M.Sc Student | Mordechay Moran |
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Subject | Optimal Measurements for Poisson Compressed Sensing |

Department | Department of Electrical Engineering |

Supervisor | Professor Yoav Schechner |

Full Thesis text |

Compressed sensing is a field in signal processing, which receives a lot of attention in recent years. In this framework, sparse signals are being measured by a given measurement matrix, such that the number of measurements is much lower than the dimension of the original signal. If the measurement matrix is chosen in a certain way, the sparse signals can be estimated well based on these compressed measurements. Usually, the measurements are assumed to be noiseless or contaminated by Gaussian noise.

Compressed sensing studies have rarely considered measurements contaminated by Poisson noise. Those who have focused on Poisson compressed sensing, suggested measurement matrix designs that either have poor reconstruction abilities or are limited in their applicability. There is a need for an optimization scheme to design the Poisson measurement matrix, with wider applicability and less limitation.

Our goal is to find an optimal nonnegative measurement matrix, such that sparse signals can be estimated well based on the noisy measurements, where the noise is Poissonian and the signals go through a known and uncontrolled mixing matrix before the measurements are taken. This situation is relevant to various optical applications.

We optimize the measurement matrix by mutual coherence minimization. Mutual coherence minimization has traditionally led to matrices that are equiangular tight frames. In Poissonian sensing, all the matrices are nonnegative. However, nonnegative equiangular frames exist only in irrelevant cases. We therefore compromise, and seek a quasi-equiangular frame, which is approximated by a tight frame.

In addition to nonnegativity, we require energy conservation, for photon sharing optical architectures.

For optical architectures which do not conserve energy, we consider a weaker constraint, limiting energy loss.

Simulations using synthetic signals, which are sparse in the canonical basis, demonstrate superior reconstruction using our optimized matrices. We also try to apply our method to Poisson compressed sensing of spectrometric signals, which are not sparse in the canonical basis. However, the reconstruction results show that generalizing our method to such signals is not trivial.