Bayesian Semi-Blind Deconvolution at Scale

TL;DR

This paper advances Bayesian semi-blind deconvolution by developing scalable Fourier-based Gibbs and Hamiltonian Monte Carlo methods, enabling efficient inference in large-scale deconvolution problems like seismic imaging.

Contribution

It extends a Bayesian hierarchical model for semi-blind deconvolution, introducing a Fourier domain Gibbs sampler and a novel marginal HMC blur update for improved scalability.

Findings

Fourier-based Gibbs and HMC updates improve sampling efficiency.

The methods scale to large seismic imaging problems with 80,000 parameters.

HMC demonstrates better exploration of the posterior compared to Gibbs.

Abstract

Blind image deconvolution refers to the problem of simultaneously estimating the blur kernel and the true image from a set of observations when both the blur kernel and the true image are unknown. Sometimes, additional image and/or blur information is available and the term semi-blind deconvolution (SBD) is used. We consider a recently introduced Bayesian conjugate hierarchical model for SBD, formulated on an extended cyclic lattice to allow a computationally scalable Gibbs sampler. In this article, we extend this model to the general SBD problem, rewrite the previously proposed Gibbs sampler so that operations are performed in the Fourier domain whenever possible, and introduce a new marginal Hamiltonian Monte Carlo (HMC) blur update, obtained by analytically integrating the blur-image joint conditional over the image. The cyclic formulation combined with non-trivial linear algebra manipulations allows a Fourier-based, scalable HMC update, otherwise complicated by the rigid constraints of the SBD problem. Having determined the padding size in the cyclic embedding through a numerical experiment, we compare the mixing and exploration behaviour of the Gibbs and HMC blur updates on simulated data and on a real geophysical seismic imaging problem where we invert a grid with $300\times50$ nodes, corresponding to a posterior with approximately $80,000$ parameters.