k
In vector form,
⎡ ⎤ ⎡
φ(x1)
[φ] = ⎣
.
⎦ , G = ⎣
φ(x¯ k )
1 x1 y1 x1y1
.
1 x¯ k y¯ k x¯ k y¯ k
⎤
⎡
⎦ , [a] = ⎣
a1
.
a¯ k
⎤
⎦
(3.18)
⇒ [a] =(G T G) −1 G T [φ] (3.19)
˜φ(x) =N GF D (x)[φ] (3.20)
where N GF D (x) =[1, x, y, xy](G T G) −1 G T
22
(3.21)

N GF D (x) is an alternative shape function for Equation (3.9). In general, if (G T G) is
not invertible, then including more nodes (which means to increase the number of rows
in Equation (3.19)) can turn it be invertible as the nodes are basically randomly dis-
tributed. Notice that G is a constant matrix so as its least square inverse, the derivatives
˜φx, ˜ φy of ˜ φ(x) can be computed quickly by taking the derivatives of the polynomial ba-
sis and then multiply (G T G) −1 G T [φ] which has already been computed for the shape
function. Second order ˜ φxx and ˜ φyy are zero, while ˜ φxy is exactly the last row of
(G T G) −1 G T .
23

3.3 Evolution process and level set re-initialization
Construct
hyper-surface
from front points
MLS / GFD
Approximation
Level Set
Update
Extract new set
of front points
Figure 3.3: Flowchart of general level set evolution process
3.3.1 Evolution process
Output
After obtaining |∇φ|, the level set function φ is to be updated by Equation (2.3), see
Figure 3.3. Update of level set value is supposed to be done node-by-node in the
NodeSet. If narrow band speed-up is applied, then only update values of nodes
within the band, NBSet (see chapter 3.1.1).
After level set update, we have an evolved implicit front. But the evolved level set
function is usually not a desired output format for other applications or display, we
would like to get the front in explicit form, see Fig. 2.1. Since the level set function φ
is defined as a signed distance function for the embedded front, the normal direction n
of the level set function, and the nearest zero level set position z for any location x are
given by:
nx = ∇φ(x)
|∇φ(x)|
zx = x − φ(x) nx
24
(3.22)
(3.23)

initial hypersurface φ
intermediate φ
final φ
Figure 3.4: Left: An illustration of a complete evolution process for embedded level set
front from top to bottom. Right:A visualization of the actual computational domain,
MLS interpolated initial, intermediate and final φ.
25

3.3.2 Level set re-initialization
In chapter 2, we stated that the update equation (2.3) is a time-domain finite difference
solution for the level set PDE given by Equation (2.2). Numerical errors exist in time-
domain discretization. Therefore regularization of the intermediate solution φ t+∆t is
frequently needed to maintain the integrity of the moving interface. Common level
set literatures usually disregard the re-initialization procedures because they assume
the zero set of a sampled function can be exacted easily [23]. Although it is true for
grid-based implementation, our grid-less approach requires more effort on this issue.
There are two ways to perform re-initialization, both aim at restoring the φ function
to be a distance map of embedded front/surface:
1. explicitly find out a set of front points (vertices or zero level set) and then re-
calculate all node-to-nearest vertex distances.
2. gradually turn ∇φ of every node xi close to 1 by gradient descent
The first way is easier in 2D because it is possible to trace out all fronts without
using a reference grid. The way we trace out the front in 2D is on coming section.
Nevertheless, if the sampling nodes has reasonable coverage over the whole problem
domain and the front motion is small, we could just extracted a set of front vertices
using Equation 3.23 and reset the φ values based on these vertices. This works for both
2D and 3D case. The second way uses the fact that a distance map should have gradient
magnitude equal to unity everywhere, so this way can avoid extracting vertices in 3D
space. But the second way takes more computations than the first one and the step size
of descent has to be carefully chosen because of our adaptive node density. By the way,
if final surface resolution is known and fixed in advance, performing marching cubes
26

to extract the vertices is also possible and could be more reliable than gradient descent
procedures. But we do not prefer this way because the usage of ”cubes” implies certain
pre-established grid existed, which violates our main purpose of suggesting grid-less
computation environment.
Re-initialize by contour extraction (2D case only)
The re-initialization can be done by first extracting the intermediate embedded front(s)
in the evolved surface φ t+∆t , then resetting all φ(xi) nodes values to be the nearest
distance to the intermediate front. To extract the front from φ t+∆t , we can start from an
arbitrary vertex zx computed from Equations (3.22) and (3.23) using an arbitrary node
xi. Here, a vertex means a point on the embedded moving front in φ. Then march along
the tangent of φ t+∆t (zx) by a curvature adaptive step size ∆s to arrive a new predicted
vertex location. This prediction is then immediately adjusted by Equation (3.23) to
become an extracted vertex. This walk-and-adjust (or project-and-update) procedure
repeats until the next vertex goes back or very near to the starting vertex. Then we can
use the extracted front to reset all node values φ(xi) within the narrow band. If there
are remaining fronts not yet extracted, some reset node values will have great changes
and very often greater then the narrow band width w, defined as the maximum |φ(xi)|
of all xi. So we can monitor the changes of node value to check for remaining fronts.
Ideally, φ(zx) should be zero. But small numerical error may exist in Equation
(3.23) using MLS approximation. To significantly reduce the error from Equation
(3.23), we can perform further one step of gradient descent on the zx location along
the nx direction, i.e:
z ′ x = zx − ˜ φ(zz) nx
27
(3.24)

