By Joachim Weickert
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2, where also many examples can be found. 4 Generalizations Higher dimensions. It is easily seen that many of the previous results can be generalized to higher dimensions. g. medical image sequences from computerized tomography (CT) or magnetic resonance imaging (MRI), or when applying diffusion filters to the postprocessing of fluctuating higher-dimensional numerical data. The first three-dimensional nonlinear diffusion filters have been investigated by Gerig et al.  in the isotropic case and by Rambaux and Gar¸con  in the anisotropic case.
61) with homogeneous Neumann boundary conditions. Equations of this type are investigated by Richardson and Mitter . 56), similar problems can arise: The functional Ff is not jointly convex in u and v, so it may have many local minima and a gradient descent algorithm may get trapped in a poor local minimum. Well-posedness results for this system have not been obtained up to now, but a maximum–minimum principle and a local stability proof have been established. Another diffusion–reaction system is studied by Shah [375, 376].
111) For this reason they call the AMSS equation also fundamental equation in image analysis. Simplifications of this axiomatic and related axioms for shape scale-spaces can be found in . The scale-space reasoning of Sapiro and Tannenbaum investigates properties of the curve evolution, see  and the references therein. Based on results of [229, 24] they point out that the Euclidean absolute curvature decreases as well as the number of extrema and inflection points of curvature. Moreover, a shape inclusion principle holds.