2,737 research outputs found
Performance bounds for expander-based compressed sensing in Poisson noise
This paper provides performance bounds for compressed sensing in the presence
of Poisson noise using expander graphs. The Poisson noise model is appropriate
for a variety of applications, including low-light imaging and digital
streaming, where the signal-independent and/or bounded noise models used in the
compressed sensing literature are no longer applicable. In this paper, we
develop a novel sensing paradigm based on expander graphs and propose a MAP
algorithm for recovering sparse or compressible signals from Poisson
observations. The geometry of the expander graphs and the positivity of the
corresponding sensing matrices play a crucial role in establishing the bounds
on the signal reconstruction error of the proposed algorithm. We support our
results with experimental demonstrations of reconstructing average packet
arrival rates and instantaneous packet counts at a router in a communication
network, where the arrivals of packets in each flow follow a Poisson process.Comment: revised version; accepted to IEEE Transactions on Signal Processin
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.Comment: 5 pages; to appear in Proc. ISIT 200
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
Poisson Matrix Completion
We extend the theory of matrix completion to the case where we make Poisson
observations for a subset of entries of a low-rank matrix. We consider the
(now) usual matrix recovery formulation through maximum likelihood with proper
constraints on the matrix , and establish theoretical upper and lower bounds
on the recovery error. Our bounds are nearly optimal up to a factor on the
order of . These bounds are obtained by adapting
the arguments used for one-bit matrix completion \cite{davenport20121}
(although these two problems are different in nature) and the adaptation
requires new techniques exploiting properties of the Poisson likelihood
function and tackling the difficulties posed by the locally sub-Gaussian
characteristic of the Poisson distribution. Our results highlight a few
important distinctions of Poisson matrix completion compared to the prior work
in matrix completion including having to impose a minimum signal-to-noise
requirement on each observed entry. We also develop an efficient iterative
algorithm and demonstrate its good performance in recovering solar flare
images.Comment: Submitted to IEEE for publicatio
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