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By Joachim Weickert

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Extensions to anisotropic vector-valued models with a common tensorvalued structure descriptor for all channels have been investigated by Weickert [422]. 5 Numerical aspects For nonlinear diffusion filtering numerous numerical methods have been applied: Finite element techniques are described in [367, 391, 34, 216]. B¨ansch and Mikula reported a significant speed-up by supplementing them with an adaptive mesh coarsening [34]. Neural network approximations to nonlinear diffusion filters are investigated by Cottet [100, 99] and Fischl and Schwartz [137].

They used convolution masks that depended on the underlying image structure. Related statistical approaches were proposed by Knutsson, Wilson and Granlund [237]. These ideas have been further developed by Nitzberg and Shiota [310], Lindeberg and G˚ arding [259], and Yang et al. [443]. 6. Anisotropic diffusion filters usually apply spatial regularization strategies11 . A general theoretical framework for spatially regularized anisotropic diffusion filters will be presented in the remaining chapters of this book.

Due to their local behaviour, they are well-suited for parallel architectures. Nevertheless, they suffer from the fact that fairly small time step sizes are needed in order to ensure stability. Semi-implicit schemes – which approximate the diffusivity or the diffusion tensor in an explicit way and the rest implicitly – are considered in [81]. They possess much better stability properties. A fast multigrid technique using a pyramid algorithm for the Perona–Malik filter has been studied by Acton et al.

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