## Poisson compressed sensing

The goal of compressive sampling or compressed sensing (CS) is to replace conventional sampling by a more efficient data acquisition framework, which generally requires fewer sensing resources and exploits sparsity and structure in the underlying signal. This paradigm is particularly enticing whenever the measurement process is costly or constrained in some sense. For example, in the context of photon-limited applications (such as low-light imaging), the photomultiplier tubes used within sensor arrays are physically large and expensive. However, photon-limited measurements are commonly modeled with a Poisson probability distribution, posing significant theoretical and practical challenges in the context of CS. My lab has addressed some of the major theoretical challenges associated with the application of compressed sensing to practical hardware systems and developed performance bounds for compressed sensing in the presence of Poisson noise. We considered two novel sensing paradigms, based on either pseudo- random dense sensing matrices or expander graphs, which satisfy physical feasibility constraints. In these settings, 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 for a fixed signal intensity, the error bound actually grows with the number of measurements or sensors. In other words, my lab found that incorporating real-world constraints into our measurement model has a significant and adverse impact on the expected performance of a Poisson CS system, suggesting that compressed sensing may be fundamentally difficult for certain optical systems and networks.

- X. Jiang, G. Raskutti, and R. Willett, “Minimax Optimal Rates for Poisson Inverse Problems with Physical Constraints,”
*IEEE Transactions on Information Theory*, arXiv:1403.6532, 2014. - M. Raginsky, S. Jafarpour, Z. Harmany, R. Marcia, R. Willett, and R. Calderbank. “Performance bounds for expander-based compressed sensing in Poisson noise.”
*IEEE Transactions on Signal Processing*, vol. 59, no. 9, 2011. - M. Raginsky, R. Willett, Z. Harmany, and R. Marcia. “Compressed sensing performance bounds under Poisson noise,”
*IEEE Transactions on Signal Processing*Volume 58, Issue 8, pp 3990-4002, 2010.