where z ′ x is the improved vertex location for each ∆s walk.
Algorithm 1 (Marching contour)
z ← pick up the zx by xi ∈ NBSet with largest φ(xi)
Rz ← the influence domain radius by Equ.(4.7)
φz ← use MLS to interpolate φ at z
tan← (-ny,nx) where nx=(nx,ny) by Equ.(3.22)
Cz ← 1 , start ← z
CurveSet ← empty set (Skip if not the first contour)
While ( not a complete loop )
ds ← min(Rz,Cz)
nextPos ← z + tan*ds
Rz ← influence domain radius by Equ.(4.7)
φz ← use MLS to interpolate φ at nextPos
n ← normal of φ at nextPos by Equ.(3.22)
tan1 ← (-ny,nx) is the tangent at nextPos
Cz ← |change of tangents| /ds
tan ← tan1
z ← (nextPos - n*φ) for numerical adjust
record z to the CurveSet
End While
For ( each node xi ∈ NBSet )
z1,z2 ← two nearest z ∈ CurveSet
store the sign of φ(xi)
reset φ(xi) by the average position of z1 and z2
restore the sign of new φ(xi) as stored
End For
Note:
Extract another contour if abnormal φ(x) is detected,
but no need to reset the CurveSet.
Re-initialize by gradient descent
This method is suggested by S. Osher in [27]. The level set update equation (2.3) does
not require knowledge of a complete set of front/surface points and theoretically can
be repeated indefinitely by its own. But the external velocity F applied on individual
sample can be arbitrary and often inconsistent with the embedded structure in level set
function. The inconsistency may due to the inadequacy of sampling nodes to represent
the desired shape implied by noisy F . In practice, the embedded front could be broken
by noisy force and fail to maintain a signed distance function with positive inside and
negative outside. As the purpose of re-initialization is to restore the level set function
28

ack to a distance function, and distance function should have a gradient magnitude
|∇φ| =1[27]. We can iteratively perform a smoothing on φ which minimizes the
error of gradient magnitude: (|∇φ|−1), but at the same time leaving the front / surface
intact. Let φ(x) be the level set function distorted by F , and ψ(x,τ) is the desired
distance function when τ →∞. So, initialize
Regularize ψ(x,τ) by
dψ
dτ
ψ(x, 0) = φ(x) (3.25)
= −sign(φ(x)) (|∇ψ(x,τ)|−1) (3.26)
where the embedded front is maintain by using
⎧
⎨ 0, if ϕ =0
sign(ϕ) = 1,
⎩
−1,
if ϕ>1
if ϕ

Chapter 4
Application to image segmentation
4.1 Deformable model
A developed strategy is to use a meshed contour, which is moving towards high gradi-
ent areas with shape changing on the way. To represent the contour in digital domain,
we could divide a continuous contour into numerous consecutive identical line pieces
with finite number of end points as vertices, see Fig.2.1. By moving individual vertex
separately, arbitrary shapes of the contour can then be achieved. This can be referred
to a classical tools to solve for various engineering problems, known as Finite Element
Methods (FEM). Snake is the first application of FEM concept on image segmentation
[16, 35].
4.1.1 FEM contour mesh
In the original FEM Snake model, we could regard the vertices as samples of the ac-
tual continuous deformable contour. And the lines between vertices are the mesh that
holding them together.
The advantage of this parametrization is that we could impose certain property on
the curve like the maximum amount of bending it can have or simply stiffness. At
the beginning of the segmentation process, the contour is like a stretched rubber band
30

placed around but outside an object on the image and intends to shrink inward so as to
reduce the amount of stretching. At the same time, high gradient areas are modelled
as resistance to the contour motion and hopefully the inward shrinking force is bal-
anced by the gradient resistance so that the contour vertices stabilized along the object
boundary. Clear edges implied by very high gradient areas will have no problem of
stopping the contour, and for those suspected broken edges, the additional imposed
contour property helps to link up clear edges by forcing the contour vertices located at
a path that minimize a pre-defined cost function. In mathematics, the contour location,
bending amount, stiffness and image gradient can be precisely derived from evaluating
the two parametric functions: {Sx(s),Sy(s)} for given s, derivative of {Sx(s),Sy(s)}
with respect to s, integration of their derivative along the contour and taking spatial
derivative of a 2D scaler function (i.e. the image intensity map) respectively. So we
can set up the cost function using the above measurements. Since we are actually mod-
elling material motion, a system of dynamic equations with energy and displacement
terms can be formulated. Therefore, the underlying principle of Snake motion is gov-
erned by the solution of a energy minimization problem. In discrete domain, a general
way to approximate the derivative and integration of {Sx(s),Sy(s)} is to use the (fi-
nite) difference and summation between {Sx(s),Sy(s)} and {Sx(s±∆s),Sy(s±∆s)},
where ∆s is a relatively small constant. This type of FEM active contour model is well
behaved because individual local error can be smoothed out by consistency enforce-
ment among neighboring vertices during the minimization process.
4.1.2 Irregular FEM contour mesh
One obvious limitation of traditional FEM Snake is that if the actual boundary length
is unknown in advance, then how many s values or simply say the number of vertices
31

