By Joachim Weickert
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Additional info for Anisotropic diffusion in image processing
14 Invariance under rotations is only satisfied for a disc centered in 0 as structuring element. 36 CHAPTER 1. PARTIAL DIFFERENTIAL EQUATIONS A useful tool for understanding this similarity and many other analogies between morphology and linear systems theory is the slope transform. This generalization of the Legendre transform is the morphological equivalent of the Fourier transform. It has been discovered simultaneously by Dorst and van den Boomgaard  and by Maragos  in slightly differing formulations.
Nevertheless, PDE formulations for classic morphological processes have been discovered recently by Brockett and Maragos , van den Boomgaard , Arehart et al.  and Alvarez et al. . This section surveys the basic ideas and elementary operations of binary and grey-scale morphology, presents its PDE representations for images and curves, and summarizes the results concerning well-posedness and scale-space properties. Afterwards numerical aspects of the PDE formulation of these processes are discussed, and generalizations are sketched which comprise the morphological equivalent of Gaussian scale-space.
72) are closely related to shape and curve evolutions. 73) where p is the parametrization and t is the evolution parameter. We assume that C evolves in outer normal direction n with speed β, which may be a function of p ,Cpp ) its curvature κ := det(C : |Cp |3 ∂t C = β(κ) · n, C(p, 0) = C0 (p). 75) One can embed the curve C(p, t) in an image u(x, t) in such a way that C is just a level curve of u. The corresponding evolution for u is given by [319, 362, 16] ∂t u = β(curv(u)) · |∇u|. 76) ∇u . 74), because it is written in terms of a fixed coordinate system.
Anisotropic diffusion in image processing by Joachim Weickert
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