Efficient regularization with wavelet sparsity constraints in PAT

TL;DR

This paper introduces a wavelet sparsity regularization method for photoacoustic tomography (PAT) that achieves statistically optimal reconstruction, especially in noisy conditions, by combining wavelet-vaguelette decomposition with soft-thresholding.

Contribution

The paper develops a novel wavelet-vaguelette decomposition for PAT and integrates it with soft-thresholding for improved regularization and noise robustness in image reconstruction.

Findings

The method is statistically optimal for white noise when the unknown is in a Besov space.

Efficient implementation enables fast image reconstruction.

Combining vaguelette soft-thresholding with TV prior enhances reconstruction quality.

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 vaguelette soft-thresholding with a TV prior. We also provide an efficient implementation of the vaguelette transform that leads to fast image reconstruction algorithms supported by numerical results.