we should use to represent the moving contour. For example, if only three vertices are
used, then at most a triangle can be represented. In general, the representable shape
variety increases as the number of vertices increases. But it does not necessary mean
that more vertices are better, computational cost and smoothness of the contour would
be the tradeoffs for having more samples. In particular, undesired jagging will occur
around sharp corners if the sampling is inappropriate. A quick solution would be a re-
sampling strategy for the moving contour [25] so that the number of samples is adaptive
towards the status of the contour, see Fig.2.1. Unfortunately, re-sampling changes the
relationship among vertices. Since the neighbors of {Sx(s),Sy(s)} are changed, the
energy measurements through their derivative and the system of the equation for energy
minimization are also affected. These relevant changes can be collectively described
as re-meshing problems.
4.2 Level set based Geometric Deformable Model (GDM)
If we move back a little bit to consider the input of our generic segmentation problem,
that is, a digital image which itself is a two dimensional function sampled by a 2D rect-
angular grid. The number of boundary points for an object on the image is constrained
and finite. So we can change the target contour representation from connected lines to
a finite point set on the image. Again the point set, which consists of boundary pixels,
should also be connectable to form a closed loop. Since these points are originally se-
lected from the 2D rectangular grid, a natural but not definite way to connect points to
form contours is to follow the spatial relationship defined by the sampling grid, that is,
we constrain the next vertex {Sx(s ± ∆s),Sy(s ± ∆s)} of vertex {Sx(s),Sy(s)} must
be one of the pre-connected neighbors of {Sx(s),Sy(s)} which could be up, down,
left or right at one fixed unit apart. So far, this change of contour representation only
32

imposed an additional constraint to reduce the size of solution domain. To avoid re-
meshing, one suggest to use a more flexible water front like modelling for the moving
contour. Instead of moving a curve in 2D, we could imagine to move a continuous
surface in 3D and the moving curve is just a particular level of the surface [1, 24, 26].
In other words, the 2D contour is embedded in a 3D surface φ and 2D contour move-
ment is governed by a 3D surface motion. If we further constrain the third dimension
z of any point (x, y, z) on the surface φ, we could model the surface motion under a
to-be-defined velocity field F , which is applied along the surface normal direction:
dφ
dt
= F · ˆn = F · ∇φ
|∇φ|
(4.1)
φ could be viewed as a signed distance function that maps a point φ :(x, y) → z where
z is the spatial distance towards the nearest embedding curve position. Therefore, lo-
cations with φ =0is the embedded contour. Further, if we design F to be a function
of ∇φ, and then look back into all other terms in the above equations (please refer
to Chapter 2 for more details of F , Equation (2.4) and (2.5) for ∇φ and |∇φ|), there
does not exist a parameter s to guide you travelling along the moving contour. Instead,
we have an imaginary surface represented by a geometric function φ(x, y). Here we
classify the FEM-like representations as parametric active contours and the level set
formulated ones as geometric active contours [36]. Note that here we are not going to
differentiate whether the motion is governed by energy minimization or front propaga-
tion, indeed they can be hybrid together for achieving certain desired properties [13, 8].
But we focus on the representations and arithmetics on those equations.
33

Data Input
↓ intensity map
chapter 4.2.3
Preprocessing
↓ gradient, GVF map
chapter 4.3
Node Distribution
↓ unstructured sampling
chapter 3.1.1 & 3.1.2
Level Set initialization
↓ distance map φ within a narrow band
chapter 3.2
Domain Interpolation ←−
↓ interpolated φx,φy,φxx...
chapter 2 & 4.2.2
Level Set Update
↓ evolved φ function
chapter 3.1.1
Narrow band Reconstruction
and Node refinement
↓ new narrow band nodes
chapter 3.3 If not stabilized
Level Set Re-initialization −→
↓ regularized distance map
Output
Figure 4.1: A complete flowchart of the segmentation process. Each block represents
a major procedure and the arrows indicates the major input/output data of each procedure.
34

4.2.1 GDM segmentation process
In previous chapters, we discussed several key components of our grid-less level set im-
plementation strategy. In order to show how this strategy can actually be implemented
for practical problem. In this chapter, we will put them altogether with additional pro-
cedures specifically for image segmentation. The overall process is summarized in
Figure 4.1. For systems blocks not discussed in this chapter, they are supposed to be
the same as in chapter 3 and please refer to the previous chapters for details. Their
chapter numbers are stated in the corresponding blocks.
Common input for segmentation would be intensity map which comprises of many
pixels or voxels [11] indicating the brightness of the captured scene or material density
of a spatial volume. They can be obtained from various image capturing devices like
ordinary camera, range scanner, X-ray or MRI machines. For computational purpose,
analog images such as those using traditional light-sensitive film will first be digitized
to form intensity maps on which computers can be processed.
Preprocessing involves the extraction of image features. We here assume our GDM
formulation is edge-based one. So the preprocessing of input data would be the extrac-
tion of edge information. Both image gradient and Gradient Vector Flow maps are
examples of edge information contained in the raw input intensity map. Nevertheless,
GDM can be region-based, see [8], and those can also be converted to a grid-less en-
vironment with only small changes in the Level Set formulation. Here, image gradient
is still computed by spatial finite difference. After that we will re-sampled the analysis
domain, and use MLS / GFD interpolation instead of finite difference.
35

4.2.2 Level set velocity field for GDM segmentation
To demonstrate the feasibility of our approach in image segmentation, we selected the
formulation for contour evolution suggested in [29]:
F = βgκ− (1 − β) g (ˆv ·∇φ)/|∇φ| (4.2)
g = 1/(1 + |∇G ∗ I|) (4.3)
Equation (4.2) consists of mainly two terms. The first term is the product of image
gradient force g and contour curvature κ, serving respectively as external and internal
energy constraints during evolution. The second term is the additional external force
using the inner product between image gradient vector flow (GVF) [35, 29] and contour
normal direction. The incorporation of GVF enhances the flexibility of choosing initial
contour position and handling concave object boundary. The parameter β ∈ [0, 1]
is just a constant weighting between the two main terms. G denotes the gaussian
smoothing kernel and finally contour curvature κ can be obtained from the divergence
of the gradient of the unit normal vector to the contour [24, 26]. In 2D case,
κ = ∇· ∇φ
|∇φ| = −φxxφ2 y − 2φxφyφxy + φyyφ2 x
(φ2 x + φ2 y) 3/2
(4.4)
where φx is the shorthanded notation for first order derivative ∂φ
∂x , φxx denotes ∂2 φ
∂x 2 and
φxy stands for ∂φx
∂y .
However, our emphasis in this thesis is to offer an alternative domain representation
scheme using point cloud such that spatial calculations can be conducted in a more
efficient and more accurate fashion. Many other F formulation can also be used.
36

