@article{FrikelHaltmeier, author = {Frikel, J{\"u}rgen and Haltmeier, Markus}, title = {Efficient regularization with wavelet sparsity constraints in photoacoustic tomography}, series = {Inverse Problems}, volume = {34}, journal = {Inverse Problems}, number = {2}, doi = {10.1088/1361-6420/aaa0ac}, pages = {1 -- 28}, abstract = {In this paper, we consider the reconstruction problem of photoacoustic tomography (PAT) with a flat observation surface. We develop a direct reconstruction method that employs regularization with wavelet sparsity constraints. To that end, we derive a wavelet-vaguelette decomposition (WVD) for the PAT forward operator and a corresponding explicit reconstruction formula in the case of exact data. In the case of noisy data, we combine the WVD reconstruction formula with soft-thresholding, which yields a spatially adaptive estimation method. We demonstrate that our method is statistically optimal for white random noise if the unknown function is assumed to lie in any Besov-ball. We present generalizations of this approach and, in particular, we discuss the combination of PAT-vaguelette soft-thresholding with a total variation (TV) prior. We also provide an efficient implementation of the PAT-vaguelette transform that leads to fast image reconstruction algorithms supported by numerical results.}, language = {en} }