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

TL;DR

This paper introduces a disjoint sparsity method for reconstructing multiple signals from their sums using sparse representations in different dictionaries, with applications to hybrid inverse problems in medical imaging.

Contribution

It generalizes morphological component analysis to multi-measurement settings, enabling stable and unique reconstruction under incoherence conditions in hybrid imaging.

Findings

Successful application to quantitative photoacoustic tomography.

Reconstruction remains effective even when multiple parameters are unknown.

Method ensures stable recovery with sufficient incoherent measurements.

Abstract

The main focus of this work is the reconstruction of the signals $f$ and $g_{i}$, $i=1,...,N$, from the knowledge of their sums $h_{i}=f+g_{i}$, under the assumption that $f$ and the $g_{i}$'s can be sparsely represented with respect to two different dictionaries $A_{f}$ and $A_{g}$. This generalizes the well-known "morphological component analysis" to a multi-measurement setting. The main result of the paper states that $f$ and the $g_{i}$'s can be uniquely and stably reconstructed by finding sparse representations of $h_{i}$ for every $i$ with respect to the concatenated dictionary $[A_{f},A_{g}]$, provided that enough incoherent measurements $g_{i}$ 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 Gr\"uneisen parameter, the optical absorption and the diffusion coefficient are all unknown.