##
**Anisotropic diffusion in image processing.**
*(English)*
Zbl 0886.68131

European Consortium for Mathematics in Industry. Stuttgart: Teubner (ISBN 3-519-02606-6). xii, 170 p. (1998).

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 (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 analyze 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 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.

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 (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 analyze 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 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.

### MSC:

68U10 | Computing methodologies for image processing |

68-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to computer science |

94A08 | Image processing (compression, reconstruction, etc.) in information and communication theory |

65M30 | Numerical methods for ill-posed problems for initial value and initial-boundary value problems involving PDEs |