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.
Highlights
Photoacoustic tomography (PAT) is a novel coupled-physics modality for non-invasive biomedical imaging that combines the high contrast of optical tomography with the high spatial resolution of acoustic imaging [7, 68, 102, 105, 107]
We present generalizations of this approach and, in particular, we discuss the combination of photoacoustic tomography (PAT)-vaguelette soft-thresholding with a total variation (TV) prior
We develop a numerically efficient reconstruction method for PAT with planar geometry that effectively deals with noisy data g = Uh + z, where regularization is achieved by enforcing sparsity constraints in the reconstruction with respect to wavelet coefficients of h
Summary
Photoacoustic tomography (PAT) is a novel coupled-physics (hybrid) modality for non-invasive biomedical imaging that combines the high contrast of optical tomography with the high spatial resolution of acoustic imaging [7, 68, 102, 105, 107]. In the case of exact data Uh, several approaches have been derived for solving the inverse problem of PAT considering different acquisition geometries This includes time reversal (see [16, 41, 62, 81, 99]), Fourier domain algorithms (see [2, 4, 13, 38, 61, 63, 67, 71, 82, 108]), analytic reconstruction formulas of back-projection type (see [40, 41, 53, 55, 60, 70, 77, 84, 106]), as well as iterative approaches [32, 59, 85, 87, 93, 95, 103, 104, 109]. To the best of our knowledge, this paper is the first to provide a WVD for PAT as well as an efficient direct implementation of sparse regularization using wavelets for that case
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