🤖 AI Summary
This work addresses the challenges of high-dimensional Bayesian inference and computational scalability in semi-blind image deconvolution, particularly for large-scale applications such as seismic imaging. The authors propose a Bayesian conjugate hierarchical model based on cyclic lattice embedding. By reformulating the Gibbs sampler in the Fourier domain and introducing a Hamiltonian Monte Carlo (HMC) strategy for blur kernel updates that analytically marginalizes over image variables, they present the first scalable Fourier-domain HMC sampling method applicable to general semi-blind deconvolution problems. Evaluated on a 300×50 seismic imaging task involving approximately 80,000 parameters, the proposed approach significantly outperforms conventional Gibbs samplers, achieving notable improvements in posterior exploration efficiency and mixing performance.
📝 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.