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Part of: Synthesis digital library of engineering and computer science.

Includes bibliographical references (pages 133-142).

1. Introduction --

2. Partial differential equations and their numerics -- 2.1 Some basics from functional analysis -- 2.2 Finite difference method -- 2.3 Weak formulation -- 2.4 Galerkin approach and finite element method -- 2.5 Time-stepping schemes --

3. Review of PDE-based image processing -- 3.1 Mathematical representation of images -- 3.2 Diffusion filtering for denoising -- 3.3 Random walker and diffusion image segmentation -- 3.4 Mumford-Shah and Ambrosio-Tortorelli segmentation -- 3.5 Level set methods for image segmentation -- 3.6 Variational methods for optical flow -- 3.7 Variational methods for registration --

4. Numerics of stochastic PDEs -- 4.1 Basics from probability theory -- 4.2 Stochastic partial differential equations -- 4.3 Polynomial chaos expansions -- 4.3.1 Calculations in the polynomial chaos -- 4.3.2 Polynomial chaos for random fields -- 4.4 Sampling-based discretization of SPDEs -- 4.5 Stochastic finite difference method -- 4.6 Stochastic finite element method -- 4.7 Generalized spectral decomposition -- 4.7.1 Using polynomial chaos with GSD --

5. Stochastic images -- 5.1 Polynomial chaos for stochastic images -- 5.2 Stochastic images from samples -- 5.2.1 Low-rank approximation -- 5.3 Stochastic images from noise models -- 5.4 Imaging modalities for stochastic images -- 5.5 Visualization of stochastic images --

6. Image processing and computer vision with stochastic images -- 6.1 Stochastic diffusion filtering -- 6.1.1 Results -- 6.2 Stochastic random Walker and stochastic diffusion image segmentation -- 6.2.1 Results -- 6.2.2 Performance evaluation -- 6.3 Stochastic Ambrosio-Tortorelli segmentation -- 6.3.1 [Gamma]-convergence of the stochastic model -- 6.3.2 Weak formulation and discretization -- 6.3.3 Results -- 6.4 Stochastic level set method for image segmentation -- 6.4.1 Derivation of a stochastic level set equation -- 6.4.2 Interpretation of stochastic level sets -- 6.4.3 Discretization of the stochastic level set equation -- 6.4.4 Reinitialization of stochastic level sets -- 6.4.5 Numerical verification -- 6.4.6 Image segmentation with stochastic level sets -- 6.4.7 Results -- 6.5 Variational method for elastic registration -- 6.5.1 Results --

7. Sensitivity analysis -- 7.1 Classical sensitivity analysis -- 7.2 Perona-Malik diffusion -- 7.3 Random Walker and diffusion image segmentation -- 7.4 Ambrosio-Tortorelli segmentation -- 7.5 Gradient-based segmentation -- 7.6 Geodesic active contours -- 7.7 Discontinuity-preserving optical flow --

8. Conclusions -- Bibliography -- Authors' biographies.

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In image processing and computer vision applications such as medical or scientific image data analysis, as well as in industrial scenarios, images are used as input measurement data. It is good scientific practice that proper measurements must be equipped with error and uncertainty estimates. For many applications, not only the measured values but also their errors and uncertainties, should be--and more and more frequently are--taken into account for further processing. This error and uncertainty propagation must be done for every processing step such that the final result comes with a reliable precision estimate. The goal of this book is to introduce the reader to the recent advances from the field of uncertainty quantification and error propagation for computer vision, image processing, and image analysis that are based on partial differential equations (PDEs). It presents a concept with which error propagation and sensitivity analysis can be formulated with a set of basic operations. The approach discussed in this book has the potential for application in all areas of quantitative computer vision, image processing, and image analysis. In particular, it might help medical imaging finally become a scientific discipline that is characterized by the classical paradigms of observation, measurement, and error awareness. This book is comprised of eight chapters. After an introduction to the goals of the book (Chapter 1), we present a brief review of PDEs and their numerical treatment (Chapter 2), PDE-based image processing (Chapter 3), and the numerics of stochastic PDEs (Chapter 4). We then proceed to define the concept of stochastic images (Chapter 5), describe how to accomplish image processing and computer vision with stochastic images (Chapter 6), and demonstrate the use of these principles for accomplishing sensitivity analysis (Chapter 7). Chapter 8 concludes the book and highlights new research topics for the future.

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Title from PDF title page (viewed on July 21, 2017).