Axis 4: Sparse, non-local, patch-based approaches and harmonic analysis

This research axis is motivated by the problems of data analysis in high dimension (signals and images) organized in the form of vectors or matrices of large size.

The first part of this axis focuses on sparse-based approaches for the analysis of high-dimensional data when they can be well approximated in a space of low dimension. We are interested in the development of variational models which naturally promote this sparsity to a preselected representation (wavelets, gradient fields, etc.) or learned on an external dataset. This representation (the space of small size) can also be selected by the model itself, it is called dictionary learning which is usually expressed as a matrix factorization problem. For examples, we focus on approximation methods based on thresholding wavelets in image processing, as well as variable selection problems with the Lasso (Least Absolute Shrinkage and Selection Operator). Beyond the development of such models, we are seeking to understand their behavior. In particular, in the first example, we are interested in conditions ensuring that a signal can be recovered, or that its reconstruction error can be bounded (robustness). In the second example, we focus on the conditions ensuring the identification of good variables.

A second part concerns the analysis of repeating patterns in the case of images or multidimensional signals. These repetitions indeed provide key information for the processing or interpretation of the signal. In particular, we are interested in non-local approaches used in image processing that rely on the similarity of patterns using patches (typically small 8 by 8 rectangular windows). The term non-local means here that only the content of the patches is relevant, regardless of the spatial location. Part of our work involves the design and choice of metrics ensuring a robust matching of patterns as well as the implementation of efficient search algorithms. We are also interested in the analysis of these repetitions, typically on the connectivity graph. Finally, we aim at developing and studying new variational models taking into account these non-local information.