Performance bounds on compressed sensing with Poisson noise

TL;DR

This paper establishes performance bounds for compressed sensing under Poisson noise, relevant for low-light imaging, by constructing positivity-preserving sensing matrices and analyzing a Poisson likelihood-based reconstruction method.

Contribution

It introduces a feasible positivity-preserving sensing matrix and analyzes a Poisson likelihood-based reconstruction approach for sparse signals under Poisson noise.

Findings

Performance bounds for Poisson noise models are derived.

A positivity-preserving sensing matrix construction is proposed.

Analysis of a Poisson likelihood-based reconstruction method.

Abstract

This paper describes performance bounds for compressed sensing in the presence of Poisson noise when the underlying signal, a vector of Poisson intensities, is sparse or compressible (admits a sparse approximation). The signal-independent and bounded noise models used in the literature to analyze the performance of compressed sensing do not accurately model the effects of Poisson noise. However, Poisson noise is an appropriate noise model for a variety of applications, including low-light imaging, where sensing hardware is large or expensive, and limiting the number of measurements collected is important. In this paper, we describe how a feasible positivity-preserving sensing matrix can be constructed, and then analyze the performance of a compressed sensing reconstruction approach for Poisson data that minimizes an objective function consisting of a negative Poisson log likelihood term and a penalty term which could be used as a measure of signal sparsity.