Image intensity Gradient magnitude Gradient vector flow
Figure 4.2: Data inputs for GDM segmentation: intensity, gradient magnitude and
Gradient Vector Flow (GVF). From top to bottom are examples of synthetic, natural
and medical images.
37

4.2.3 Gradient Vector Flow
Enhancements have been made to make the evolving curve less sensitive to its ini-
tial position, such as the introduction and adoption of the gradient vector flow (GVF)
as data constraints [29, 35]. The GVF is computed as a diffusion of image gradient
vectors and it increases the domain of influence for image-derived edge information.
Every pixel p of the image is affected by all edge points at different levels, where the
influence of different edge points is inversely proportional to the Euclidean distance
from p to the edge, i.e. closer edge point has stronger effect. The output of the GVF
process is a smooth vector field with each vector indicating the most likely direction
towards an edge and the vector length denoting how close the edge is, longer means
closer (see Figure 4.2 for examples). Incorporation of the GVF as data constraints into
the level set formulation has been attempted [29], and it has shown greater flexibility
on initial contour positioning. The main advantage is that the evolving contour can
move in the reversed normal direction by taking the inner product between the front
normal N = ∇φ/|∇φ| in Equation (4.2) and the GVF vector v on the same location
as the active contour evolving velocity Ct =(v · N)N.
4.3 Adaptive unstructured sampling of computational
domain
The formulas in the previous section lay down the background of our active contour
model, but the main discrepancy of our new method is the way we obtain the solution
for those formulas, in terms of both data structures and procedures.
Here, we want discuss about the data structure for the level set function φ(x). A
common and simple way to store the φ function for computation is using a matrix
which may have the same size as the input image. In particular, a matrix element
38

epresents a sample of the φ function at the corresponding location on the image.
Then φ values are available only at φ(ih, jw) for i, j = 1, 2, 3, ... and some con-
stants h, w ∈ R.We may not aware that, by the matrix arrangement, we have implicity
used a rectangular grid to sample the level set φ function. The sampling rate and spa-
tial relationship of φ have been pre-established through the use of a matrix or other
special grids, like triangular, refined or deformed level set grids. However, these pre-
establishments are not necessary in our new method. So from now on, we will elimi-
nate the level set grid and use only a finite set of points xi, where i is the index of the
point in the set, to sample the φ(x) function. We would like to name xi as the level set
supporting nodes or simply nodes to differentiate them from other types of points on
the image. Note that φ(xi) → R, and there is no spatial relationship between xi and
xi±n for any integer n.
4.3.1 Sampling node distribution
As we are expecting that our final contour will reside at high image gradient areas, it
is essential to get a high precision representation at those regions. It can be done by
improving the sampling rate over there. We quantify our requirement by setting the
node density of a point x ∈ R 2 on the image proportional to the image gradient of that
point 1 :
Node Density at x ∝ |∇I| (4.5)
Nodes
Area
= ρ|∇I|, where ρ is constant (4.6)
In Equation (4.6), determining the Area for node density measurement at any point x
would be a issue of selecting proper scale, so we would like to let the interpolant of φ
to control it by fixing the number of Nodes within an Area to be the minimum amount
1 Bilinear interpolation of image gradient is needed as x may not be a pixel point.
39

of neighboring nodes xi the interpolant needed to interpolate the φ(x) value. Assume
Area must be circular, we get
Area = πR 2 = k
ρ |∇I|
or R =
k
(πρ) |∇I|
(4.7)
where k is the minimum neighboring nodes amount for interpolate φ(x), and R is the
radius that we are going to spread k nodes around x. Before going to the algorithm,
there is a little problem with Equation (4.7) when |∇I| →0, which implies low image
gradient at x (such as image background or smooth regions), the spreading radius goes
unbounded. To guarantee a reasonable coverage of samples at low gradient areas, we
first initialize a base set of nodes which have a uniform separation Rmin apart:
k
Rmin =
(πρ) mean(|∇I|)/2
(4.8)
To further speed up the distributing process, we also like to include the highest 1-
2% gradient pixels in the base set. Note that the base set might look like grid points
(see Figure 4.3), again we do not assume any relationship among the nodes in the set.
Therefore slightly modifying Rmin or adding other points with no special reason could
be fine.
4.3.2 Feature-adaptive node distribution algorithm
If the image size is large, exhaustive put-and-check for node density at every pixel
would be slow. Here, we simplify an existing mesh refining algorithm [22] to a node
distribution algorithm. The original meshing algorithm assumes the actual boundary of
the to-be-meshed object is known and the node density at an image point is controlled
by how far that point is away from the known boundary. Using this metrics, regions
closer to actual boundary will have finer triangulation in the final mesh.
40

The two main differences of our simplified version are that, first, the point-to-
boundary measurement is replaced by the image gradient magnitude since actual bound-
ary is unavailable for segmentation problem. Second, the triangulation is not per-
formed as the mesh is not needed in our method. Nevertheless, our method is still
recursive and requires a initial coarse node distribution. The idea is that we check ev-
ery initial nodes on the image by counting the amount of neighboring nodes already
existed within an area of radius R defined in Equ. (4.7). If any node failed to have k
neighbors, then that node is removed and additional new nodes are randomly added to
fill up the shortage. During the checking, all newly created nodes are stored in a list
for next iteration. After checking all initial nodes, the same checking procedures are
performed on the list of new nodes and produces another list of new nodes. The pro-
cess terminates until an empty list is found at the end of an iteration which implies no
new node has been added in the last iteration. Then all nodes have at least k neighbors
in its influence domain, which has been discussed in chapter 3.2.
Algorithm 2 (Gradient-Adaptive Node Distribution)
CurrList ← Base Node Set
NodeSet ← Base Node Set
While( CurrList is not empty )
Randomize the order of node in CurrList
NextList ← Empty
For every node x in CurrList
Rx ← min(sqrt(k/(pi*ρ*|∇I(x)|)),Rmin)
N ← No. of Nodes in NodeSet within Rx circle
If( k-N > 0 )
Remove x from NodeSet
NewNodes ← Random (k-N) Nodes in Rx circle
Add NewNodes into NextList
Add NewNodes into NodeSet
End If
End For
CurrList ← NextList
End While
41

