Empirical Bayesian image restoration by Langevin sampling with a denoising diffusion implicit prior

Score-based diffusion methods provide a powerful strategy to solve image restoration tasks by flexibly combining a pre-trained foundational prior model with a likelihood function specified during test time. Such methods are predominantly derived from two stochastic processes: reversing Ornstein-Uhlenbeck, which underpins the celebrated denoising diffusion probabilistic models (DDPM) and denoising diffusion implicit models (DDIM), and the Langevin diffusion process. The solutions delivered by DDPM and DDIM are often remarkably realistic, but they are not always consistent with measurements because of likelihood intractability issues and the associated required approximations. Alternatively, using a Langevin process circumvents the intractable likelihood issue, but usually leads to restoration results of inferior quality and longer computing times. This paper presents a novel and highly computationally efficient image restoration method that carefully embeds a foundational DDPM denoiser within an empirical Bayesian Langevin algorithm, which jointly calibrates key model hyper-parameters as it estimates the model's posterior mean. Extensive experimental results on three canonical tasks (image deblurring, super-resolution, and inpainting) demonstrate that the proposed approach improves on state-of-the-art strategies both in image estimation accuracy and computing time.

Do Bayesian imaging methods report trustworthy probabilities?

Bayesian statistics is a cornerstone of imaging sciences, underpinning many and varied approaches from Markov random fields to score-based denoising diffusion models. In addition to powerful image estimation methods, the Bayesian paradigm also provides a framework for uncertainty quantification and for using image data as quantitative evidence. These probabilistic capabilities are important for the rigorous interpretation of experimental results and for robust interfacing of quantitative imaging pipelines with scientific and decision-making processes. However, are the probabilities delivered by existing Bayesian imaging methods meaningful under replication of an experiment, or are they only meaningful as subjective measures of belief? This paper presents a Monte Carlo method to explore this question. We then leverage the proposed Monte Carlo method and run a large experiment requiring 1,000 GPU-hours to probe the accuracy of five canonical Bayesian imaging methods that are representative of some of the main Bayesian imaging strategies from the past decades (a score-based denoising diffusion technique, a plug-and-play Langevin algorithm utilising a Lipschitz-regularised DnCNN denoiser, a Bayesian method with a dictionary-based prior trained subject to a log-concavity constraint, an empirical Bayesian method with a total-variation prior, and a hierarchical Bayesian Gibbs sampler based on a Gaussian Markov random field model). We find that, a few cases, the probabilities reported by modern Bayesian imaging techniques are in broad agreement with long-term averages as observed over a large number of replication of an experiment, but existing Bayesian imaging methods are generally not able to deliver reliable uncertainty quantification results.

Unsupervised Training of Convex Regularizers using Maximum Likelihood Estimation

Unsupervised learning is a training approach in the situation where ground truth data is unavailable, such as inverse imaging problems. We present an unsupervised Bayesian training approach to learning convex neural network regularizers using a fixed noisy dataset, based on a dual Markov chain estimation method. Compared to classical supervised adversarial regularization methods, where there is access to both clean images as well as unlimited to noisy copies, we demonstrate close performance on natural image Gaussian deconvolution and Poisson denoising tasks.

Statistical modelling and Bayesian inversion for a Compton imaging system: application to radioactive source localisation

This paper presents a statistical forward model for a Compton imaging system, called Compton imager. This system, under development at the University of Illinois Urbana Champaign, is a variant of Compton cameras with a single type of sensors which can simultaneously act as scatterers and absorbers. This imager is convenient for imaging situations requiring a wide field of view. The proposed statistical forward model is then used to solve the inverse problem of estimating the location and energy of point-like sources from observed data. This inverse problem is formulated and solved in a Bayesian framework by using a Metropolis within Gibbs algorithm for the estimation of the location, and an expectation-maximization algorithm for the estimation of the energy. This approach leads to more accurate estimation when compared with the deterministic standard back-projection approach, with the additional benefit of uncertainty quantification in the low photon imaging setting.

