Diffusion and regularization of vector- and matrix-valued images

The goal of this paper is to present a unified description of diffusion and regularization techniques for vector-valued as well a matrix-valued data fields. In the vector-valued setting, we first review a number of existing methods and classify them into linear and nonlinear as well as isotropic and anisotopic methods. For these approaches we present corresponding regularization methods. This taxonomy is applied to the design of regularization methods for variational motion analysis in image sequences. Our vector-valued framework is then extended to the smoothing of positive semidefinite matrix fields. In this context a novel class of anisotropic diffusion and regularization methods is derived and it is shown that suitable algorithmic realizations preserve the positive semidefinitness of the matrix field without any additional constraints. As an application, we present an anisotopic nonlinear structure tensor and illustrate its advantages over the linear structure tensor