  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.
