Michael Dr. Zwanger

Riemannian Anisotropic Diffusion for Tensor Valued Images

Tensor valued images, for instance originating from diffusion tensor magnetic resonance imaging (DT-MRI), have become more and more important over the last couple of years. Due to the nonlinear structure of such data it is nontrivial to adapt well-established image processing techniques to them. In this contribution we derive anisotropic diffusion equations for tensor-valued images based on the intrinsic Riemannian geometric structure of the space of symmetric positive tensors. In contrast to anisotropic diffusion approaches proposed so far, which are based on the Euclidian metric, our approach considers the nonlinear structure of positive definite tensors by means of the intrinsic Riemannian metric. Together with an intrinsic numerical scheme our approach overcomes a main drawback of former proposed anisotropic diffusion approaches, the so-called eigenvalue swelling effect. Experiments on synthetic data as well as real DT-MRI data demonstrate the value of a sound differential geometric formulation of diffusion processes for tensor valued data.