Disjoint sparsity for signal separation and applications to hybrid inverse problems in medical imaging

The main focus of this work is the reconstruction of the signals f and gi, i=1,…,N, from the knowledge of their sums hi=f+gi, under the assumption that f and the gi's can be sparsely represented with respect to two different dictionaries Af and Ag. This generalizes the well-known “morphological component analysis” to a multi-measurement setting. The main result of the paper states that f and the gi's can be uniquely and stably reconstructed by finding sparse representations of hi for every i with respect to the concatenated dictionary [Af,Ag], provided that enough incoherent measurements gi are available. The incoherence is measured in terms of their mutual disjoint sparsity. This method finds applications in the reconstruction procedures of several hybrid imaging inverse problems, where internal data are measured. These measurements usually consist of the main unknown multiplied by other unknown quantities, and so the disjoint sparsity approach can be directly applied. As an example, we show how to apply the method to the reconstruction in quantitative photoacoustic tomography, also in the case when the Grüneisen parameter, the optical absorption and the diffusion coefficient are all unknown.