Segmentation of Stochastic Images using ... - Jacobs University

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
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