Synthetic image Real image Medical image
Figure 4.3: From top to bottom: process of the recursive gradient-adaptive node distribution
algorithm. The gradient magnitudes are used as the background for all images
since they are the actually inputs of the algorithm.
42

Chapter 5
Experimental results and conclusion
5.1 General GDM segmentation results in 2D
◮ ◮ ◮
◮ ◮ ◮
Figure 5.1: Segmentation of the endocardium (top) and epicardium (bottom) from
cardiac magnetic resonance image
In this chapter, we show some experimental results of grid-less GDM segmenta-
tion on synthetic and selected real images. All GDMs in this section used spatial image
gradient magnitude as the only source for shape information, and therefore can be re-
garded as boundary-based GDM. The flowchart of these GDM processes can be found
at Figure 4.1. We have not used other prior shape information which cannot be derived
from input image intensity map except the initial circular contours’ sizes and positions.
43

◮ ◮ ◮
Figure 5.2: Segmentation of the brain ventricles from noisy BrainWeb [6] image
Segmentation of the endocardium and epicardium from cardiac magnetic resonance
image are given in Figure 5.1. GDM process for brain ventricles segmentation is given
in Figure 5.2. The GDM formulation for brain ventricles segmentation does not in-
clude the number of ventricles as prior information except initialization. Due to the
small separation between two ventricles, the gap between two separate initial contours
are smooth out by MLS and two contours merged together during GDM evolution pro-
cess. Improved result using Topology Preserving Level Set Surface (TPLSS) will be
described in Appendix A.
We also want to demonstrate our grid-less strategy is valid for different GDM for-
mulations. Hence in Figure 5.3, we show some results after slight modification to the
force term in Equation (2.3). In practice, there are many alternative GDM formulations
which are often application-dependent.
Comparison among different implementation strategies is also one of our studies.
Images with ground truth of boundaries were used to enable error measurements for
quantitative comparison. Selected results are given in Figure 5.4 to show the quality
of segmentations under different implementation strategy and input noise level. The
measurements are given in Table 5.1.
The effect of sampling node density on GDM precision is explained in Figure 5.5.
44

◮ ◮ ◮
◮ ◮ ◮
◮ ◮ ◮
◮ ◮ ◮
Figure 5.3: Examples of point-based GDM level set contour evolution. Top two:
Real and synthetic image. Middle: Inward shrinking front by modified force F − 1
(SNR=6.9dB). Bottom: Outward expanding by F +1(SNR=6.9dB)
45

Images with different levels of noise:
SNR:10dB 6.9dB 5.2dB 3.9dB
Point-based Level Set Results with MLS Interpolation
Point-based Level Set Results with GFD Interpolation
Regular Grid Level Set Results with Finite Difference
Figure 5.4: Comparison of segmentation results among different implementation
strategies. The first row are examples of noise-corrupted images in different levels.
Start from the 2nd row, results shown at the same column are using input image with
the same amount of noise, while at the same row are implemented using the same
interpolation scheme.
46

Table 5.1: Error statistics comparison between point-based and regular grid level set
implementations
2 Circles SNR MSE S.D. Min Err Max Err Time(s)
MLS 10dB 0.095 0.152 0.002 0.678 60.048
MLS 6.9dB 0.396 0.4 0.005 1.417 57.175
MLS 5.2dB 0.402 0.374 0.000 1.529 77.187
MLS 3.9dB 0.459 0.42 0.002 1.61 82.813
GFD 10dB 0.376 0.231 0.004 1.173 43.577
GFD 6.9dB 1.054 0.571 0.001 2.055 51.062
GFD 5.2dB 1.432 0.623 0.005 2.838 46.619
GFD 3.9dB 1.937 0.718 0.026 3.36 50.551
FD-LS 10dB 1.153 0.533 0.080 2.351 73.425
FD-LS 6.9dB 1.990 0.709 0.080 3.348 70.591
FD-LS 5.2dB 1.921 0.548 0.304 3.384 72.855
FD-LS 3.9dB 2.360 0.694 0.136 3.141 73.145
3 Objects Noise MSE S.D. Min Err Max Err Time(s)
MLS 10dB 0.222 0.324 0.006 2.093 372
MLS 6.9dB 0.402 0.4 0.002 2.084 295
MLS 5.2dB 0.405 0.410 0.002 1.820 224
MLS 3.9dB 0.504 0.464 0.003 2.133 317
GFD 10dB 0.480 0.443 0.000 2.166 164
GFD 6.9dB 1.324 0.606 0.001 2.954 106
GFD 5.2dB 1.218 0.654 0.006 3.322 148
GFD 3.9dB 2.119 1.039 0.014 6.057 126
FD-LS 10dB 1.074 0.617 0.008 3.318 341
FD-LS 6.9dB 2.11 0.999 0.000 5.298 340
FD-LS 5.2dB 2.128 0.994 0.002 5.952 344
FD-LS 3.9dB 5.343 1.300 0.080 7.183 294
MLS: Moving Least Square approximation
GFD: Generalized Finite Difference
FD-LS: Finite Difference Level Set
47

