  This paper provides performance bounds for compressed sensing in the presence
of Poisson noise using expander graphs. The Poisson noise model is appropriate
for a variety of applications, including low-light imaging and digital
streaming, where the signal-independent and/or bounded noise models used in the
compressed sensing literature are no longer applicable. In this paper, we
develop a novel sensing paradigm based on expander graphs and propose a MAP
algorithm for recovering sparse or compressible signals from Poisson
observations. The geometry of the expander graphs and the positivity of the
corresponding sensing matrices play a crucial role in establishing the bounds
on the signal reconstruction error of the proposed algorithm. We support our
results with experimental demonstrations of reconstructing average packet
arrival rates and instantaneous packet counts at a router in a communication
network, where the arrivals of packets in each flow follow a Poisson process.
