135 research outputs found
Compressed sensing performance bounds under Poisson noise
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 non-additive 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 -- 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
signal-dependent part of 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.Comment: 12 pages, 3 pdf figures; accepted for publication in IEEE
Transactions on Signal Processin
Sparse Poisson Intensity Reconstruction Algorithms
The observations in many applications consist of counts of discrete events,
such as photons hitting a dector, which cannot be effectively modeled using an
additive bounded or Gaussian noise model, and instead require a Poisson noise
model. As a result, accurate reconstruction of a spatially or temporally
distributed phenomenon (f) from Poisson data (y) cannot be accomplished by
minimizing a conventional l2-l1 objective function. The problem addressed in
this paper is the estimation of f from y in an inverse problem setting, where
(a) the number of unknowns may potentially be larger than the number of
observations and (b) f admits a sparse approximation in some basis. The
optimization formulation considered in this paper uses a negative Poisson
log-likelihood objective function with nonnegativity constraints (since Poisson
intensities are naturally nonnegative). This paper describes computational
methods for solving the constrained sparse Poisson inverse problem. In
particular, the proposed approach incorporates key ideas of using quadratic
separable approximations to the objective function at each iteration and
computationally efficient partition-based multiscale estimation methods.Comment: 4 pages, 4 figures, PDFLaTeX, Submitted to IEEE Workshop on
Statistical Signal Processing, 200
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