A general fractional total variation-Gaussian (GFTG) prior for Bayesian inverse problems

讲座时间 Datetime: 2022年9月13日,星期二,15:30-16:30

地点 Venue: 腾讯线上会议：677-738-019

主持人 Host：时聪

报告人 Speaker: 郑光辉 

单位 Affiliation: 湖南大学

报告摘要 Abstract:

The Bayesian inference is widely used in many scientific and engineering problems, especially in the inverse problems in infinite dimensional setting where the unknowns are functions.In this talk, we discuss the imaging inverse problem by employing an infinite dimensional Bayesian inference method with a general fractional total variation-Gaussian (GFTG) prior. This novel hybrid prior is a development for the total variation-Gaussian (TG) prior and the non-local total variation-Gaussian (NLTG) prior, which is a combination of the Gaussian prior and a general fractional total variation regularization term, which contains a wide class of fractional derivative. Compared to the TG prior, the GFTG prior can effectively reduce the staircase effect, enhance the texture details of the images and also provide a complete theoretical analysis in the infinite dimensional limit similarly to TG prior. We give the well-posedness and infinite-dimensional approximation of the posterior measure of the Bayesian inverse problem based on the GFTG prior, and then the samples are extracted from the posterior distribution by using the preconditioned Crank-Nicolson (pCN) algorithm. Finally, we give several numerical examples of image reconstruction under liner and nonlinear models to illustrate the advantages of the proposed improved prior.