| This thesis proposes an alternative to partial differential equations (PDEs) for the solution of some problems in computer vision based on the heat transfer equation. Traditionally, the method for solving such physics-based problems is to discretize and solve a PDE by a purely mathematical process. Instead of using the PDE, we propose to use the global heat equation and to decompose it into basic laws. We show that some of these laws admit an exact global version since they arise from balance principles. We also show that the assumptions made on the other basic laws can be made wisely, taking into account knowledge about the problem and the domain. We use a computational algebraic topology-based image model which allows us to encode a physical conservative law by linking a global value on a domain with values on its boundary. The numerical scheme is derived in a straightforward way from the problem modeled. It thus provides a physical explanation of each solving step in the solution. We apply the scheme to various applications: image reconstruction from the Laplacian, optical flow computation, denoising by graylevel and multispectral diffusion and inpainting which are all modeled with the heat transfer equation. |
