Diffusion-Like Reconstruction Schemes from Linear Data Models

In this paper we extend anisotropic diffusion with a diffusion tensor to be applicable to data that is well modeled by linear models. We focus on its variational theory, and investigate simple discretizations and their performance on synthetic data fulfilling the underlying linear models. To this end, we first show that standard anisotropic diffusion with a diffusion tensor is directly linked to a data model describing single orientations. In the case of spatio-temporal data this model is the well known brightness constancy constraint equation often used to estimate optical flow. Using this observation, we construct extended anisotropic diffusion schemes that are based on more general linear models. These schemes can be thought of as higher order anisotropic diffusion. As an example we construct schemes for noise reduction in the case of two orientations in 2d images. By comparison to the denoising result via standard single orientation anisotropic diffusion, we demonstrate the better suited behavior of the novel schemes for double orientation data.