Bayesian Spatially Varying Multi-Regularization Image Deblurring
Many scientific experiments such as those found in astronomy, geology, microbiology, and X-ray radiography require the use of high-energy instruments to capture images. Due to the imaging system, blur and added noise are inevitably present. Oftentimes the captured images must be deblurred to extract valuable information. In the presence of noise, image deblurring is an ill-posed inverse problem in which regularization is required to obtain useful reconstructions. Choosing the appropriate strength of the regularization, however, is difficult. Moreover, many images contain some mixture of smooth features and edges which requires the use of multi-regularization, i.e., the type of regularization (total variation or Tikhonov) varies across the image. We address these two issues by formulating the image deblurring problem within a hierarchical Bayesian framework, varying both the strength of the regularization, as well as the regularization type across the image. In this way, both the image and the strength of the regularization, which varies across the image, are described by a hierarchical posterior distribution which we can sample by Markov chain Monte Carlo (MCMC), in particular Gibbs samplers that make use of conditional distributions for efficient sampling. We compute the means of the image and parameter samples for simulated test images, and we compare our results with existing non-spatially-varying Bayesian methods to show that our new method both increases the quality and decreases the error of the image reconstruction.