The use of sparse signal and image representations in inverse problems and in new artwork classification methods.

"Compressed Sensing" (CS) is a new paradigm for compression and sampling of large quantities of data. With this technique it is possible to reconstruct a signal from far fewer measuments than is required traditionally. A key ingredient for this method to work is that the required solution is sparse in a known basis. Reconstruction is then performed through minimization of a cost function with L1 penalty. A first theme of this proposal will consist of extending the CS framework to those operators that are encountered in realistic inverse problems (instead of the usual random matrix setting). A second problem that stands in the way of an immediate application of the CS paradigm is the slowness of the algorithms for L1 minimization (as compared to the usual linear methods that are often used in inverse problems). The development of better performing algorithms is thus also a crucial part of this research programme. In a second part, sparse representations will be used for the analysis of paintings, in particular for authentication and dating. First we will focus on a theoretical study of directional transforms (complex wavelets, curvelets, contourlets, shearlets,...) that are capable of describing geometrical properties of images. When we have gained a clear understanding of the properties that are needed for these applications, we will develop and implement a suitable directional transform, useable in a automized analysis process for paintings.
This research runs in close collaboration with Prof. I. Daubechies (Princeton University) who will spend about six months at the VUB in the 2010. This will also be a good occasion to organize a workshop on the theme of sparsity in applied mathematics in Brussels.