Masked unbiased principles for parameter selection in variational image restoration under Poisson noise

In this paper we address the problem of automatically selecting the regularization parameter in variational models for the restoration of images corrupted by Poisson noise. More specifically, we first review relevant existing unmasked selection criteria which fully exploit the acquired data by considering all pixels in the selection procedure. Then, based on an idea originally proposed by Carlavan and Blanc-Feraud to effectively deal with dark backgrounds and/or low photon-counting regimes, we introduce and discuss the masked versions—some of them already existing—of the considered unmasked selection principles formulated by simply discarding the pixels measuring zero photons. However, we prove that such a blind masking strategy yields a bias in the resulting principles that can be overcome by introducing a novel positive Poisson distribution correctly modeling the statistical properties of the undiscarded noisy data. Such distribution is at the core of newly proposed masked unbiased counterparts of the discussed strategies. All the unmasked, masked biased and masked unbiased principles are extensively compared on the restoration of different images in a wide range of photon-counting regimes. Our tests allow to conclude that the novel masked unbiased selection strategies, on average, compare favorably with unmasked and masked biased counterparts.