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Variance-stabilization-based compressive inversion under Poisson or Poisson–Gaussian noise with analytical bounds

Most existing bounds for signal reconstruction from compressive measurements make the assumption of additive signal-independent noise. However in many compressive imaging systems, the noise statistics are more accurately represented by Poisson or Poisson–Gaussian noise models. In this paper, we derive upper bounds for signal reconstruction error from compressive measurements which are corrupted by Poisson or Poisson–Gaussian noise. The features of our bounds are as follows: (1) the bounds are derived for a computationally tractable convex estimator with statistically motivated parameter selection. The estimator penalizes signal sparsity subject to a constraint that imposes a novel statistically motivated upper bound on a term based on variance stabilization transforms to approximate the Poisson or Poisson–Gaussian distributions by distributions with (nearly) constant variance. (2) The bounds are applicable to signals that are sparse as well as compressible in any orthonormal basis, and are derived for compressive systems obeying realistic constraints such as non-negativity and flux-preservation. Our bounds are motivated by several properties of the variance stabilization transforms that we develop and analyze. We present extensive numerical results for signal reconstruction under varying number of measurements and varying signal intensity levels. Ours is the first piece of work to derive bounds on compressive inversion for the Poisson–Gaussian noise model. We also use the properties of the variance stabilizer to develop a principle for selection of the regularization parameter in penalized estimators for Poisson and Poisson–Gaussian inverse problems.