  We study the problem of deconvolution for light-sheet microscopy, where the
data is corrupted by spatially varying blur and a combination of Poisson and
Gaussian noise. The spatial variation of the point spread function (PSF) of a
light-sheet microscope is determined by the interaction between the excitation
sheet and the detection objective PSF. First, we introduce a model of the image
formation process that incorporates this interaction, therefore capturing the
main characteristics of this imaging modality. Then, we formulate a variational
model that accounts for the combination of Poisson and Gaussian noise through a
data fidelity term consisting of the infimal convolution of the single noise
fidelities, first introduced in L. Calatroni et al. "Infimal convolution of
data discrepancies for mixed noise removal", SIAM Journal on Imaging Sciences
10.3 (2017), 1196-1233. We establish convergence rates in a Bregman distance
under a source condition for the infimal convolution fidelity and a discrepancy
principle for choosing the value of the regularisation parameter. The inverse
problem is solved by applying the primal-dual hybrid gradient (PDHG) algorithm
in a novel way. Finally, numerical experiments performed on both simulated and
real data show superior reconstruction results in comparison with other
methods.
