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Deblurring is a fundamental inverse problem in bioimaging. It requiresmodelling the point spread function (PSF), which captures the opticaldistortions entailed by the image formation process. The PSF limits the spatialresolution attainable for a given microscope. However, recent applicationsrequire a higher resolution, and have prompted the development ofsuper-resolution techniques to achieve sub-pixel accuracy. This requirementrestricts the class of suitable PSF models to analog ones. In addition,deblurring is computationally intensive, hence further requiringcomputationally efficient models. A custom candidate fitting both requirementsis the Gaussian model. However, this model cannot capture the rich tailstructures found in both theoretical and empirical PSFs. In this paper, we aimat improving the reconstruction accuracy beyond the Gaussian model, whilepreserving its computational efficiency. We introduce a new class of analog PSFmodels based on Gaussian mixtures. The number of Gaussian kernels controls boththe modelling accuracy and the computational efficiency of the model: the lowerthe number of kernels, the lower accuracy and the higher efficiency. To explorethe accuracy--efficiency trade-off, we propose a variational formulation of thePSF calibration problem, where a convex sparsity-inducing penalty on the numberof Gaussian kernels allows trading accuracy for efficiency. We derive anefficient algorithm based on a fully-split formulation of alternating splitBregman. We assess our framework on synthetic and real data and demonstrate abetter reconstruction accuracy in both geometry and photometry in point sourcelocalisation---a fundamental inverse problem in fluorescence microscopy.

Modelling Point Spread Function in Fluorescence Microscopy with a Sparse Combination of Gaussian Mixture: Trade-off between Accuracy and Efficiency