Inverse Problems Regularized by Sparsity
Pier Luigi Dragotti (Imperial College London, UK)

Modelling signals as sparse in a proper domain has proved useful in many signal processing tasks and, in this talk, we show how sparsity can be used to solve inverse problems.

We first recall that many inverse problems involve the reconstruction of continuous-time or continuous-space signals from discrete measurements and show how to relate
the discrete measurements to some properties of the original signal (e.g., its Fourier transform at specific frequencies or its first L moments). Given this partial knowledge of the original signal, we then solve the inverse problem using sparsity.

We focus on two specific problems which have important practical implications: localization of diffusion sources from sensor measurements and reconstrction of planar domains from samples.

We localize diffusion sourses using a variation of the 'reciprocity gap' method and use it also to estimate the activation time of the source. We validate the method by estimating the location and activation time of a heat source from real measurements. Finally, we show how to reconstruct specific planar domains which are driven by sparsity models and apply this approach to
enhance the resolution of natural images.

This is joint work with T. Blu (CUHK), H. Pan (CUHK), J. A. Uriguen (ICL), J. Onativia Bravo (ICL) and J. Murray-Bruce (ICL).
This work is supported by the European Research Council (ERC) starting investigator award Nr. 277800 (RecoSamp).