Weickert J. Anisotropic Diffusion in Image Processing

Издательство B.G. Teubner, 1998, -182 pp.Partial differential equations (PDEs) have led to an entire new field in image processing and computer vision. Hundreds of publications have appeared in the last decade, and PDE-based methods have played a central role at several conferences and workshops.
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:
Deep mathematical results with respect to well-posedness are available, such that stable algorithms can be found. PDE-based methods are one of the mathematically best-founded techniques in image processing.
They allow a reinterpretation of several classical methods under a novel unifying framework. This includes many well-known techniques such as Gaussian convolution, median filtering, dilation or erosion.
This understanding has also led to the discovery of new methods. They can offer more invariances than classical techniques, or describe novel ways of shape simplification, structure preserving filtering, and enhancement of coherent line-like structures.
The PDE formulation is genuinely continuous. Thus, their approximations aim to be independent of the underlying grid and may reveal good rotational invariance.
PDE-based image processing techniques are mainly used for smoothing and restoration purposes. Many evolution equations for restoring images can be derived as gradient descent methods for minimizing a suitable energy functional, and the restored image is given by the steady-state of this process. Typical PDE techniques for image smoothing regard the original image as initial state of a parabolic (diffusion-like) process, and extract filtered versions from its temporal evolution. The whole evolution can be regarded as a so-called scale-space, an embedding of the original image into a family of subsequently simpler, more global representations of it. Since this introduces a hierarchy into the image structures, one can use a scale-space representation for extracting semantically important information.
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 [326]. 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.Image smoothing and restoration by PDEs
Continuous diffusion filtering
Semidiscrete diffusion filtering
Discrete diffusion filtering
Examples and applications
Conclusions and perspectives