Figure 5.5: An illustration of the relationship between node distribution and the precision
of implicit contour. Top: Use only image gradient as the node distribution criteria.
Higher gradient areas have more sampling nodes. The gradient-to-nodes ratio is small.
Middle: Similar to the top one, but the gradient-to-nodes ratio is relatively higher.
Bottom: Use both image gradient and image iso-curvature as node distribution criteria.
The sampling ratio is small. The result of the bottom case is comparable to the
middle one but it saves time because less nodes are used.
48

5.2 Segmentation in 3D
Input intensity data
Intensity gradient
magnitude
Adaptive level set nodes
Initial surface Narrow band of nodes Initial implicit surface
Level Set Evolution Level Set Evolution Level Set Evolution
Converged implicit
surface
Putting more nodes to
refine the surface
Refined output
Figure 5.6: Meshfree GDM Segmentation of a synthetic cube. The figures are organized
from left to right, top to bottom. Input data size: 100x100x100, cube
size:30x30x30, sampling nodes: 3.5% of data size, running time: 238 sec on Pentium4
1.5GHz with 768MB memory
49

Input intensity data
Intensity gradient and
target layer
Adaptive level set nodes
Initial surface Level Set Evolution Level Set Evolution
Segmented layer Densify surface points Refined surface
Figure 5.7: Meshfree GDM Segmentation on BrainWeb [6] image. The figures are
organized from left to right, top to bottom. Input data size: 181x271x181, sampling
nodes: ∼10% of data size, running time: ∼2hrs on Pentium4 1.5GHz with 768MB
memory.
50

Intensity gradient and
target layer
Reconstruct narrow band
and evolve
Segmented surface under
coarse sampling
Refined surface
Adding more nodes
around the surface
Further adding more
nodes
Level Set Evolution Level Set Evolution Densify surface
Figure 5.8: Solution refinement using unstructured point cloud. The input data is
the same as Figure 5.7 but with different parameter for the F term. The figures are
organized from left to right, top to bottom.
51

Figure 5.9: Top: Enlarged view of result from BrainWeb [6] image in Figure 5.8.
Bottom: Overlay results from both Figure 5.7 and 5.8
5.3 Conclusion
To conclude, we have not introduced new formula for front propagation. Instead, we
have simply connected three simple concepts, namely the sampling rate, influence do-
main and surface fitting into old formulations and improve their flexibility and effi-
ciency. To aim for higher precision, we randomly add more sampling nodes onto the
solution domain. If holding the order of surface polynomial constant, areas with more
52

Figure 5.10: Left: Narrow band of nodes, zero level set (the contour) and normal
directions of the zero level set. Right: Overview of variable-sized and overlapping
influence domains within the narrow band of nodes same as the left figure.
nodes can have smaller influence domains since the amount of nodes needed for sur-
face reconstruction is relatively constant. Areas of the solution (or the front) which
have smaller influence domains will have higher precisions due to the increased node
density (Figure 5.5 and 5.10), the equivalent effect as grid refinement. The difference is
that our partitions is overlapping (Figure 5.10) so that they are loosely interrelated and
individual error would be smoothed out by neighboring values in its influence domain
when applying MLS approximation. The smoothing scale is controlled by the influ-
ence domain size Rx (Figure 3.2) which is inversely proportional to the node density.
Therefore, the smoothing would be automatically adjusted whenever the node density
changes. By making the sampling rate adaptive to image gradient or other derived
features, we can have greater smoothing for the contour at low gradient area and less
for edge-likely location. For images in Figure 5.4, we need to apply large filters for
de-noising the image. But boundary leakage occurs when clear edges are missing. It
does not happened to point-based strategy because node distribution also governs the
de-noising capability of MLS approximation. Although MLS is computational expen-
sive, working on fewer nodes can make the speed of point based strategy comparable
to traditional regular grid finite difference level set (FD-LS). Moreover, the running
speed is also adaptive simply because fewer node values computation around low im-
53

age gradient areas, meaning a fast marching and level set refinement are performed
concurrently. Moreover, continuous φ representation enhances the accuracy of ∇φ
which is critical for both the evolution and final result in all level set formulations.
5.4 Future work
The extension of grid-less approach to 3D GDM has been attempted (Figure 5.6 and
5.7. The result also prove the grid-less approach can flexibly enhance the accuracy
of 3D GDM implementation (Figure 5.8). But the challenge we are now facing is
the extra computation brought by using gradient descent to re-initialize 3D GDM (see
chapter 3.3). As a result, re-initialization consumes a lot of time and significantly
degrades the efficiency of using grid-less approach comparing to the normal grid-based
3D implementation which allows efficient surface extraction using marching cubes
algorithm [23] before re-initialize by computing actual node-to-surface distance. But
we noticed that re-initialization is only for stabilizing purpose. ∇φ =1is not a definite
condition for the GDM evolution to proceed. Therefore, we could look for a weak
solution ψ where most of the nodes xi have ∇ψ(xi) close to desired value 1 and then
we replace φ by new ψ.
Up to now, we have used only simple velocity field F in our experiments that may
not be able to give very good results. We would like to look for a more specific F
formulation for particular type of image that can fully demonstrate the effectiveness of
our strategy.
If we consider MLS is a smoothing filter for φ on point cloud, then similar filtering
can also be applied to the F field so that the stability of the level set evolution can
be improved. Extension to 3D+T is also possible and we would be able to offer a
continuous 3D segmentation over time using our point cloud strategy.
54

