Discontinuity Preserving Filtering over Analytic Manifolds

Discontinuity preserving filtering of images is an important low-level vision task. With the development of new imaging techniques like diffusion tensor imaging (DTI), where the data does not lie in a vector space, previous methods like the original mean shift are not applicable. In this paper, we use the nonlinear mean shift algorithm to develop filtering methods for data lying on analytic manifolds. We work out the computational details of using mean shift on Sym_n ⁺, the manifold of n times n symmetric positive definite matrices. We apply our algorithm to chromatic noise filtering, which requires mean shift over the Grassmann manifold G_3,1, and obtain better results then standard mean shift filtering. We also use our method for DTI filtering, which requires smoothing over Sym₃ ⁺.