Performance bounds for expander-based compressed sensing in the presence of poisson noise

Authors:
Sina Jafarpour;Rebecca Willett;Maxim Raginsky;Robert Calderbank
Affiliations:
Computer Science, Princeton University, Princeton, NJ;Electrical and Computer Engineering, Duke University, Durham, NC;Electrical and Computer Engineering, Duke University, Durham, NC;Electrical Engineering, Princeton University, Princeton, NJ
Venue:
Asilomar'09 Proceedings of the 43rd Asilomar conference on Signals, systems and computers
Year:
2009

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Abstract

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streaming, where the signal-independent and/or bounded noise models used in the compressed sensing literature are no longer applicable.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. The geometry of the expander graphs makes them provably superior to random dense sensing matrices, such as Gaussian or partial Fourier ensembles, for the Poisson noise model. We support our results with experimental demonstrations.