Nonlinear Tensor Diffusion in Image Processing

Abstract

The paper deals with the nonlinear tensor diffusion which enhances image structure coherence. First we briefly describe the diffusion model and provide its basic properties. Further we build a semi-implicit finite volume scheme for the above mentioned model with the help of a co-volume mesh. This method is well-known as diamond-cell method owing to the choice of co-volume as a diamond-shaped polygon. The scheme design is shown for $2$D as well as $3$D case, cf. [3,4]. Then the convergence and error estimate analysis for $2$D scheme is presented, cf. [3,2]. The convergence proof is based on Kolmogorov's compactness theorem and a bounding of a gradient in tangential direction by means a gradient in normal direction. We proved that the error of the numerical solution in $L^2$-norm is of order $h$, where $h$ is a spatial step under the natural relation $k \approx h^2$ with a time step $k$. Last part is devoted to results of computational experiments. They confirm the usefulness this diffusion type especially as pre-processed algorithm. Let us note that this numerical technique was successfully applied within the framework of EU projects in such a way for the structure segmentation in zebrafish embryogenesis, cf [5].