Appendix A
Topology-preserving GDM through
domain partitioning
A.1 Introduction
The key feature of the geometric deformable models (GDMs) is that they naturally
allow topological changes during the curve evolution process. However, for multi-
ple objects with extremely small spatial separations, these GDMs would not be able
to form separate final contour for each of the object because of the spatial limitation
imposed by the finite difference computational grid. Further, a GDM contour could
possibly leak out from the weaker edges of the target object and evolve towards nearby
stronger edges of another object. In this chapter, we present a topology constrain-
ing geometrical deformable model scheme to address these two situations. Utilizing
the prior knowledge on the number of targets and their rough spatial positioning, we
add an additional domain partitioning level set surface (DPLSS) which seeks between-
object gaps and hence constrains the each boundary finding level set surface (BFLSS)
to reside within its own designated region and evolve towards the respective object
boundary. Relying on adaptive meshfree particle representations of the analysis do-
mains for DPLSS and BFLSSs, the relative spatial separation between each BFLSS
and DPLSS can be flexibly and effectively magnified. Further, the evolution of the
55

DPLSS and BFLSSs is processed in piecewise continuous fashion, through moving
lease squares (MLS) approximations, to ensure high accuracy.
Geometrical deformable model (GDM) is represented implicitly as the level set
of a higher-dimensional distance function and evolves in an Eulerian fashion [7, 24].
Especially because of its topological adaptiveness, which greatly simplifies the curve
initialization problems, GDM has found enormous interests from the computer vision
and medical image analysis communities [30, 38].
Nevertheless, there are cases where the topological flexibility is more a burden than
a blessing [13]. For example, when the topological structure of the targeting object is
known a priori, it is often more robust to search for the correct object composition
instead of letting the GDM freely evolve. This is especially important in medical im-
age segmentation where many of the objects of interests have specific topologically
consistent anatomy. Further, for multiple objects with extremely small spatial separa-
tions, such as the two brain ventricles shown in Fig. A.1, GDM would not be able to
form separate final contour for each of the ventricle because of weak edge information
between them (weak separation), and because of the spatial limitation imposed by the
finite difference computational grid. Depending on the formulations for the image in-
formation, there are also many situations where GDMs would leak out from the weaker
edges of the target object and evolve towards nearby stronger edges of another object.
Based on the simple point concept from digital topology, topology-preserving level set
methods are proposed to specifically maintain pre-determined topological composition
of the evolving GDMs on standard finite difference grid [13] and the refined moving
grid [12]. Several other GDM efforts on surface-coupling [38], shape-prior constraints
[19], and neighbor-constraints [33, 37] are also possibly extendable to impose topo-
logical stableness. We have also reviewed these representation schemes in Chapter
56

2.2.
Figure A.1: Brain image and the closeup view of the ventricles.
In the following sections, we employ a weak domain partitioning constraint, through
the added domain partitioning level set surface (DPLSS), to enforce topology preser-
vation of the boundary finding level set surfaces (BFLSSs). This formulation allows
us to handle un-balanced close edges (one weak, one strong). Further, relying on the
adaptive meshfree particle representations of the analysis domains for both DPLSS and
BFLSSs, it can separate objects with, theoretically, infinitely close spatial distance, an
impossibility for finite difference schemes.
A.2 Problem definition
Our goal is to solve the evolutions of multiple geometric deformable active contours,
based on level set formulations and on the same image, under an additional topological
constraint that all the individually segmented areas, from their corresponding GDMs,
are mutually exclusive.
Let each GDM be represented by the zero level set of a higher dimensional hyper-
surface φ. Further, region inside the contour has values φ0. The evolution of each active contour is then governed by the level
57

set partial differential equation [30]:
∂φ
∂t
= F |∇φ| (A.1)
where F is the image-data dependent velocity of the φ surface in the normal direction,
and ∇ denotes the gradient operator.
Assuming that we are going to segment n objects, we start with n GDMs embedded
in n boundary finding level set surfaces (BFLSSs). We solve for the above level set
equation (Eqn. (A.1)) for the arbitrary c-th GDM by taking the finite difference in time
domain:
φ t+∆t
c = φ t c − ∆tF|∇φc| (A.2)
where c =1, 2...n. Define the region inside an active contour as Φc = {φc < 0}, we
impose the additional constraint on Eqn. A.2 for all c that:
n
Φc = {0} (A.3)
c=1
where denotes the intersect operator among all Φc, and {0} denotes an empty set
(we changed the empty set notation to avoid confusion with the level set function).
A.3 Domain representation
To handle the small object separation issue, we need to emphasize that we cannot use
the grid-based finite difference schemes to compute the spatial gradient value ∇φ in
Equ. A.2. Instead, we will first distribute a set of point clouds onto the image and the
density of the point clouds is designed to be proportional to the image gradient magni-
tude (see Fig. A.4), so that we would have more points near potential boundary areas
including the gap between objects. We called this set of points the supporting nodes
58

xi of the level set function. We adopted this point-based strategy because there is need
to have higher precision of contour representation when two boundary are too close
to each other. Higher precision of the GDMs requires more sampling along boundary
areas for the level set functions. Since the refinement of nodes distribution is much
easier comparing with traditional grid refinement strategy [12], point-based represen-
tation offers higher flexibility for us to tackle the small boundary gaps problems. At
later stage, point-based strategy also benefits the implementation of the domain parti-
tioning level set surface (DPLSS) since we only need a few nodes to implement the
DPLSS.
A.4 Domain partitioning
The closure of the BFLSS embedded active contour is enforced in both initialization
and evolution phase so that we can always decide the inside and outside of the contour
using the sign of the level set function. Conventionally, φ0 is outside (or the background). Since the φ function maps each point on
the image to only a single value, the separated object region and the background region
are mutually exclusive under the sign convention of BFLSS.
Domain partitioning level set surface can be regarded as a variant of traditional
BFLSS. Constructed from a similar idea, the only difference for DPLSS is that the
embedded curve is assumed to be extended to infinity and therefore it is not a closed
contour. The separation drawn by the DPLSS is now called left side and right side
instead. Using DPLSS, we assign the BFLSS supporting nodes into different groups
by checking the sign of the DPLSS at those node locations using MLS. Each group
covers a unique region on the image and maintains its own level set function φc. The
BFLSS supporting nodes grouped by DPLSS are also mutually exclusive. Since MLS
59

