Derivative Free Regularization for Filtering and Enhancing

Image denoising and texture enhancement are of central interest in medical image processing. Variational regularization methods are common tools for such applications. These methods aim for decomposing an image into two or more components (cartoon part, texture and noise) by minimizing appropriate energy functionals. Such functionals consist of the sum of a similarity term, that compares data with smoothed images, and a regularization term, that ensures smoothness of the filtered image. A frequently used regularization term is total variation, it ensures that edges in images are preserved. Recently Y. Meyer introduced the G-norm to capture textures in images. The calculation of this norm is not straight forward, hence we aim for approximations of such texture capturing norms, which can be calculated numerically more easily. The main idea is based on the fact that Sobolev semi norms and the total variation can be written as the limit of a derivative free but singular double integral. The mathematical foundations of this result have been developed by J. Bougain, H. Brezis, P. Mironescu, and J. Davila. One aim is to establish such approximations for the dual semi norms, that characterize texture features in images. The theoretical results are useful to develop new numerical methods for texture enhancement and noise filtering. In this way for instance the famouse Chambolle algorithm can be put on a greater perspective and enable for efficient generalization. We are interested in convergence, existence and characterization of minimizing elements of such variational methods.