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Chapter 8 Segmentation of Classical Images Using Stochastic Parameters Figure 8.8: Result of the geodesic active contour segmentation with stochastic parameters for the liver data set. On the left the expected value of a detail of the image and contour realizations and on the right the variance of the level set with contour realizations are shown. regions, where the object to segment does not have the “elliptic” contour behavior. In these regions, the gradient differs from the remaining parts of the image. The geodesic active contour method has different attractors depending on the particular value of α and β in these regions. This is visible from Fig. 8.7, because the contour realizations are far from each other, and the variance is high in a region in the upper part of the object boundary. Remark 19. Note that for level sets there is, in contrast to the random walker method, a one-to-one correspondence between the distance between the contour realizations and the variance, because we use a stochastic equivalent of the signed distance function. Thus, deviations in the level set position are related to the variance directly. For the liver data set, Fig. 8.8 shows the results with the same stochastic parameters. Again, the results are close together for parameter realizations, because we have one attractor for the level set only (the object boundary). The differences in the lower part of the object are due to the weak attraction of the liver boundary for some realizations of the parameter α. Conclusion The presented sensitivity analysis is a natural extension of the stochastic image processing framework presented in this thesis. With the sensitivity analysis, we investigate the robustness of the classical image segmentation methods with respect to parameter changes. This additional stochastic information is available for the costs of a few Monte Carlo runs. However, we do not use the Monte Carlo method, but have to solve a couple of deterministic problems when using the GSD method. A possible application of this kind of sensitivity analysis is to warn the user, when the segmentation result is sensitive to parameter changes. This can be done via background calculations, i.e. the system computes the stochastic solution while the user examines the deterministic result. When necessary, the system informs the user about the stochastic result and makes additional information like the variance or contour realizations available. 106
Chapter 9 Summary, Discussion, and Conclusion In this thesis, we presented extensions of PDE-based segmentation methods to stochastic images, i.e. images whose pixels are random variables. The characterization of such stochastic images is based on the recently developed generalized polynomial chaos expansion. With this expansion, we developed extensions of the well-known finite element and finite difference schemes for the discretization of the PDE to the stochastic dimensions, leading to stochastic PDEs. To demonstrate the power of using stochastic images, we extended the well-known segmentation methods proposed by Mumford-Shah and the related approximation by Ambrosio-Tortorelli as well as the random walker method and three methods based on a level set formulation. The input for the stochastic segmentation is constructed via computing the leading random variables via a principal component analysis of samples of the input scene and a projection on the polynomial chaos basis. Furthermore, we used the stochastic images and model extensions to perform a sensitivity analysis of the methods by identifying the parameters with random variables. 9.1 Discussion The work presented in this thesis is a complete framework for the important task of error propagation in mathematical image processing [36, 106]. For every step of the mathematical image processing pipeline (data acquisition, data representation, operator modeling, discretization, solution strategies and visualization) methods for the solution of the particular problems are presented. Besides the development of the framework, theoretical justifications of the methods are presented as well. In particular, these are the extensions of the Γ-convergence proof for the stochastic Ambrosio-Tortorelli model and the proof of the existence of SPDE solutions used in this thesis. This thesis applies the error propagation framework to mathematical operators for image segmentation, but the thesis can also be seen as a case study to demonstrate the applicability of the methods in image processing. Other image processing operators based on a PDE formulation can be extended by the presented methods easily, because the framework and the implementation of all steps around the operator extension are available. The only step that remains is the stochastic operator extension. Furthermore, the stochastic parameter study presented in this thesis sensitizes users to be skeptic about the segmentation results if these are not robust with respect to parameter changes. 9.2 Conclusion We presented methods for all tasks along the stochastic image processing pipeline, but some of the methods presented in this thesis can be improved to get more stable and more accurate results. For example, the projection step for the estimation of the input distribution (cf. Section 5.2) is based on a Monte Carlo sampling (it is based on the uncorrelated image samples) and the method has the poor convergence speed O(1/ √ N) of the Monte Carlo method. Stefanou et al. [141] presented two methods based on an optimization problem. These methods are computationally much more expensive, but lead to a better convergence speed. Furthermore, the complete stochastic pipeline is restricted to a few basic random variables, which might be problematic, because image noise has 107
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Segmentation of Stochastic Images u
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Abstract The task of segmentation,
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7.5 Segmentation of Stochastic Imag
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Chapter 5 Stochastic Images
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