Scalable Bayesian uncertainty quantification with data-driven priors for radio interferometric imaging

Next-generation radio interferometers like the Square Kilometer Array have the potential to unlock scientific discoveries thanks to their unprecedented angular resolution and sensitivity. One key to unlocking their potential resides in handling the deluge and complexity of incoming data. This challenge requires building radio interferometric imaging methods that can cope with the massive data sizes and provide high-quality image reconstructions with uncertainty quantification (UQ). This work proposes a method coined QuantifAI to address UQ in radio-interferometric imaging with data-driven (learned) priors for high-dimensional settings. Our model, rooted in the Bayesian framework, uses a physically motivated model for the likelihood. The model exploits a data-driven convex prior, which can encode complex information learned implicitly from simulations and guarantee the log-concavity of the posterior. We leverage probability concentration phenomena of high-dimensional log-concave posteriors that let us obtain information about the posterior, avoiding MCMC sampling techniques. We rely on convex optimisation methods to compute the MAP estimation, which is known to be faster and better scale with dimension than MCMC sampling strategies. Our method allows us to compute local credible intervals, i.e., Bayesian error bars, and perform hypothesis testing of structure on the reconstructed image. In addition, we propose a novel blazing-fast method to compute pixel-wise uncertainties at different scales. We demonstrate our method by reconstructing radio-interferometric images in a simulated setting and carrying out fast and scalable UQ, which we validate with MCMC sampling. Our method shows an improved image quality and more meaningful uncertainties than the benchmark method based on a sparsity-promoting prior. QuantifAI's source code: https://github.com/astro-informatics/QuantifAI.

Equivariant Bootstrapping for Uncertainty Quantification in Imaging Inverse Problems

Scientific imaging problems are often severely ill-posed, and hence have significant intrinsic uncertainty. Accurately quantifying the uncertainty in the solutions to such problems is therefore critical for the rigorous interpretation of experimental results as well as for reliably using the reconstructed images as scientific evidence. Unfortunately, existing imaging methods are unable to quantify the uncertainty in the reconstructed images in a manner that is robust to experiment replications. This paper presents a new uncertainty quantification methodology based on an equivariant formulation of the parametric bootstrap algorithm that leverages symmetries and invariance properties commonly encountered in imaging problems. Additionally, the proposed methodology is general and can be easily applied with any image reconstruction technique, including unsupervised training strategies that can be trained from observed data alone, thus enabling uncertainty quantification in situations where there is no ground truth data available. We demonstrate the proposed approach with a series of numerical experiments and through comparisons with alternative uncertainty quantification strategies from the state-of-the-art, such as Bayesian strategies involving score-based diffusion models and Langevin samplers. In all our experiments, the proposed method delivers remarkably accurate high-dimensional confidence regions and outperforms the competing approaches in terms of estimation accuracy, uncertainty quantification accuracy, and computing time.

Accelerated Bayesian imaging by relaxed proximal-point Langevin sampling

This paper presents a new accelerated proximal Markov chain Monte Carlo methodology to perform Bayesian inference in imaging inverse problems with an underlying convex geometry. The proposed strategy takes the form of a stochastic relaxed proximal-point iteration that admits two complementary interpretations. For models that are smooth or regularised by Moreau-Yosida smoothing, the algorithm is equivalent to an implicit midpoint discretisation of an overdamped Langevin diffusion targeting the posterior distribution of interest. This discretisation is asymptotically unbiased for Gaussian targets and shown to converge in an accelerated manner for any target that is $\kappa$-strongly log-concave (i.e., requiring in the order of $\sqrt{\kappa}$ iterations to converge, similarly to accelerated optimisation schemes), comparing favorably to [M. Pereyra, L. Vargas Mieles, K.C. Zygalakis, SIAM J. Imaging Sciences, 13, 2 (2020), pp. 905-935] which is only provably accelerated for Gaussian targets and has bias. For models that are not smooth, the algorithm is equivalent to a Leimkuhler-Matthews discretisation of a Langevin diffusion targeting a Moreau-Yosida approximation of the posterior distribution of interest, and hence achieves a significantly lower bias than conventional unadjusted Langevin strategies based on the Euler-Maruyama discretisation. For targets that are $\kappa$-strongly log-concave, the provided non-asymptotic convergence analysis also identifies the optimal time step which maximizes the convergence speed. The proposed methodology is demonstrated through a range of experiments related to image deconvolution with Gaussian and Poisson noise, with assumption-driven and data-driven convex priors.