surface approximation for the BFLSS is performed using exclusive node set, neighbor-
ing φc are not going to interfere each others no matter how close they are. Note that
approximated DPLSS using MLS is continuous which allows us to check for its value
at any point x ∈ R 2 .
Figure A.2: Illustration of the saddle map detecting the crossing of intensity derivative
and corresponding Gradient Vector Flow. The intensity images are on the left and the
saddle maps are on the right. On the saddle maps, bright regions imply high likelihood
of being boundary gaps and the vector field provide the guidance for the DPLSS
A.5 Movable partitioning
The key feature of DPLSS is that it is movable and also gap-seeking. This ability of
DPLSS allows flexible initialization (weak domain partitioning) for both BFLSS and
DPLSS which are supposed to work interactively. To define the velocity field F for
DPLSS, we create a saddle map S(x) by detecting the crossing of intensity derivative
60

Ix,Iy along x, y direction respectively:
Sx(x, y)
Sy(x, y)
|Ix(x + δx, y) − Ix(x − δx, y)|
=
0,
|Iy(x, y + δy) − Iy(x, y − δy)|
=
0,
if Ix(x, y) =0
otherwise
if Iy(x, y) =0
otherwise
(A.4)
(A.5)
S(x) ≡ S(x, y) =Sx(x, y)+Sy(x, y) (A.6)
S(x) is a scalar gap likelihood map which is then further processed by gradient vector
flow operation (GVF, see chapter 2 and [35]) to obtain the force field of DPLSS (see
Fig. A.2). The update of DPLSS is the same as the ordinary BFLSS evolution but
based on a different force field, seeking the gaps between objects.
A.6 Separation enforcement
MLS approximation can extrapolate contour away from its node clouds, so supporting
nodes partitioning only ensures individual contour would not merge, but they can still
be overlapping which is a violation of the physical phenomenon. To ensure the BFLSS
active contours do not cross over the DPLSS embedded domain partitioning curve, we
check the nearest zero set location of each BFLSS supporting nodes to see whether the
BFLSS active contour moves across the partitioning curve. This checking is simple
because we can make use of the sign of the DPLSS function which can tell the left or
right side immediately. If the nearest zero violates the separation rule, we can simply
increase the φc value of that node until the DPLSS function returns a desired sign which
matches with all other nodes in the same group. Define the nearest zero set location of
BFLSS node φ(xi) to be:
zx = xi − ∇φ
|∇φ| φ(xi) (A.7)
61

Let the DPLSS separator be φs, adjust the φ(xi) value if φs(zx) > 0 (for left side node)
using:
φ ′ (xi) =φ(xi)+|φs(zx)| (A.8)
until φs(zx) ≤ 0. For right side nodes, the same adjustment should be applied with the
two inequalities flipped over.
A.7 TPLSS results
Initialization Separation-unconstrained Separation-constrained
Figure A.3: Comparison between separation-unconstrained and separation-constrained
segmentation on synthetic and real images.
We test our algorithm on various images to perform topology preserving segmen-
tation. Particular interesting to us are those cases where the objects are extremely close
to each other, and/or with un-balanced edges. Figs. A.4 show the curve evolution pro-
cesses of the BFLSSs and DPLSS on synthetic and real images, which demonstrate the
ability of our effort. The final segmentation results of these two experiment are shown
in Fig. A.3. The running time of the experiment in Fig. A.4 using MATLAB is 79.5s
62

Synthetic image with unbalanced edges without separation constraint:
with separation constraint:
◮ ◮ ◮
◮ ◮ ◮
Brain ventricle image with improper initialization without separation constraint:
with separation constraint:
◮ ◮ ◮
◮ ◮ ◮
Figure A.4: Comparison between separation-unconstrained (1st and 3rd rows) and
separation-constrained (2nd and 4th rows) GDM evolution processes on synthetic and
real image. The synthetic example shows close objects with unbalanced edges (one
strong, one weak). The real example shows the effect of improper initialization and
how the separation enforcement correct the problem.
63

(3rd row without DPLSS) and 105.5s (4th row with DPLSS) on a Pentium-M 1.3GHz
computer with 256MB RAM. This shows the overhead time for DPLSS is comparable
to an additional level set contour.
A.8 Summary of the algorithm
1. Pre-process: compute the image gradient magnitude, gradient vector flow of the
saddle map S(x), and distribute sampling nodes according to image gradient.
2. Initialize: initialize several closed contours and establish their signed distance
functions BFLSSs φc(xi); also initialize the separators in-between all initial con-
tours (through Voronoi diagrams) and the corresponding DPLSSs φs(xj)
3. Update: update all φc(xi) and φs(xj) independently according to different force
fields and MLS approximated φc, ∇φc, φs and ∇φs.
4. Enforce Separation: make sure all nearest zeros of BFLSS nodes in the same
group are located in the same region classified by all DPLSSs.
5. Extract Contour: extract all φc(x) =0from the evolved BFLSS by marching on
the MLS approximated φc(x); explicit DPLSS separator can also be extracted in
similar way.
6. Output/Reset: if the curve is not stabilized, reset all BFLSS and DPLSS func-
tions by the extracted φc(x) =0and φs(x) =0positions, then go back to the
Update procedure; otherwise, output φc(x) =0.
64

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