Reconstruction Error Bounds for Compressed Sensing under Poisson or Poisson-Gaussian Noise Using Variance Stabilization Transforms

TL;DR

This paper develops upper bounds for signal reconstruction error in compressive sensing systems affected by Poisson or Poisson-Gaussian noise, using variance stabilization transforms and convex estimators, applicable to sparse and compressible signals.

Contribution

It introduces new error bounds for Poisson and Poisson-Gaussian noise models using a convex estimator with variance stabilization, applicable to realistic compressive sensing scenarios.

Findings

Bounds are applicable to sparse and compressible signals.

Estimator effectively handles non-negativity and flux-preservation constraints.

Numerical results demonstrate the bounds' effectiveness across measurement and signal intensity variations.

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 probabilistically motivated, computationally tractable convex estimator with principled parameter selection. The estimator penalizes signal sparsity subject to a constraint that imposes an upper bound on a term based on variance stabilization transforms to approximate the Poisson or Poisson-Gaussian negative log-likelihoods. (2) They 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. We present extensive numerical results for signal reconstruction under varying number of measurements and varying signal intensity levels.