Negative Binomial Optimization for Low-Count Overdispersed Sparse Signal Reconstruction

Yu Lu, Roummel F. Marcia

Abstract: Low-photon count imaging has been typically modeled by Poisson statistics. This discrete probability distribution model assumes that the mean and variance of a signal are equal. In the presence of greater variability in a dataset than what is expected, the negative binomial distribution is a suitable overdispersed alternative to the Poisson distribution. In this work, we present a framework for reconstructing sparse signals in these low-count overdispersed settings. Specifically, we describe a gradient-based sequential quadratic optimization approach that minimizes the negative log-likelihood corresponding to the negative binomial distribution coupled with a sparsity-promoting regularization term. Numerical experiments on 1D and 2D sparse/compressible signals are presented.