Segmentation of Stochastic Images using ... - Jacobs University

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Chapter 9 Summary, Discussion, and Conclusion Figure 9.1: Visualization of the PDF of the object boundary in the case of the segmentation of the stochastic US image. a short correlation length in many applications. Thus, multiple random variables are required to characterize the input adequately. A possibility to deal with huge numbers of random variables, and therefore a high dimensional polynomial chaos, is to use adaptive methods for the stochastic dimensions. A starting point are the methods presented in [22, 80, 152]. In addition, the use of the parabolic approximation of the level set equation might be problematic in applications where sharp corners and shocks are important. A direct implementation of the stochastic level set equation could be based on the work for hyperbolic SPDEs in [92, 147], but these methods are computationally too expensive and not accurate enough for this task, but the discretization of hyperbolic SPDEs is still an active field of research. Another important task is the development of intuitive visualization techniques for the high dimensional stochastic output. This thesis presented ideas for the visualization of the stochastic results, but in cooperation with visualization experts, these techniques can be improved. The availability of such visualization techniques is helpful to convince the image processing community of this kind of error propagation and to use the error propagation in applications. The user, e.g. a physician, needs an intuitive access to the stochastic data. A starting point might be the visualization of stochastic boundaries depicted in Fig. 9.1. 9.3 Outlook and Future Work Besides the improvement of the methods presented in this thesis, there are possibilities for future research in the field of image processing with SPDEs. For example, advanced segmentation methods based on level set formulations can be investigated for stochastic extensions. Furthermore, it is planned to investigate registration methods [103] for stochastic extensions. Moreover, the stochastic extensions presented in [130] have to be adapted to the new ansatz space. Further work directions are the development of efficient methods for stochastic finite difference schemes, especially for nonlinear operations, because the calculation of the square root of a polynomial chaos expansion is a bottleneck for most algorithms in this thesis. In addition, the investigation 108
9.3 Outlook and Future Work of level set schemes, which do not need a reinitialization step, is important for efficient stochastic methods, because at the moment 80% of the computation time is spent for the reinitialization. In addition, the emerging field of tensor-structured methods [19, 60, 81] is important for the efficient solution of the presented SPDEs. Tensor-structured methods represent the data and the operators in a compressed form with a storage requirement linear in the number of dimensions, instead of the exponential dependence when storing the uncompressed data. Up to now, there are first numerical examples available in the literature [19, 81] and the methods are not applied on problems arising in applications like image processing. A big challenge for the future is to bring this error-aware image processing pipeline into applications. To be able to achieve this, it is necessary to use problem-dependent basic random variables for the polynomial chaos. For example, for the modeling of magnetic resonance images it is advantageous to use Rice distributed basic random variables, because the noise of gradient magnitude images is Rice distributed. To use a compatible basis leads to more accurate results with fewer basic random variables. Other input data require different basic random variables. Therefore, it might be a good idea to construct the basis on the fly if the input data is available based on the method from [157]. 109
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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 1 Introduction The developm
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the presentation of the stochastic
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Chapter 2 Image Segmentation and Li
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Chapter 3 SPDEs and Polynomial Chao
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Chapter 4 Discretization of SPDEs F
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Chapter 4 Discretization of SPDEs a
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Chapter 5 Stochastic Images
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Chapter 5 Stochastic Images Figure
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Chapter 5 Stochastic Images like qu
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Chapter 5 Stochastic Images Figure
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