Bayesian inference and uncertainty quantification for image reconstruction with Poisson data
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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 first 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. Third 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.
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