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

• Main Authors: Garg, Deepak, Bohra, Pakshal, Gurumoorthy, Karthik S, Rajwade, Ajit
• Format: Journal Article
• Language: English
• Published: 03.07.2017
• Subjects: Computer Science - Information Theory; Mathematics - Information Theory
• DOI: 10.48550/arxiv.1707.00475

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.