This paper proposes an alternative to partial differential equations (PDEs) for the solution of the optical flow problem. The problem is modeled using the heat transfer process. Instead of using PDEs, we propose to use the global equation of heat conservation. We use a computational algebraic topology-based image model which allows us to encode some underlying physical laws 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 and provides a physical explanation of each solving step. Experimental results are presented.

A new method for the deformation of curves is presented. It is based upon a decomposition of the linear elasticity problem into basic physical laws.

this paper shows that LH performs better than the Quadratic Tetrahedron (10 nodes) even in a static linear elastic analysis

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.

This paper proposes an alternative to partial differential equations (PDEs) for the solution of diffusion (Perona and Malik scheme), using the heat transfer problem. Traditionally, the method for solving such physics-based problems is to discretize and solve a PDE by a mathematical process. We propose to use the global heat equation and decompose it into simpler laws. Some of these laws admit an exact global version since they arise from conservation principles while the assumptions on the others can be made wisely, taking into account knowledge about the problem. A computational algebraic topology-based image model allows us to write directly discrete equations. 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. Finally, we present results for non linear diffusion.