Variational semi-blind sparse deconvolution with orthogonal kernel bases and its application to MRFM

• We proposed a novel variational solution to a semi-blind sparse deconvolution problem.
• Our method uses Bayesian inference for image and PSF restoration with a sparsity-inducing image prior via the variational Bayes approximation.
• Our algorithm correctly produces sparse image estimates.
• The performance of our method competes with MCMC methods in sparse image estimation.
• The benefits of our solution compared to MCMC solutions are faster convergence, stability of the method, and memory efficiency.

We present a variational Bayesian method of joint image reconstruction and point spread function (PSF) estimation when the PSF of the imaging device is only partially known. To solve this semi-blind deconvolution problem, prior distributions are specified for the PSF and the 3D image. Joint image reconstruction and PSF estimation is then performed within a Bayesian framework, using a variational algorithm to estimate the posterior distribution. The image prior distribution imposes an explicit atomic measure that corresponds to image sparsity. Importantly, the proposed Bayesian deconvolution algorithm does not require hand tuning. Simulation results clearly demonstrate that the semi-blind deconvolution algorithm compares favorably with previous Markov chain Monte Carlo (MCMC) version of myopic sparse reconstruction. It significantly outperforms mismatched non-blind algorithms that rely on the assumption of the perfect knowledge of the PSF. The algorithm is illustrated on real data from magnetic resonance force microscopy (MRFM).