Bayesian Semi-Blind Deconvolution at Scale

TL;DR

This work addresses Bayesian semi-blind deconvolution (SBD) where both the latent image and the blur are unknown. It extends a cyclic-lattice Bayesian model to general SBD, reformulates the Gibbs sampler to exploit Fourier-domain computations, and introduces a marginal Hamiltonian Monte Carlo (HMC) update for the blur that integrates over the image. The approach leverages circulant and Kronecker-structured covariances to achieve scalable computations, including FFT-based sampling and efficient evaluation of log-determinants and quadratic forms. Results on simulated data and a real seismic dataset with about $80{,}000$ posterior parameters show that HMC improves mixing in multimodal regimes, while Gibbs sampling remains competitive when image observations sufficiently constrain the solution; the method provides uncertainty quantification for blur, image, and hyperparameters. Overall, the paper advances scalable Bayesian inference for SBD and demonstrates practical impact in geophysical imaging where reliable uncertainty estimates are crucial.

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.

Figures (8)

• Figure 1: DAG of the model.
• Figure 1: RMSE distribution of the multivariate parameters $\bm{\omega}$ and $\bm{c}_u^{\star}$ and scalar RMSE for the univariate parameters $\sigma_c^2$, $\sigma_w^2$, and $\zeta$ estimated from a simulated dataset. Based on the plots for $\bm{\omega}$ and $\bm{c}_u^{\star}$, the RMSE stabilizes at $m_v= 12 = n_v^o / 2$ in the presence of horizontal padding, i.e. when $m_h \in \{ 6, 12\}$.
• Figure 2: Realizations from the marginal blur posteriors obtained with $\alpha=1$ and $\alpha=1$ for the $24 \times 1$ dataset simulated from the model, for increasing number $m$ of exact image observations. Each line represents a posterior sample, with the posterior mean depicted by thicker solid lines. Gibbs struggles to visit both sign-shift modes.
• Figure 2: Blur traceplots for the posterior realizations in the Figure \ref{['fig:posterior_blur_constraints']}, corresponding to the numerical experiment used to determine the padding size.
• Figure 3: ESS and MSJD distributions for all parameters as a function of the number of exact image observations $m$, for $\alpha=0$ and $\alpha=1$. HMC has better ability to jump between the modes in the multimodal regime, while Gibbs might have equal or improved exploration abilities when the solution space is adequately constrained.