NLTG Prior For Bayesian Imaging Inverse Problems And Applications

In recent years,Bayesian inference methods play an increasingly important role in the field of inverse problems,such as medical imaging,astronomy,and geological exploration.Due to the ill-conditioness of the forward operator or partial observation of data,most of inverse problems are ill-posed,that is,even small disturbances to the observation data may lead to large errors in the inference results.Bayesian methods allow to alleviate the stability problem by introducing prior information.Based on the framework of Bayesian methods,this thesis proposes a new prior distribution for image restoration problems,and establish the well-posedness of the resulting posterior distribution and illustrate the numerical performance through two applications: limited CT image restoration and image fusion.In addition,for the computation problem with gaussian prior,we propose a fast sparse precision matrix estimation method.The theoretical asymptotic approximation properties and numerical results show that the proposed model has advantages over other classical methods.In the first part of the thesis,we present the proposed hybrid prior for image restoration problems,namely NLTG,by combining nonlocal total variation regularization(NLTV)and Gaussian-prior.This hybrid prior distribution has the following two advantages: firstly,the NLTG prior inherits the advantages of non-local total variation,on extracting textural and structures in the images;secondly,the Gaussian prior in the NLTG prior allows to extract structural information from a reference image for reducing the uncertainty of the problems.Theoretically,we show the well-posedness of the proposed prior in infinite dimensional space.Eor the proposed NLTG prior,we adapt two common point estimation methods,maximum a posterior estimation(MAP)and conditional mean(CM)estimation,for reconstructing images from given partial data.CT reconstruction with limited data,NLTG prior outperforms other traditional models in the MAP estimation;from the conditional mean method,we find that our proposed prior is also superior to other models,and its uncertainty for unknown parameters are well mitigated.In addition,based on the iterative process of the NLTG we propose to update the Gaussian prior which can further improve the image reconstruction results.For Gaussian prior introduced in Bayesian inference problems,the computation of the inverse of the covariance matrix,that is,the precision matrix problem,is a major challenge in numerical implementation.In the second part,we propose a fast sparse precision matrix estimation method by combining CLIME model and GISS~? algorithm.In detail,we reformulate the constrained optimization problem in the CLIME model as a sparsity recovery problem with appropriate stopping criteria and use a greedy strategy to improve the computation speed.Theoretically,we show that the model maintains the asymptotic convergence results obtained in the CLIME model under relatively weak condition assumptions.The numerical performance of the proposed algorithm is demonstrated with three settings of covariance matrix and detail comparisons to other methods.In the third part,we further explore the application of hybrid prior to image fusion problems.We propose to apply NLTG prior as a regularization for recovering a fused image from multi-modality images.Through comparisons with other models,we find that the NLTG model still have some advantages for image fusion.The preliminary results verify the feasibility of the NLTG hybrid prior for recovering a fused image,and provide an experimental basis for further study of the related uncertainty quantification for image fusion problems.In summary,under the framework of Bayesian inference for imaging problems,we proposed a new well-defined prior,and established the theoretical properties and showed the advantages from the numerical aspects.In addition,in order to solve the problem of high-dimensional sparse precision matrix estimation for Gaussian prior,we proposed an efficient method that is shown to improve the computation speed and accuracy with extensive numerical experiments.