Teresa Klatzer

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 κ-strongly log-concave (i.e., requiring in the order of √κ 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 κ-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.

Demosaicing is an important first step for color image acquisition. For practical reasons, demosaicing algorithms have to be both efficient and yield high quality results in the presence of noise. The demosaicing problem poses several challenges, e.g. zippering and false color artifacts as well as edge blur. In this work, we introduce a novel learning based method that can overcome these challenges. We formulate demosaicing as an image restoration problem and propose to learn efficient regularization inspired by a variational energy minimization framework that can be trained for different sensor layouts. Our algorithm performs joint demosaicing and denoising in close relation to the real physical mosaicing process on a camera sensor. This is achieved by learning a sequence of energy minimization problems composed of a set of RGB filters and corresponding activation functions. We evaluate our algorithm on the Microsoft Demosaicing data set in terms of peak signal to noise ratio (PSNR) and structured similarity index (SSIM). Our algorithm is highly efficient both in image quality and run time. We achieve an improvement of up to 2.6 dB over recent state-of-the-art algorithms.