TY - JOUR A1 - Frikel, Jürgen A1 - Quinto, Eric Todd T1 - Artifacts in Incomplete Data Tomography with Applications to Photoacoustic Tomography and Sonar JF - SIAM Journal on Applied Mathematics N2 - We develop a paradigm using microlocal analysis that allows one to characterize the visible and added singularities in a broad range of incomplete data tomography problems. We give precise characterizations for photoacoustic and thermoacoustic tomography and sonar, and provide artifact reduction strategies. In particular, our theorems show that it is better to arrange sonar detectors so that the boundary of the set of detectors does not have corners and is smooth. To illustrate our results, we provide reconstructions from synthetic spherical mean data as well as from experimental photoacoustic data. KW - computed tomography KW - Fourier integral operators KW - Lambda tomography KW - limited angle tomography KW - microlocal analysis KW - photoacoustic tomography KW - radon transforms KW - sonar KW - spherical mean KW - thermoacoustic tomography Y1 - 2015 U6 - https://doi.org/10.1137/140977709 SN - 1095-712X SN - 0036-1399 VL - 75 IS - 2 SP - 703 EP - 725 PB - Society for Industrial and Applied Mathematics CY - Philadelphia, Pa. ER - TY - JOUR A1 - Frikel, Jürgen A1 - Haltmeier, Markus T1 - Efficient regularization with wavelet sparsity constraints in photoacoustic tomography JF - Inverse Problems N2 - 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. KW - Image reconstruction KW - photoacoustic tomography KW - sparsity constraints KW - variational regularization KW - wavelet-vaguelette decomposition Y1 - 2018 U6 - https://doi.org/10.1088/1361-6420/aaa0ac N1 - Corresponding authors: Jürgen Frikel and Markus Haltmeier VL - 34 IS - 2 SP - 1 EP - 28 ER -