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SUMMARY:Bayesian Approaches to Inverse Problems with deep learning-based p
 riors
DTSTART:20260915T091500Z
DTEND:20260915T101500Z
DTSTAMP:20260811T001300Z
UID:indico-event-16696@indico.math.cnrs.fr
DESCRIPTION:Speakers: Mame Diarra Fall (LITIS\, Université de Rouen Norma
 ndie)\n\nInverse problems are ubiquitous in signal and image processing. A
 s inverse problems are known to be ill-posed or\, at least\, ill-condition
 ed\, they require regularization through the introduction of additional co
 nstraints to mitigate the lack of information provided by the observations
 . A common difficulty lies in selecting an appropriate regularizer\, which
  has a decisive influence on the quality of the reconstruction. Another ch
 allenge concerns the level of confidence we may have in the reconstructed 
 signal or image. In other words\, it is desirable for a method to quantify
  the uncertainty associated with the reconstructed image in order to promo
 te more principled decision-making.\nThese two tasks - regularization and 
 uncertainty quantification - can be addressed simultaneously within the Ba
 yesian statistical framework. This approach makes it possible to incorpora
 te additional information by specifying a marginal distribution for the im
 age\, known as the prior distribution. The traditional approach consists i
 n defining the prior analytically\, as a hand-crafted explicit function ch
 osen to promote specific desired properties of the recovered image. Follow
 ing the recent surge in deep learning\, data-driven regularization using p
 riors specified by neural networks has become widespread in image inverse 
 problems. Popular approaches within this framework include Plug-and-Play (
 PnP) [1] and Regularization by Denoising (RED) [2].\nIn the first part of 
 the talk\, I will present the probabilistic approach to the RED framework 
 we have  introduced in [3]\, which defines a new probability distribution
  based on a RED potential that can be used as the prior distribution in a 
 Bayesian inversion task. We also propose a dedicated Markov chain Monte Ca
 rlo (MCMC) sampling algorithm that is particularly well suited for high-di
 mensional sampling of the resulting posterior distribution. In addition\, 
 we provide a theoretical analysis guaranteeing convergence to the target d
 istribution and quantifying the convergence rate. The effectiveness of the
  proposed approach is illustrated on various linear inverse restoration ta
 sks such as image deblurring\, inpainting\, and super-resolution.\nThe sec
 ond part of the talk will be devoted to a novel approach we proposed in [4
 ]\, for solving  Poisson inverse problems. We also develop  a Monte Carl
 o sampling algorithm that accounts for the underlying non-Euclidean geomet
 ry of the problem. The proposed approach has been evaluated on different t
 asks such as denoising\, deblurring\, and positron emission tomography (PE
 T) reconstruction.\n\nReferences\n [1] S. V. Venkatakrishnan et al. « P
 lug-and-Play priors for model based reconstruction ». In IEEE Global Con
 f. on Signal and Information Processing\, pp 945-948\, 2013.\n [2] Y. Rom
 ano\, M. Elad and P. Milanfar\, « The little engine that could: Regulariz
 ation by denoising (RED)\, » SIAM Journal on Imaging Sciences\, 10(4):18
 04–1844\, 2017\n[3] E.C. Faye\, M.D. Fall and N. Dobigeon. « Regulari
 zation by denoising: Bayesian model and Langevin-within-split Gibbs sampli
 ng »\, IEEE Transactions on Image Processing\,  vol 34\, pages 221-234\
 , 2024\n [4] E.C. Faye\, M.D. Fall\, N. Dobigeon and  É. Barat « Bregm
 an geometry-aware split Gibbs sampling for Bayesian Poisson inverse proble
 ms »\, Submitted.\n\nhttps://indico.math.cnrs.fr/event/16696/
LOCATION:Salle K. Johnson (1R3\, 1er étage)
URL:https://indico.math.cnrs.fr/event/16696/
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