  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.
