Young researcher ANR program: Numerical Schemes using Lattice Basis Reduction (NSLBR)

In this project, we will develop, study, and distribute in open source software, new methods for the discretization of anisotropic Partial Differential Equations (PDEs) on two or three dimensional grids. We will focus on the setting where at least one of the operators involved in the PDE is anisotropic, in the sense that it involves position dependent, non axis-aligned, privileged directions. Grid discretizations are natural in medical image or volume processing, and numerous fundamental isotropic PDE operators are known to have efficient representations in this form.

Here is below an example of application of the techniques developped in our project. The orginal fingerprint image is obviously noisy, which can be problematic for its automatic analysis. Standard isotropic diffusion techniques to denoise the image will blur the image textures, which is also problematic here as the textures are fine. Wiser anisotropic diffusion techniques blur the image orthogonaly to the textures, which reduces the noise and preserves the textures. If the level of anisotropy is locally strong, the numerical models to perform properly anisotropic diffusion can however be particularly time consuming. The solution developped in our project consists in using Lattice Basis Reduction (LBR) to make the computational burden resonable in this context.
 Image before anisotropic filtering    
Image after anisotropic filtering
 Original fingerprint image
   Image after LBR based anisotropic diffusion

A taste of the strategy is given below. For different unit balls showing how local image intensities are diffused, we show the neighbors, in the pixel grid, which are used to perform local diffusion (More details here). The idea to only pick-up points in the grid, and not to refine the grid, is central in the project but opens different questions in mathematics. Such strategies can also be used for different applications (see the publications).