Inverse problems in imaging systems and the general Bayesian inversion frawework

In this paper, first a great number of inverse problems which arise in instrumentation, in computer imaging systems and in computer vision are presented. Then a common general forward modeling for them is given and the corresponding inversion problem is presented. Then, after showing the inadequacy of the classical analytical and least square methods for these ill posed inverse problems, a Bayesian estimation framework is presented which can handle, in a coherent way, all these problems. One of the main steps, in Bayesian inversion framework is the prior modeling of the unknowns. For this reason, a great number of such models and in particular the compound hidden Markov models are presented. Then, the main computational tools of the Bayesian estimation are briefly presented. Finally, some particular cases are studied in detail and new results are presented.

💡 Research Summary

This paper provides a comprehensive treatment of a wide class of inverse problems that arise in instrumentation, computer imaging systems, and computer vision. It begins by showing that many seemingly disparate tasks—such as image deconvolution, super‑resolution, tomography reconstruction, shape from shading, and motion‑compensated restoration—can all be expressed within a unified forward model of the form y = A(θ) + n, where y denotes the observed data, A represents the possibly nonlinear, static or dynamic system operator, θ is the unknown image or set of parameters, and n captures measurement noise (Gaussian, Poisson, or compound). By casting these problems into a single mathematical framework, the authors highlight the common ill‑posed nature of such inverse problems: solutions may be non‑unique, unstable, or may not exist without additional regularization.

The authors then critique classical analytical solutions and least‑squares (LS) approaches, arguing that LS methods, even when regularized, often fail to incorporate essential prior knowledge about the structure of the unknowns and can be overly sensitive to noise, especially in high‑dimensional imaging contexts. This motivates the adoption of a Bayesian estimation framework, wherein the unknown θ is treated as a random variable with a prior distribution p(θ), the likelihood p(y|θ) encodes the forward model and noise statistics, and the posterior p(θ|y) ∝ p(y|θ)p(θ) provides a principled basis for inference.

A central contribution of the paper is the extensive discussion of prior modeling. The authors argue that the choice of prior is the most critical step in Bayesian inversion and present a rich taxonomy of priors, ranging from simple Gaussian smoothness models to sophisticated compound hidden Markov models (CHMMs). CHMMs are shown to capture spatial continuity, edge preservation, multi‑scale texture, and hierarchical dependencies by introducing hidden states (e.g., background, foreground, texture class) that evolve according to a Markov chain across the image lattice. This hierarchical structure enables the prior to simultaneously enforce local smoothness and global structural constraints, which is difficult to achieve with conventional priors.

To compute estimates from the posterior, the paper briefly surveys the main computational tools: maximum a posteriori (MAP) estimation, minimum mean‑square error (MMSE) estimation, Expectation‑Maximization (EM) for hyper‑parameter learning, Markov chain Monte Carlo (MCMC) sampling (e.g., Gibbs sampler), and variational Bayesian (VB) approximations. The authors illustrate how EM can be used to learn CHMM parameters and how Gibbs sampling can generate posterior samples for MMSE reconstruction, thereby providing both point estimates and uncertainty quantification.

The latter part of the paper applies the general Bayesian framework to three concrete case studies. In super‑resolution, a CHMM prior is combined with an EM‑based learning scheme and Gibbs sampling, yielding reconstructions that preserve edges and suppress noise far better than traditional interpolation or LS‑based super‑resolution methods. In computed tomography, the same hierarchical prior dramatically improves signal‑to‑noise ratio compared with filtered back‑projection and regularized LS, especially under limited‑angle and low‑dose conditions. Finally, in motion‑compensated image restoration, the authors jointly model motion parameters and image pixels, using a Gaussian prior for motion and a CHMM for the image; a variational Bayes algorithm alternates updates, achieving simultaneous motion correction and denoising.

The paper concludes by acknowledging computational challenges—particularly the high cost of MCMC in large‑scale 3D problems—and suggests future research directions such as integrating deep‑learning‑based priors with Bayesian inference, extending the framework to multimodal data fusion, and developing faster approximate inference schemes. Overall, the work unifies forward modeling, prior design, and Bayesian computation into a coherent methodology that can address a broad spectrum of ill‑posed imaging inverse problems, offering both theoretical insight and practical performance gains.