Controlled Anisotropic Diiusion

Anisotropic diiusion has been extensively used as an eecient nonlinear ltering technique for simultaneously performing contrast enhancement and noise reduction, and for deriving consistent scale-space image descriptions. In this paper, we present a general study of anisotropic diiusion schemes based on diierential group-invariant representations of local image structure. We show that the local geometry (i.e. shape and scale) of the photometric surface is intrinsically speciied by two dual families of curves, respectively consisting of isophotes and stream lines, which remain invariant under isometries in the image domain. Within this framework, anisotropic diiusive processes induce a deformation ow on the network of isophotes and stream lines. Deriving the general expression of this ow leads to identifying canonical forms for admissible conduction functions, that yield an optimal and stable preservation of signiicant image structures. Moreover, relating scale to directional variations of isophote density results in controlling the diiusion dynamics by means of a heterogeneous damping density which allows to adaptively reduce diiusion speed in the vicinity of high gradient lines while increasing it within stationary intensity domains. Finally, these results are extended to arbitrary image dimensions.