Several methods in image processing are based on PDEs, that need to be solved numerically. We are interested in PDE-based methods for multidimensional images. The scope is to use the geometric properties of the data as a base for the construction of the numerical integrator. An important application is biomedical imaging (diffusion tensor imaging, image registration).

 

In diffusion tensor imaging, each voxel corresponds to a 3x3 symmetric matrix that can be represented by an ellipsoid. Below, an example of a synthetic field, to which we impose noise and then denoise, by a method that preserves eigenvalues (images by master student Sjur Kvammen). Diffusion tensor imaging is used, among other things to track neuronal fibers within the brain (image from the web).