Efficient multivariate Skellam shrinkage for denoising photon-limited image data: An Empirical Bayes approach

In this article we address the issue of denoising photon-limited image data by deriving new and efficient multivariate Bayesian estimators that approximate the conditional expectation of Haar wavelet and filterbank transform coefficients of Poisson data - coefficients that take the so-called Skellam distribution. We show that in this setting, the posterior mean under a Bayesian model forms the solution to a linear differential equation, owing in part to the recursive property of the Skellam distribution. We then propose a practical approach to solve - approximately - this differential equation, and arrive at a near mean-square-optimal Skellam mean estimator that is both computationally efficient and amenable to an empirical Bayes approach. We then derive three approaches to shrinkage based on smoothing the marginal likelihood of the data, and demonstrate their superior performance relative to state-of-the-art approaches for both natural test images and examples from computed tomography scans.