Bayesian Image Reconstruction With Uncertainty Quantification

Image reconstruction has been a hot topic in image processing,that requires to recover hidden multidimensional model parameters from observational data with noise.We treat image reconstruction as a statistical inverse problem.Bayesian inversion uses statistical methods to solve the inverse problems.Based on Bayes rule,Bayesian inversion infers the posterior,provided the information of the prior and the likelihood function.Bayesian inference provides a means of both estimation and uncertainty quantification for the unknown,and thus is the suitable method for solving image reconstruction problems.Bayesian inference in image reconstruction is faced with many challenges,such as computational efficiency,hyperparameters optimization,determination of prior,development of sampling methods and so on.This paper aims at improving the quality of the reconstructed image and increasing the computational efficiency.To solve linear inverse problems in which the prior or the likelihood function depends on unspecified hyperparameters,we provide a hyperparameters optimization method.In practice,these hyperparameters are often determined via an empirical Bayesian method that maximizes the marginal likelihood function,i.e.the probability density of the data conditioned on the hyperparameters.Evaluating the marginal likelihood,however,is computationally challenging for large-scale problems.In this work,we present a method to approximately evaluate marginal likelihood functions,based on a low-rank approximation of the update from the prior covariance to the posterior covariance.We show that this approximation is optimal in a minimax sense.Moreover,we provide an efficient algorithm to implement the proposed method,based on a combination of the randomized SVD and a spectral approximation method to compute square roots of the prior covariance matrix.For posterior without closed form,we develop a highly efficient dimensional independent sampling method.The preconditioned Crank-Nicolson(pCN)method is a Markov Chain Monte Carlo(MCMC)scheme,specifically designed to perform Bayesian inferences in function spaces.Unlike many standard MCMC algorithms,the pCN method can preserve the sampling efficiency under the mesh refinement,a property referred as being dimensional independent.In this work we consider an adaptive strategy to further improve the efficiency of pCN.In particular,we develop a hybrid adaptive MCMC method: the algorithm performs an adaptive Metropolis scheme in a chosen finite dimensional subspace,and a standard pCN algorithm in the complement space of the chosen subspace.We show that the proposed algorithm satisfies certain important ergodicity conditions.We provide a complete framework for performing infinite-dimensional Bayesian inference and uncertainty quantification for image reconstruction with Poisson data.In particular,we address the following issues to make the Bayesian framework applicable in practice.We introduce a positivity-preserving reparametrization,and we prove that under the reparametrization and a hybrid prior,the posterior distribution is well-posed in the infinite dimensional setting.Second,we provide a dimension-independent MCMC algorithm,based on the preconditioned Crank-Nicolson Langevin method,in which we use a primal-dual scheme to compute the offset direction.Then we give a method based on the model discrepancy to determine the regularization parameter in the hybrid prior.Finally,we propose to use the obtained posterior distribution to detect artifacts in a recovered image.We provide an application example to demonstrate the effectiveness of the proposed method.