Efficient regularization with wavelet sparsity constraints in photoacoustic tomography. (17th January 2018)

• Record Type: Journal Article
• Authors: Frikel, Jürgen; Haltmeier, Markus
• Is Part Of: Inverse problems. Volume 34:Number 2(2018:Feb.)
• Publication Date: 2018-01-17
• Subjects: photoacoustic tomography -- wavelet-vaguelette decomposition -- variational regularization -- sparsity constraints -- image reconstruction; Inverse problems (Differential equations) -- Periodicals; 515.357
• DOI: 10.1088/1361-6420/aaa0ac
• Languages: English
• ISSNs: 0266-5611

Abstract

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.