Compressed sensing performance bounds under Poisson noise

TL;DR

This paper establishes performance bounds for compressed sensing with Poisson noise, addressing the challenges of non-additive, signal-dependent noise and practical measurement constraints, and analyzing how reconstruction error behaves with signal intensity and measurements.

Contribution

It introduces a feasible positivity-preserving sensing matrix and analyzes the reconstruction error bounds for Poisson noise, highlighting how error scales with signal intensity and measurement count.

Findings

Error bounds decay with increasing signal intensity for fixed measurements.

Signal-dependent error component grows with the number of measurements at fixed intensity.

Constructed sensing matrices adhere to physical constraints like nonnegativity and flux preservation.

Abstract

This paper describes performance bounds for compressed sensing (CS) where the underlying sparse or compressible (sparsely approximable) signal is a vector of nonnegative intensities whose measurements are corrupted by Poisson noise. In this setting, standard CS techniques cannot be applied directly for several reasons. First, the usual signal-independent and/or bounded noise models do not apply to Poisson noise, which is non-additive and signal-dependent. Second, the CS matrices typically considered are not feasible in real optical systems because they do not adhere to important constraints, such as nonnegativity and photon flux preservation. Third, the typical $\ell_2$--$\ell_1$ minimization leads to overfitting in the high-intensity regions and oversmoothing in the low-intensity areas. In this paper, we describe how a feasible positivity- and flux-preserving sensing matrix can be constructed, and then analyze the performance of a CS reconstruction approach for Poisson data that minimizes an objective function consisting of a negative Poisson log likelihood term and a penalty term which measures signal sparsity. We show that, as the overall intensity of the underlying signal increases, an upper bound on the reconstruction error decays at an appropriate rate (depending on the compressibility of the signal), but that for a fixed signal intensity, the signal-dependent part of the error bound actually grows with the number of measurements or sensors. This surprising fact is both proved theoretically and justified based on physical intuition.