Anisotropic $\alpha$-Kernels and Associated Flows

Abstract

The Laplacian raised to fractional powers can be used to generate scale spaces as was shown in recent literature by Duits, Felsberg, Florack, and Platel [$\alpha$ scale spaces on a bounded domain, in Scale Space Methods in Computer Vision, L. D. Griffin and M. Lillholm, eds., Lecture Notes in Comput. Sci. 2695, Springer, Berlin, Heidelberg, 2003, pp. 494–510] and Duits, Florack, de Graaf, and ter Haar Romeny [J. Math. Imaging Vision, 20 (2004), pp. 267–298]. In this paper, we study the anisotropic diffusion processes by defining new generators that are fractional powers of an anisotropic scale space generator. This is done in a general framework that allows us to explain the relation between a differential operator that generates the flow and the generators that are constructed from its fractional powers. We then generalize this to any other function of the operator. We discuss important issues involved in the numerical implementation of this framework and present several examples of fractional versions of the Perona–Malik and Beltrami flows along with their properties.