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

Abstract

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 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 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 , Ag], 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 Grüneisen parameter, the optical absorption and the diffusion coefficient are all unknown.

6. A variational method for quantitative photoacoustic tomography with piecewise constant coefficients

“…However, in [1], the authors were able to recover µ (independently of the light transfer model used) from a single measurement of E (again using a sparsity method, assuming different degrees of smoothness of the coefficients and the fluence). In [6,7] it was shown that this problem is uniquely solvable if two measurements of E (corresponding to well-chosen boundary illuminations g 1 , g 2 ) are available.…”

Multispectral reconstruction methods for quantitative photoacoustic tomography

“…10,11 However, PA images are not directly representative of the underlying tissue structure because spatial variations in the illumination pattern cause the optical energy to be absorbed non-uniformly throughout the tissue. If φ(r) is the optical fluence generated by the laser within the tissue, then we have…”

Efficient Inclusion of Total Variation Type Priors in Quantitative Photoacoustic Tomography

“…Numerically speaking, multiplying a vector by the Jacobian J h κ,μ or its transpose is relatively cheap: one only needs to perform elementwise multiplications of vectors and matrix-vector multiplications with sparse matrices, and to operate with the inverse of the FEM system matrix K on a vector. However, when the number of degrees of freedom and/or illuminations is very high, the sizes of the matrices G (1) and G (2) may become impractically large. In this case, one can continue the analysis to avoid forming the matrices G (1) and G (2) altogether, but the details are omitted here for brevity.…”

“…However, when the number of degrees of freedom and/or illuminations is very high, the sizes of the matrices G (1) and G (2) may become impractically large. In this case, one can continue the analysis to avoid forming the matrices G (1) and G (2) altogether, but the details are omitted here for brevity.…”