Bayesian imaging using Plug & Play priors: when Langevin meets Tweedie
| Autor: | Rémi Laumont, Valentin De Bortoli, Andrés Almansa, Julie Delon, Alain Durmus, Marcelo Pereyra |
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| Přispěvatelé: | Université de Paris - UFR Mathématiques et informatique [Sciences], Université de Paris (UP), Mathématiques Appliquées Paris 5 (MAP5 - UMR 8145), Institut National des Sciences Mathématiques et de leurs Interactions (INSMI)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), University of Oxford [Oxford], Ecole Normale Supérieure Paris-Saclay (ENS Paris Saclay), CB - Centre Borelli - UMR 9010 (CB), Service de Santé des Armées-Institut National de la Santé et de la Recherche Médicale (INSERM)-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS)-Ecole Normale Supérieure Paris-Saclay (ENS Paris Saclay)-Université de Paris (UP), School of Mathematical and Computer Sciences (MATHEMATICS DEPARTMENT OF HERIOT-WATT UNIVERSITY), Heriot-Watt University [Edinburgh] (HWU), Laumont, Rémi, Repenser la post-production d'archives avec des méthodes à patch, variationnelles et par apprentissage - - PostProdLEAP2019 - ANR-19-CE23-0027 - AAPG2019 - VALID, Université Paris Cité - UFR Mathématiques et informatique [Sciences], Université Paris Cité (UPCité), Institut National des Sciences Mathématiques et de leurs Interactions (INSMI)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), University of Oxford, Service de Santé des Armées-Institut National de la Santé et de la Recherche Médicale (INSERM)-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS)-Ecole Normale Supérieure Paris-Saclay (ENS Paris Saclay)-Université Paris Cité (UPCité), ANR-19-CE23-0027,PostProdLEAP,Repenser la post-production d'archives avec des méthodes à patch, variationnelles et par apprentissage(2019) |
| Jazyk: | angličtina |
| Rok vydání: | 2022 |
| Předmět: |
FOS: Computer and information sciences
Computer Vision and Pattern Recognition (cs.CV) General Mathematics Inverse Problems Computer Science - Computer Vision and Pattern Recognition Machine Learning (stat.ML) Mathematics - Statistics Theory Statistics Theory (math.ST) Methodology (stat.ME) [STAT.ML]Statistics [stat]/Machine Learning [stat.ML] Statistics - Machine Learning FOS: Electrical engineering electronic engineering information engineering FOS: Mathematics Inpainting Monte-Carlo Markov Chain Statistics - Methodology Plug-and-Play [STAT.TH] Statistics [stat]/Statistics Theory [stat.TH] Applied Mathematics Image and Video Processing (eess.IV) Langevin Algorithm [STAT.TH]Statistics [stat]/Statistics Theory [stat.TH] Electrical Engineering and Systems Science - Image and Video Processing [STAT.ML] Statistics [stat]/Machine Learning [stat.ML] [INFO.INFO-TI] Computer Science [cs]/Image Processing [eess.IV] Deblurring [INFO.INFO-TI]Computer Science [cs]/Image Processing [eess.IV] 65K10 65K05 65D18 62F15 62C10 68Q25 68U10 90C26 |
| Zdroj: | SIAM Journal on Imaging Sciences SIAM Journal on Imaging Sciences, Society for Industrial and Applied Mathematics, 2022 SIAM Journal on Imaging Sciences, 2022, ⟨10.1137/21m1406349⟩ |
| ISSN: | 1936-4954 |
| Popis: | Since the seminal work of Venkatakrishnan et al. [83] in 2013, Plug & Play (PnP) methods have become ubiquitous in Bayesian imaging. These methods derive Minimum Mean Square Error (MMSE) or Maximum A Posteriori (MAP) estimators for inverse problems in imaging by combining an explicit likelihood function with a prior that is implicitly defined by an image denoising algorithm. The PnP algorithms proposed in the literature mainly differ in the iterative schemes they use for optimisation or for sampling. In the case of optimisation schemes, some recent works guarantee the convergence to a fixed point, albeit not necessarily a MAP estimate. In the case of sampling schemes, to the best of our knowledge, there is no known proof of convergence. There also remain important open questions regarding whether the underlying Bayesian models and estimators are well defined, well-posed, and have the basic regularity properties required to support these numerical schemes. To address these limitations, this paper develops theory, methods, and provably convergent algorithms for performing Bayesian inference with PnP priors. We introduce two algorithms: 1) PnP-ULA (Plug & Play Unadjusted Langevin Algorithm) for Monte Carlo sampling and MMSE inference; and 2) PnP-SGD (Plug & Play Stochastic Gradient Descent) for MAP inference. Using recent results on the quantitative convergence of Markov chains, we establish detailed convergence guarantees for these two algorithms under realistic assumptions on the denoising operators used, with special attention to denoisers based on deep neural networks. We also show that these algorithms approximately target a decision-theoretically optimal Bayesian model that is well-posed. The proposed algorithms are demonstrated on several canonical problems such as image deblurring, inpainting, and denoising, where they are used for point estimation as well as for uncertainty visualisation and quantification. |
| Databáze: | OpenAIRE |
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