Compressed Sensing Performance Bounds Under Poisson Noise

Abstract

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 2-1 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.

Sparsity and Structure in Hyperspectral Imaging : Sensing, Reconstruction, and Target Detection

“…Some of the major theoretical challenges associated with the application of CS to linear optical systems in the presence of Poisson noise have been addressed in the recent literature [38,39]. These works considered two novel sensing paradigms, based on either pseudo-random dense sensing matrices (akin to the shifted and scaled dense sensing matrix described above) or expander graph constructions, both of which satisfy the nonnegativity and flux preservation constraints.…”

Photon-limited Sensing and Surveillance

“…Most prior theoretical results in compressed sensing and related inverse problems apply to idealized settings where the noise is i.i.d., and do not account for signal-dependent noise and physical sensing constraints. Prior results on Poisson compressed sensing with signal-dependent noise and physical constraints in [16] provided upper bounds on mean squared error performance for a specific class of estimators. However, it was unknown whether those bounds were tight or if other estimators could achieve significantly better performance.…”

Compressive sensing computational ghost imaging

“…The successive steps of the proposed iterative algorithm are based on the above solutions for (21) and (22). In our implementation of the algorithm the analysis and synthesis operations, thresholding and design of the analysis and synthesis frames (matrixes and ) are integrated in a single block, which we call BM3D-…lter.…”

“…Some of the basic principles discussed in these works can be tracked in our technique. In the pa- Raginsky et al [22].…”