This paper presents a scheme for video denoising by diffusion of gray levels in the video domain, based on the Computational Algebraic Topology (CAT) image model. Contrary to usual approaches, using the heat transfer PDE and discretizing and solving it by a purely mathematical process, our approach considers the global expression of the heat transfer and decomposes it into elementary physical laws. Some of these laws link global quantities, integrated on some domains. They are called conservative relations, and lead to error-free expressions. The other laws depend on metric quantitites and require approximations to be expressed in this scheme. However, as every step of the resolution process has a physical interpretation, the approximations can be chosen wisely depending of the wanted behavior of the algorithm. We propose in this paper a nonlinear diffusion algorithm based on the extension to video of an existing 2D algorithm thanks to the flexibility of the topological support. After recalling the physical model for diffusion and the decomposition into basic laws, these laws are modeled in the CAT image model, yielding a numerical scheme. Finally, this model is validated with experimental results and extensions of this work are proposed.
One physical process involved in many computer vision problems is the heat diffusion process. Such Partial differential equations are continuous and have to be discretized by some techniques, mostly mathematical processes like finite differences or finite elements. The continuous domain is subdivided into sub-domains in which there is only one value. The diffusion equation comes from the energy conservation then it is valid on a whole domain. We use the global equation instead of discretize the PDE obtained by a limit process on this global equation. To encode these physical global values over pixels of different dimensions, we use a computational algebraic topology (CAT)-based image model. This model has been proposed by Ziou and Allili and used for the deformation of curves and optical flow. It introduces the image support as a decomposition in terms of points, edges, surfaces, volumes, etc. Images of any dimensions can then be handled. After decomposing the physical principles of the heat transfer into basic laws, we recall the CAT-based image model and use it to encode the basic laws. We then present experimental results for nonlinear graylevel diffusion for denoising, ensuring thin features preservation.