Parallel Optimization: Theory, Algorithms and Applications
Parallel Optimization: Theory, Algorithms and Applications
Elements of Information Theory (Wiley Series in Telecommunications and Signal Processing)
Elements of Information Theory (Wiley Series in Telecommunications and Signal Processing)
A proximal iteration for deconvolving Poisson noisy images using sparse representations
IEEE Transactions on Image Processing
Nested Iterative Algorithms for Convex Constrained Image Recovery Problems
SIAM Journal on Imaging Sciences
Weighted superimposed codes and constrained integer compressed sensing
IEEE Transactions on Information Theory
Sparse density estimation with l1 penalties
COLT'07 Proceedings of the 20th annual conference on Learning theory
Sparse signal recovery with exponential-family noise
Allerton'09 Proceedings of the 47th annual Allerton conference on Communication, control, and computing
Performance bounds for expander-based compressed sensing in the presence of poisson noise
Asilomar'09 Proceedings of the 43rd Asilomar conference on Signals, systems and computers
Multiscale Photon-Limited Spectral Image Reconstruction
SIAM Journal on Imaging Sciences
Decoding by linear programming
IEEE Transactions on Information Theory
Just relax: convex programming methods for identifying sparse signals in noise
IEEE Transactions on Information Theory
IEEE Transactions on Information Theory
Signal Reconstruction From Noisy Random Projections
IEEE Transactions on Information Theory
Near-Optimal Signal Recovery From Random Projections: Universal Encoding Strategies?
IEEE Transactions on Information Theory
Multiscale Poisson Intensity and Density Estimation
IEEE Transactions on Information Theory
Performance bounds for expander-based compressed sensing in the presence of poisson noise
Asilomar'09 Proceedings of the 43rd Asilomar conference on Signals, systems and computers
A Novel Sparsity Reconstruction Method from Poisson Data for 3D Bioluminescence Tomography
Journal of Scientific Computing
Compressive sensing based sub-mm accuracy UWB positioning systems: A space-time approach
Digital Signal Processing
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This paper describes performance bounds for compressed sensing (CS) where the underlying sparse or compressible (sparsely approximable) signal is a vector of nonnegative intensities whose measurements are corrupted by Poisson noise. In this setting, standard CS techniques cannot be applied directly for several reasons. First, the usual signal-independent and/or bounded noise models do not apply to Poisson noise, which is nonadditive and signal-dependent. Second, the CS matrices typically considered are not feasible in real optical systems because they do not adhere to important constraints, such as nonnegativity and photon flux preservation. Third, the typical l2 - l1 minimization leads to overfitting in the high-intensity regions and oversmoothing in the low-intensity areas. In this paper, we describe how a feasible positivity-and flux-preserving sensing matrix can be constructed, and then analyze the performance of a CS reconstruction approach for Poisson data that minimizes an objective function consisting of a negative Poisson log likelihood term and a penalty term which measures signal sparsity. We show that, 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 that for a fixed signal intensity, the error bound actually grows with the number of measurements or sensors. This surprising fact is both proved theoretically and justified based on physical intuition.