Tensor-based image diffusions derived from generalizations of the Total Variation and Beltrami Functionals

We introduce a novel functional for vector-valued images that generalizes several variational methods, such as the Total Variation and Beltrami Functionals. This functional is based on the structure tensor that describes the geometry of image structures within the neighborhood of each point. We first generalize the Beltrami functional based on the image patches and using embeddings in high dimensional spaces. Proceeding to the most general form of the proposed functional, we prove that its minimization leads to a nonlinear anisotropic diffusion that is regularized, in the sense that its diffusion tensor contains convolutions with a kernel. Using this result we propose two novel diffusion methods, the Generalized Beltrami Flow and the Tensor Total Variation. These methods combine the advantages of the variational approaches with those of the tensor-based diffusion approaches.