Sparsifying transformations of photoacoustic signals enabling compressed sensing algorithms

Compressed sensing allows performing much fewer measurements than advised by the Shannon sampling theory. This is surprising because it requires the solution of a system of equations with much fewer equations than unknowns. This is possible if one can assume sparsity of the solution, which means that only a few components of the solution are significantly different from zero. An important ingredient for compressed sensing is the restricted isometry property (RIP) of the sensing matrix, which is satisfied for certain types of random measurement ensembles. Then a sparse solution can be found by minimizing the ℓ¹-norm. Using standard approaches, photoacoustic imaging generally neither satisfies sparsity of the data nor the RIP. Therefore, no theoretical recovery guarantees could be given. Despite ℓ¹-minimization has been used for photoacoustic image reconstruction, only marginal improvements in comparison to classical photoacoustic reconstruction have been observed. We propose the application of a sparsifying temporal transformation to the detected pressure signals, which allows obtaining theoretical recovery guarantees for our compressed sensing scheme. Such a sparsifying transform can be found because spatial and temporal evolution of the pressure wave are not independent, but connected by the wave equation. We give an example of a sparsifying transform and apply our compressed sensing scheme to reconstruct images from simulated data.