Efficient regularization with wavelet sparsity constraints in photoacoustic tomography

• Published in: Inverse problems Vol. 34; no. 2; pp. 24006 - 24033
• Main Authors: Frikel, Jürgen, Haltmeier, Markus
• Format: Journal Article
• Language: English
• Published: IOP Publishing 01.02.2018
• Subjects: image reconstruction; photoacoustic tomography; sparsity constraints; variational regularization; wavelet-vaguelette decomposition
• ISSN: 0266-5611; 1361-6420
• DOI: 10.1088/1361-6420/aaa0ac

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.