Two penalized estimators based on variance stabilization transforms for sparse compressive recovery with Poisson measurement noise

In this paper, we consider compressive inversion of a signal/image that is sparse in typical orthonormal bases used in image processing, given its measurements that have been corrupted by Poisson noise. The square-root operation is known to convert a Poisson random variable into one that is approximately Gaussian distributed with a constant variance. We present two different computationally tractable, penalized estimators with a data-fidelity term based on the aforementioned square-root based `variance stabilization transform'. The first estimator has been proposed earlier in the literature, but this is the first paper to analyze its theoretical performance in compressed sensing. Our second estimator is completely novel, and also has the interesting statistical property of being an approximately pivotal estimator. For both estimators, we specifically consider the case of a physically realistic sensing matrix in our analysis. We present detailed performance bounds on the l₂ recovery error for purely sparse signals for both estimators, making use of many different Poisson concentration inequalities. Several numerical results are presented, showing the practicality of the proposed estimators.