Variance Stabilization Based Compressive Inversion under Poisson or Poisson-Gaussian Noise with Analytical Bounds

Ajit Rajwade

Abstract:

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 nonnegativity 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.