On Maximum a Posteriori Estimation with Plug & Play Priors and Stochastic Gradient Descent

Bayesian methods to solve imaging inverse problems usually combine an explicit data likelihood function with a prior distribution that explicitly models expected properties of the solution. Many kinds of priors have been explored in the literature, from simple ones expressing local properties to mor...

Full description

Saved in:
Bibliographic Details
Published in:Journal of mathematical imaging and vision 2023, Vol.65 (1), p.140-163
Main Authors: Laumont, Rémi, De Bortoli, Valentin, Almansa, Andrés, Delon, Julie, Durmus, Alain, Pereyra, Marcelo
Format: Article
Language:English
Subjects:
Algorithms
Applications of Mathematics
Artificial neural networks
Bayesian analysis
Computer Science
Engineering Sciences
Image Processing
Image Processing and Computer Vision
Inverse problems
Machine Learning
Mathematical Methods in Physics
Mathematics
Numerical Analysis
Optimization and Control
Plug & play
Redundancy
Regularization
Signal and Image processing
Signal,Image and Speech Processing
Statistics
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:Bayesian methods to solve imaging inverse problems usually combine an explicit data likelihood function with a prior distribution that explicitly models expected properties of the solution. Many kinds of priors have been explored in the literature, from simple ones expressing local properties to more involved ones exploiting image redundancy at a non-local scale. In a departure from explicit modelling, several recent works have proposed and studied the use of implicit priors defined by an image denoising algorithm. This approach, commonly known as Plug & Play (PnP) regularization, can deliver remarkably accurate results, particularly when combined with state-of-the-art denoisers based on convolutional neural networks. However, the theoretical analysis of PnP Bayesian models and algorithms is difficult and works on the topic often rely on unrealistic assumptions on the properties of the image denoiser. This papers studies maximum a posteriori (MAP) estimation for Bayesian models with PnP priors. We first consider questions related to existence, stability and well-posedness and then present a convergence proof for MAP computation by PnP stochastic gradient descent (PnP-SGD) under realistic assumptions on the denoiser used. We report a range of imaging experiments demonstrating PnP-SGD as well as comparisons with other PnP schemes.
ISSN:0924-9907
1573-7683
DOI:10.1007/s10851-022-01134-7