The success of these techniques is not really surprising, since PDEs have proved their usefulness in areas such as physics and engineering sciences for a very long time. In image processing and computer vision, they offer several advantages:
One of the two goals of this book is to give an overview of the state-of-the-art of PDE-based methods for image enhancement and smoothing. Emphasis is put on a unified description of the underlying ideas, theoretical results, numerical approximations, generalizations and applications, but also historical remarks and pointers to open questions can be found. Although being concise, this part covers a broad spectrum: it includes for instance an early Japanese scale-space axiomatic, the Mumford-Shah functional for image segmentation, continuous-scale morphology, active contour models and shock filters. Many references are given which point the reader to useful original literature for a task at hand.
The second goal of this book is to present an in-depth treatment of an interesting class of parabolic equations which may bridge the gap between scale-space and restoration ideas: nonlinear diffusion filters. Methods of this type have been proposed for the first time by Perona and Malik in 1987. In order to smooth an image and to simultaneously enhance important features such as edges, they apply a diffusion process whose diffusivity is steered by derivatives of the evolving image. These filters are difficult to analyse mathematically, as they may act locally like a backward diffusion process. This gives rise to well-posedness questions. On the other hand, nonlinear diffusion filters are frequently applied with very impressive results; so there appears the need for a theoretical foundation.
We shall develop results in this direction by investigating a general class of nonlinear diffusion processes. This class comprises linear diffusion filters as well as spatial regularizations of the Perona-Malik process, but it also allows processes which replace the scalar diffusivity by a diffusion tensor. Thus, the diffusive flux does not have to be parallel to the grey value gradient: the filters may become anisotropic. Anisotropic diffusion filters can outperform isotropic ones with respect to certain applications such as denoising of highly degraded edges or enhancing coherent flow-like images by closing interrupted one-dimensional structures. In order to establish well-posedness and scale-space properties for this class, we shall investigate existence, uniqueness, stability, maximum-minimum principles, Lyapunov functionals, and invariances. The proofs present mathematical results from the nonlinear analysis of partial differential equations. Since digital images are always sampled on a pixel grid, it is necessary to know if the results for the continuous framework carry over to the practically relevant discrete setting. These questions are an important topic of the present book as well. A general characterization of semidiscrete and fully discrete filters, which reveal similar properties as their continuous diffusion counterparts, is presented. It leads to a semidiscrete and fully discrete scale-space theory for nonlinear diffusion processes. Mathematically, this comes down to the study of nonlinear systems of ordinary differential equations and the theory of nonnegative matrices.
Chapter 1 surveys the fundamental ideas behind PDE-based smoothing and restoration methods. This general overview sketches their theoretical properties, numerical methods, applications and generalizations. The discussed methods include linear and nonlinear diffusion filtering, coupled diffusion-reaction methods, PDE analogues of classical morphological processes, Euclidean and affine invariant curve evolutions, and total variation methods.
The subsequent three chapters explore a theoretical framework for anisotropic diffusion filtering. Chapter 2 presents a general model for the continuous setting where the diffusion tensor depends on the structure tensor (interest operator, second-moment matrix), a generalization of the Gaussian-smoothed gradient allowing a more sophisticated description of local image structure. Existence and uniqueness are discussed, and stability and an extremum principle are proved. Scale-space properties are investigated with respect to invariances and information-reducing qualities resulting from associated Lyapunov functionals.
Chapter 3 establishes conditions under which comparable well-posedness and scale-space results can be proved for the semidiscrete framework. This case takes into account the spatial discretization which is characteristic for digital images, but it keeps the scale-space idea of using a continuous scale parameter. It leads to nonlinear systems of ordinary differential equations. We shall investigate under which conditions it is possible to get consistent approximations of the continuous anisotropic filter class which satisfy the abovementioned requirements.
In practice, scale-spaces can only be calculated for a finite number of scales, though. This corresponds to the fully discrete case which is treated in Chapter 4. The investigated discrete filter class comes down to solving linear systems of equations which may arise from semi-implicit time discretizations of the semidiscrete filters. We shall see that many numerical schemes share typical features with their semidiscrete counterparts, for instance well-posedness results, extremum principles, Lyapunov functionals, and convergence to a constant steady-state. This chapter also shows how one can design efficient numerical methods which are in accordance with the fully discrete scale-space framework and which are based on an additive operator splitting (AOS).
Chapter 5 is devoted to practical topics such as filter design, examples and applications of anisotropic diffusion filtering. Specific models are proposed which are tailored towards smoothing with edge enhancement and multiscale enhancement of coherent structures. Their qualities are illustrated using images arising from computer aided quality control and medical applications, but also fingerprint images and expressionistic paintings shall be processed. The results are juxtaposed to related methods from Chapter 1.
Finally, Chapter 6 concludes the book by giving a summary and discussing possible future perspectives for nonlinear diffusion filtering.
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