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Chapter 1 Introduction Figure 1.2: Noisy images from an ultrasound device (left) showing a structure in the forearm and a computed tomography (right) of a vertebra in a human spine. The aim of this thesis is to provide a representation for images containing error information and to provide a framework for the error propagation of image processing operators. The representation of images containing error information is based on a concept presented by Preusser et al. [130]. This thesis identifies pixels by random variables. We call images containing random variables as pixels stochastic images. The discretization of stochastic images uses the generalized polynomial chaos developed by Xiu and Karniadakis [160] to approximate the random variables at the pixels in a numerically meaningful way. This way of image representation is possible when information about the distribution of the gray value for a pixel is available. Repeated acquisitions of the same scene with the same imaging device or the usage of noise models can generate this information. The repeated acquisitions are only possible in rare situations, where a still scene is available and the repeated acquisition is ethically maintainable. Typically, the generation of medical images violates these conditions, because the human under investigation is alive and acquisition devices like computed tomography use high-energy radiation. Thus, for medical applications there is only a limited area for the application of these methods. For other areas like quality control, it is easy to generate samples of the same typically still scene. A possibility to overcome the need for multiple samples is the application of noise models in combination with a single image. However, the available image sample has to be as close as possible to the expected value of multiple acquisitions to get meaningful results. This is hard to achieve. Nevertheless, we present a possibility to generate a stochastic image from a sinogram of a computed tomography. Replacing the classical images in the PDE based operators by their stochastic counterparts achieves error propagation for image processing operators, but leads to stochastic partial differential equations (SPDEs). The numerical solution of SPDEs is a rapidly growing field, because these equations arise in the modeling of physical processes with uncertain parameters like heat propagation [24] or fluid dynamics [84, 93, 109]. Uncertain parameters are e.g. the thermal conductivity or the speed, because it is impossible to estimate these parameters exactly, but sometimes information about the probability density function (PDF) is available for these parameters. In the classical modeling with PDEs, one uses the expected value of the parameters for the calculation, yielding results that seem accurate, but lose the information about the distribution of the input parameters. It is a great advantage to have this information also in the output of such a calculation. The simplest method to get information about this distribution is to perform a Monte Carlo simulation [101], i.e. to perform deterministic calculations with a parameter sampled from the known input distribution. This is timeconsuming, due to the high number of runs needed to achieve a sufficient precision. To overcome this problem, methods have been developed ranging from stochastic collocation [158], a technique to use 2
Figure 1.3: This thesis combines findings from image processing with findings about SPDEs to yield segmentation algorithms acting on stochastic images. special sampling points in the random space, to the stochastic finite element method (SFEM) [54], a discretization with finite elements in the random and spatial dimensions. Furthermore, the generalized spectral decomposition (GSD) [113] allows breaking down the huge equation system of the SFEM into a series of smaller systems. The main goal of this thesis is to investigate, whether it is possible to combine the results from the SPDE context with the results from image processing (see Fig. 1.3), especially for the task of image segmentation. In other words: Which methods from image processing benefit from a stochastic modeling of the input or model parameters, and how can we interpret the results from stochastic modeling? The combination of stochastic images and SPDE, in the way presented in this thesis, was never done before in the literature, although this approach yields new insights for the PDE based segmentation of images: • It determines regions where the segmentation is reliable also in the presence of image noise and regions where the image noise has a great impact on the result of a segmentation. • It quickly investigates the influence of parameters on the segmentation results, when using SFEM or similar techniques for the computation. The development of SPDE methods for image segmentation is based on existing PDE segmentation methods. However, there are various methods proposed in the literature (see [144] for a review), and we limit the analysis for a stochastic extension to a few of them. Namely, these are random walker segmentation [59], Mumford-Shah segmentation [107] with the related Ambrosio-Tortorelli approximation [14], gradient-based segmentation [29], geodesic active contours [30, 82], and Chan- Vese segmentation [31]. The latter three are based on a level set formulation. Random walker segmentation [59] in contrast to other segmentation methods based on PDEs is a supervised segmentation method, meaning that the user influences the segmentation result by interactive input. For random walker segmentation, the user input consists of defining seed regions, i.e. regions where the user specifies whether they belong to the object or not. The idea of the random walker segmentation is to start random walks from the unseeded pixels and to give every pixel a probability to belong to the object dependent on the fraction of random walks reaching the seed region of the object. The direction the random walker chooses is dependent on the image gradient between neighboring pixels, i.e. the probability to walk from one pixel to another is higher, when the image gradient between the pixels is low. An implementation of the random walker algorithm uses a different strategy, because it is unnecessary to compute random walks for every pixel to compute the probabilities. Doyle and Snell [45] showed the equivalence to a Dirichlet problem. This reduces the complexity to the solution of an elliptic PDE with an unknown for every unseeded pixel. 3
Page 1: Segmentation of Stochastic Images u
Page 5: Abstract The task of segmentation,
Page 8 and 9: 7.5 Segmentation of Stochastic Imag
Page 11: Chapter 1 Introduction The developm
Page 15: the presentation of the stochastic
Page 18 and 19: Chapter 2 Image Segmentation and Li
Page 36 and 37: Chapter 3 SPDEs and Polynomial Chao
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Chapter 5 Stochastic Images like qu
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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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6.2 Ambrosio-Tortorelli Segmentatio
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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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