Preface Through many centuries physics has been one of the most fruitful sources of inspiration for mathematics. As a consequence, mathematics has become an economic language providing a few basic principles which allow to explain a large variety of physical phenomena. Many of them are described in terms of partial diierential equations (PDEs). In recent years, however, mathematics also has been stimulated by other novel elds such as image processing. Goals like image segmentation, multiscale image representation, or image restoration cause a lot of challenging mathematical questions. Nevertheless, these problems frequently have been tackled with a pool of heuristical recipes. Since the treatment of digital images requires very much computing power, these methods had to be fairly simple. With the tremendous advances in computer technology in the last decade, it has become possible to apply more sophisticated techniques such as PDE-based methods which have been inspired by physical processes. Among these techniques, parabolic PDEs have found a lot of attention for smoothing and restoration purposes, see e.g. 113]. To restore images these equations frequently arise from gradient descent methods applied to variational problems. Image smoothing by parabolic PDEs is closely related to the scale-space concept where one embeds the original image into a family of subsequently simpler , more global representations of it. This idea plays a fundamental role for extracting semantically important information. The pioneering work of Alvarez, Guichard, Lions and Morel 11] has demonstrated that all scale-spaces fulllling a few fairly natural axioms are governed by parabolic PDEs with the original image as initial condition. Within this framework, two classes can be justiied in a rigorous way as scale-spaces: the linear diiusion equation with constant dif-fusivity and nonlinear so-called morphological PDEs. All these methods satisfy a monotony axiom as smoothing requirement which states that, if one image is brighter than another, then this order is preserved during the entire scale-space evolution. An interesting class of parabolic equations which pursue both scale-space and restoration intentions is given by nonlinear diiusion lters. Methods of this type have been proposed for the rst time by Perona and Malik in 1987 190]. In v vi PREFACE order to smooth the image and to simultaneously enhance semantically important features such as edges, they apply a diiusion process whose diiusivity is steered by local image properties. These lters are diicult to analyse mathematically , as they may act locally like a backward diiusion process. …
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