Segmentation of Stochastic Images using ... - Jacobs University

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Chapter 1 Introduction Ambrosio-Tortorelli segmentation [14] is a regularization of the segmentation approach proposed by Mumford and Shah [107]. The idea is to compute a smooth representation of the image and the corresponding edges, respectively a phase field approximation of the edges. For the Ambrosio- Tortorelli model, the author developed a stochastic extension [1, 3], allowing to propagate information about the measurement error to the result, the smooth image and the phase field. Level set based segmentation is based on the evolution of a contour, represented by a level set function, i.e. the contour is given as the zero level set of a higher-dimensional function. A speed function controls the evolution of the contour. A typical choice for the speed function is to make it dependent on the image gradient [29, 96]. Caselles et al. [30] and simultaneously Kichenassamy et al. [82] developed improvements by adding a term that forces the contour to stay at edges. Furthermore, Chan and Vese [31] developed a segmentation method that is able to segment objects without sharp edges to the background. Instead of using gradient information, they proposed a functional that segments homogeneous regions in the image. Besides the development of stochastic segmentation algorithms, the investigation of pre- and postprocessing steps is essential to end up with a complete framework for error propagation in image processing. For example, it is necessary to develop a technique to acquire stochastic images, i.e. images whose pixels are random variables when image samples are available. This step benefits from techniques available in the literature [41, 130, 141] or from the modeling of the noise distribution. In addition, this thesis investigates the visualization of the stochastic segmentation results. Furthermore, it is possible to change the perspective and use the segmentation methods developed for stochastic images for a sensitivity analysis of the segmentation methods with respect to the segmentation parameters. The sensitivity analysis uses segmentation parameters that are random variables. The segmentation result is a stochastic image that contains information about the influence of the segmentation parameters. Thus, the stochasticity comes from the parameters and not from the input image, but the equations are nearly the same. Structure of the Thesis The thesis has the following structure: Chapter 2 presents segmentation methods for images based on PDEs. In particular, these are random walker segmentation, Ambrosio-Tortorelli segmentation, and methods based on level sets. Besides the presentation of these classical methods, this chapter discusses the drawbacks, especially for the propagation of errors. Furthermore, we review related work and highlight the differences between the related work and the methods proposed here. Chapter 3 contains an introduction into SPDEs and provides a theoretical background for the treatment of SPDEs. Furthermore, it presents the polynomial chaos expansion, a widely used tool for the approximation of random variables. The polynomial chaos expansion is the key for the numerical treatment of SPDEs and random variables, because this expansion converts the abstract idea of random variables into a series expansion with deterministic coefficients. A computer can work with these coefficients, which enables the development of numerical methods for random variables. At the end of this chapter we highlight the advantages of the polynomial chaos over interval arithmetic [64]. Chapter 4 investigates the discretization of SPDEs based on the polynomial chaos. The presented methods range from sampling based methods like Monte Carlo simulation and stochastic collocation to methods based on the polynomial chaos approximation for random variables. For the polynomial chaos, this chapter presents a finite difference method as well as SFEM and the GSD method. After the presentation of discretization methods for random variables and SPDEs in the previous chapters, Chapter 5 presents stochastic images. The concept of stochastic images is crucial for this thesis, because all methods developed in this thesis act on stochastic images. The main idea is to replace a pixel from a classical image by a random variable. Using the notion from stochastics, a stochastic image is a random field indexed by the position of the pixels inside the image. Besides 4
the presentation of the stochastic images, Section 5.2 describes a possibility to generate stochastic images from image samples. This stochastic image generation is based on the method presented by Desceliers et al. [41], who applied an empirical Karhunen-Loève expansion on the centered covariance matrix. Section 5.4 investigates the visualization of 2D and 3D stochastic images. Chapter 6 generalizes two segmentation algorithms based on elliptic PDEs to stochastic segmentation methods acting on stochastic images. Section 6.1 deals with the extension of the random walker segmentation to stochastic images. The idea of the random walker segmentation is to prescribe a set of seed points for the objects and the background. Then a random walk starts at every unseeded point and the probability that the random walker goes from one point to another is dependent on the image intensity difference. In a stochastic image, this difference is a stochastic quantity and the probabilities for the walk of the random walker are stochastic quantities. The discretization of this method is based on the solution of a diffusion equation, because diffusion is the limit process of an infinite number of random walks. Section 6.2 investigates a stochastic extension of the Ambrosio-Tortorelli model for segmentation. The author presented this work at the European Conference on Computer Vision (ECCV) 2010 [3] and received the “ECCV 2010 Best Student Paper Award”. The idea is to replace all quantities in the Ambrosio-Tortorelli approach by their stochastic counterparts, yielding to two coupled SPDEs as the stochastic Euler-Lagrange equations for the computation of an energy minimizer. Using the GSD for the solution of the discretized SPDEs, the Ambrosio-Tortorelli method segments stochastic images computed from samples acquired via devices like digital camera or ultrasound imaging. Chapter 7 presents the last segmentation method for stochastic images investigated in this thesis, the segmentation of stochastic images with stochastic level sets. First, this chapter presents the extension of level sets to stochastic level sets, i.e. level sets evolving under an uncertain velocity. This extension is based on a parabolic approximation of the original level set equation. Having the stochastic level set extension at hand, it is possible to develop methods for the segmentation based on stochastic level sets. A method where the speed for the stochastic level set evolution is based on the image gradient and stochastic extensions of the geodesic active contour approach developed simultaneously by Caselles et al. [30] and Kichenassamy et al. [82] and the Chan-Vese approach [31]. Chapter 8 deals with a sensitivity analysis of segmentation methods with respect to parameter changes. The sensitivity analysis uses the stochastic framework developed in the previous chapters, but applies it on a single deterministic input image. The stochasticity comes from the segmentation parameters that are random variables. With this modeling, we investigate the influence of the parameters on the result with the same segmentation framework developed for stochastic images. Chapter 9 contains a summary of the thesis along with a discussion. Furthermore, the chapter draws conclusions and gives directions for future work. 5
Page 1: Segmentation of Stochastic Images u
Page 5: Abstract The task of segmentation,
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Chapter 5 Stochastic Images Figure
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Chapter 6 Segmentation of Stochasti
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6.1 Random Walker Segmentation on S
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Chapter 7 Stochastic Level Sets whe
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Chapter 8 Segmentation of Classical
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Chapter 9 Summary, Discussion, and
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List of Figures 1.1 Left: CT image
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List of Figures 6.2 Mean and varian
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List of Figures 7.12 Mean (left) of
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Appendix A Publications Written Dur
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Bibliography [14] L. Ambrosio and M
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Bibliography [46] B. Echebarria, R.
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Bibliography [81] B. N. Khoromskij
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Bibliography [113] A. Nouy. A gener
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Bibliography [147] J. Tryoen, O. Le
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