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Well-posedness results are not known up to now, since the theory of viscosity solutions is no longer applicable. All these preceding modifications are at the expense of renouncing the morphological invariance of the genuine operators (and also affine invariance in the case of [364, 365, 304], unless an “affine invariant gradient” [231, 315] is used). If one wants to stay within the morphological framework one can combine different morphological processes, for instance MCM and dilation/erosion. This leads to a process which is useful for analysing components of shape [228, 230, 383, 384, 453], and which is called entropy scale-space or reaction–diffusion scale-space.

2 In analogy to the fact that Gaussian-type functions k(x, t) = a exp( |x| ) are the 4t only rotationally symmetric kernels which are separable with respect to convolution, van den Boomgaard proves that the quadratic structuring functions b(x, t) are the only rotationally invariant structuring functions which are separable with respect to dilation [50, 51]. 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.

The temporal evolution of an image under such an evolution resembles mean curvature motion; see Fig. 6(a). Besides the name affine invariant heat flow, this equation is also called affine shortening flow, affine morphological scale-space (AMSS), and fundamental equation in image processing. This image evolution equation has been discovered independently of and simultaneously with the curve evolution approach of Sapiro and Tannenbaum by Alvarez, Guichard, Lions and Morel [12] via an axiomatic scale-space approach.

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