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Multiscale Analysis for Images on Riemannian Manifolds

In this paper we study multiscale analyses for images defined on Riemannian manifolds and extend the axiomatic approach proposed by Alvarez, Guichard, Lions, and Morel to this general case. This covers the case of two- and three-dimensional images and video sequences. After obtaining the general classification, we consider the case of morphological scale spaces, which are given in terms of geometric equations, and the linear case given by the Laplace--Beltrami flow. We consider in some detail the case of image metrics given in terms of the structure tensor and compute some cases of such a tensor for video. Then we comment on the connections with variational formulations of image diffusion comparing the anisotropies that appear. Finally, we include numerical experiments illustrating some of the models. Namely, we compare some examples for still images using the Laplace--Beltrami flow and some variational models. We also consider several examples in video: the mean curvature motion and the extension of the ...