Joachim Weickert wrote: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.