Proximal nested sampling with data-driven priors for physical scientists

Proximal nested sampling was introduced recently to open up Bayesian model selection for high-dimensional problems such as computational imaging. The framework is suitable for models with a log-convex likelihood, which are ubiquitous in the imaging sciences. The purpose of this article is two-fold. First, we review proximal nested sampling in a pedagogical manner in an attempt to elucidate the framework for physical scientists. Second, we show how proximal nested sampling can be extended in an empirical Bayes setting to support data-driven priors, such as deep neural networks learned from training data.

The split Gibbs sampler revisited: improvements to its algorithmic structure and augmented target distribution

This paper proposes a new accelerated proximal Markov chain Monte Carlo (MCMC) methodology to perform Bayesian computation efficiently in imaging inverse problems. The proposed methodology is derived from the Langevin diffusion process and stems from tightly integrating two state-of-the-art proximal Langevin MCMC samplers, SK-ROCK and split Gibbs sampling (SGS), which employ distinctively different strategies to improve convergence speed. More precisely, we show how to integrate, at the level of the Langevin diffusion process, the proximal SK-ROCK sampler which is based on a stochastic Runge-Kutta-Chebyshev approximation of the diffusion, with the model augmentation and relaxation strategy that SGS exploits to speed up Bayesian computation at the expense of asymptotic bias. This leads to a new and faster proximal SK-ROCK sampler that combines the accelerated quality of the original SK-ROCK sampler with the computational benefits of augmentation and relaxation. Moreover, rather than viewing the augmented and relaxed model as an approximation of the target model, positioning relaxation in a bias-variance trade-off, we propose to regard the augmented and relaxed model as a generalisation of the target model. This then allows us to carefully calibrate the amount of relaxation in order to simultaneously improve the accuracy of the model (as measured by the model evidence) and the sampler's convergence speed. To achieve this, we derive an empirical Bayesian method to automatically estimate the optimal amount of relaxation by maximum marginal likelihood estimation. The proposed methodology is demonstrated with a range of numerical experiments related to image deblurring and inpainting, as well as with comparisons with alternative approaches from the state of the art.

Learned reconstruction with convergence guarantees

In recent years, deep learning has achieved remarkable empirical success for image reconstruction. This has catalyzed an ongoing quest for precise characterization of correctness and reliability of data-driven methods in critical use-cases, for instance in medical imaging. Notwithstanding the excellent performance and efficacy of deep learning-based methods, concerns have been raised regarding their stability, or lack thereof, with serious practical implications. Significant advances have been made in recent years to unravel the inner workings of data-driven image recovery methods, challenging their widely perceived black-box nature. In this article, we will specify relevant notions of convergence for data-driven image reconstruction, which will form the basis of a survey of learned methods with mathematically rigorous reconstruction guarantees. An example that is highlighted is the role of ICNN, offering the possibility to combine the power of deep learning with classical convex regularization theory for devising methods that are provably convergent. This survey article is aimed at both methodological researchers seeking to advance the frontiers of our understanding of data-driven image reconstruction methods as well as practitioners, by providing an accessible description of convergence concepts and by placing some of the existing empirical practices on a solid mathematical foundation.