Inverse Problems Regularized by

Abstract: Modelling signals as sparse in a proper domain has proved useful in many signal processing tasks and, in this paper, we show how sparsity can be used to solve inverse prob-lems. 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 reconstrcution of planar domains from samples. We localize diffusion sources using a variation of the ‘reciprocity gap ’ method and use it also to estimate the acti-vation 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