  Deblurring is a fundamental inverse problem in bioimaging. It requires
modelling the point spread function (PSF), which captures the optical
distortions entailed by the image formation process. The PSF limits the spatial
resolution attainable for a given microscope. However, recent applications
require a higher resolution, and have prompted the development of
super-resolution techniques to achieve sub-pixel accuracy. This requirement
restricts the class of suitable PSF models to analog ones. In addition,
deblurring is computationally intensive, hence further requiring
computationally efficient models. A custom candidate fitting both requirements
is the Gaussian model. However, this model cannot capture the rich tail
structures found in both theoretical and empirical PSFs. In this paper, we aim
at improving the reconstruction accuracy beyond the Gaussian model, while
preserving its computational efficiency. We introduce a new class of analog PSF
models based on Gaussian mixtures. The number of Gaussian kernels controls both
the modelling accuracy and the computational efficiency of the model: the lower
the number of kernels, the lower accuracy and the higher efficiency. To explore
the accuracy--efficiency trade-off, we propose a variational formulation of the
PSF calibration problem, where a convex sparsity-inducing penalty on the number
of Gaussian kernels allows trading accuracy for efficiency. We derive an
efficient algorithm based on a fully-split formulation of alternating split
Bregman. We assess our framework on synthetic and real data and demonstrate a
better reconstruction accuracy in both geometry and photometry in point source
localisation---a fundamental inverse problem in fluorescence microscopy.
