Performance bounds on compressed sensing with Poisson noise

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.

[1]  R. White,et al.  Image recovery from data acquired with a charge-coupled-device camera. , 1993, Journal of the Optical Society of America. A, Optics and image science.

[2]  H. Chernoff.  A Measure of Asymptotic Efficiency for Tests of a Hypothesis Based on the sum of Observations , 1952 .

[3]  R. DeVore,et al.  A Simple Proof of the Restricted Isometry Property for Random Matrices , 2008 .

[4]  S. Mendelson,et al.  Reconstruction and Subgaussian Operators in Asymptotic Geometric Analysis , 2007 .

[5]  Andrew R. Barron,et al.  Mixture Density Estimation , 1999, NIPS.

[6]  Emmanuel J. Candès,et al.  Near-Optimal Signal Recovery From Random Projections: Universal Encoding Strategies? , 2004, IEEE Transactions on Information Theory.

[7]  Alexandros G. Dimakis,et al.  Sparse Recovery of Positive Signals with Minimal Expansion , 2009, ArXiv.

[8]  M. Rudelson.  Random Vectors in the Isotropic Position , 1996, math/9608208.

[9]  Y. Censor,et al.  Parallel Optimization:theory , 1997 .

[10]  Emmanuel J. Candès,et al.  Decoding by linear programming , 2005, IEEE Transactions on Information Theory.

[11]  E. Candès,et al.  Stable signal recovery from incomplete and inaccurate measurements , 2005, math/0503066.

[12]  Robert D. Nowak,et al.  Signal Reconstruction From Noisy Random Projections , 2006, IEEE Transactions on Information Theory.

[13]  Xiaoming Huo,et al.  Uncertainty principles and ideal atomic decomposition , 2001, IEEE Trans. Inf. Theory.

[14]  I. Csiszár.  Why least squares and maximum entropy? An axiomatic approach to inference for linear inverse problems , 1991 .

[15]  M E Gehm,et al.  Single-shot compressive spectral imaging with a dual-disperser architecture. , 2007, Optics express.

[16]  Y. Censor,et al.  Parallel Optimization: Theory, Algorithms, and Applications , 1997 .

[17]  R. Nowak,et al.  Multiscale likelihood analysis and complexity penalized estimation , 2004, math/0406424.

[18]  R. Tibshirani.  Regression Shrinkage and Selection via the Lasso , 1996 .

[19]  Olgica Milenkovic,et al.  Weighted Superimposed Codes and Constrained Integer Compressed Sensing , 2008, IEEE Transactions on Information Theory.

[20]  Joel A. Tropp,et al.  Just relax: convex programming methods for identifying sparse signals in noise , 2006, IEEE Transactions on Information Theory.