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Chapter 2 Image Segmentation and Limitations Random Level Set Functions Stefanou et al. [141] presented a method to obtain a random level set, i.e. a polynomial chaos expansion of a level set function. The proposed work relates to this work, but [141] obtains the polynomial chaos expansion of the level set from a classical level set approach on the samples and an estimation of the polynomial chaos coefficient afterwards. Computing the level set equation on the samples is exactly the objective we want to overcome in this work by applying a stochastic level set equation on a stochastic image, obtained from the samples. This results in a significant speed-up, because in the stochastic framework the level set method has to be applied only once on the stochastic image, not on every sample. Stochastic Active Contours The work presented in this thesis should not be confounded with the method presented by Juan et al. [75] named “Stochastic Active Contour”. Although the title suggests a close relation between the stochastic level set equation and the work presented in [75], they are opposed. Juan et al. [75] proposed a technique to overcome drawbacks of the classical level set approach by adding fictive noise to the image. This relates to the simulated annealing technique [83]. The idea of the stochastic level set framework is the development of a stochastic active contour moving under an uncertain velocity, obtained from the uncertain gray values of the stochastic pixels. A Multiresolution Stochastic Level Set Method for Mumford-Shah Image Segmentation Law et al. [89] use a method called “Stochastic Level Set Method” for the segmentation of objects. This also should not be confounded with the stochastic level set work presented in this thesis, because Law et al. proposed a method to overcome the drawback of the level set segmentation to run into local minima. They developed a method that “jumps” out of these local minima to get a global minimum of the solution. Again, this method is related to the simulated annealing technique [83]. Error-in-Variables Likelihood Functions for Motion Estimation Nestares et al. [111,112] where able to compute confidence measures for error propagation in motion estimation [71]. Their method allows to estimate the influence of the image noise on the computed motion field. To achieve this, they combined Bayesian estimation [77] and likelihood functions to solve a total-least squares problem [58]. Nevertheless, their investigations are restricted to independent, identically distributed Gaussian random variables. Thus, their proposed framework can be used in rare situations only. Conclusion We presented the basics for mathematical image processing and gave a short overview over segmentation methods based on PDEs. All these methods produce good results on single images, but are unable to deal with error propagation. Furthermore, the robustness of the methods with respect to image noise and parameter changes is unknown. This highlights the need for error-aware methods for segmentation and image processing in general. Before we are able to present these methods, we have to provide background from stochasticity for the representation of random variables and about SPDEs. This is the task of the next chapter. 24
Chapter 3 SPDEs and Polynomial Chaos Expansions This chapter deals with the fundamentals required to develop stochastic images. First, we review notation and results from probability theory. Afterwards, we introduce SPDEs and the polynomial chaos expansion, the main ingredient for the numerical approximation of random variables. 3.1 Basics from Probability Theory This section provides background from probability theory for the presentation of the stochastic images and SPDEs. First, we introduce the basic ingredients, probability measures, probability spaces and random variables. Definition A probability space (Ω,A ,Π) is a triple consisting of a sample space Ω containing all possible outcomes, a σ-algebra of events A ⊂ 2 Ω and a probability measure Π. The probability measure Π is defined on the σ-algebra A and has the following properties: • Π is non-negative: Π(A) ≥ 0 for all A ∈ A . • The measure of the sample space Ω is one: Π(Ω) = 1. • Π is countable additive, i.e. for a countable number of pairwise disjoint sets A i ⊂ A we have Π(∪A i ) = ∑(Π(A i )). On the probability space (Ω,A ,Π) we define functions from this space into the real numbers. Definition A random variable f : Ω → IR is a function from the sample space Ω into the real numbers that is measurable with respect to the σ-algebras A and B, where B is the Borel measure. Random variables are an important object for the definition of stochastic images. In Chapter 5, we will see that every pixel of a stochastic image is a random variable. For random variables, it is possible to define the probability density function (PDF): Definition The function ρ is called probability density function (PDF) of the random variable f if it satisfies Π(a < f < b) = ∫ b a ρ(x)dx for all a,b ∈ IR. Having the probability density at hand, we define further properties of random variables. The most important property of random variables is the expected value: Definition The expected value or first moment of a random variable X : Ω → IR with PDF ρ is ∫ ∫ ∫ E(X) = X(ω)dω = xρ(x)dx = xdΠ . (3.1) Ω In (3.1) we used dΠ = f dx to characterize integration with respect to the PDF. Knowing the probability density of a random variable allows us to transform the integral over the sample space Ω into an easier computable integral over the real numbers weighted by the probability density. Using this equality, it is also possible to compute higher-order moments of random variables: IR IR 25
Page 1: Segmentation of Stochastic Images u
Page 5: Abstract The task of segmentation,
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Page 11 and 12: Chapter 1 Introduction The developm
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Page 18 and 19: Chapter 2 Image Segmentation and Li
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6.2 Ambrosio-Tortorelli Segmentatio
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6.2 Ambrosio-Tortorelli Segmentatio
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Chapter 7 Stochastic Level Sets whe
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Chapter 9 Summary, Discussion, and
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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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