Skellam Shrinkage: Wavelet-Based Intensity Estimation for Inhomogeneous Poisson Data

The ubiquity of integrating detectors in imaging and other applications implies that a variety of real-world data are well modeled as Poisson random variables whose means are in turn proportional to an underlying vector-valued signal of interest. In this article, we first show how the so-called Skel...

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Bibliographic Details
Published in:IEEE transactions on information theory 2012-02, Vol.58 (2), p.1080-1093
Main Authors: Hirakawa, K., Wolfe, P. J.
Format: Article
Language:English
Subjects:
Algorithms
Applied sciences
Approximation methods
Bayesian analysis
Detection, estimation, filtering, equalization, prediction
Estimation
Estimators
Exact sciences and technology
Image processing
Information theory
Information, signal and communications theory
Mathematical models
Nonhomogeneous media
Random variables
Reactive power
Risk
Shrinkage
Signal and communications theory
Signal processing
Signal, noise
Simulation
Telecommunications and information theory
Theorems
Transforms
Wavelet
Wavelet transforms
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Summary:The ubiquity of integrating detectors in imaging and other applications implies that a variety of real-world data are well modeled as Poisson random variables whose means are in turn proportional to an underlying vector-valued signal of interest. In this article, we first show how the so-called Skellam distribution arises from the fact that Haar wavelet and filterbank transform coefficients corresponding to measurements of this type are distributed as sums and differences of Poisson counts. We then provide two main theorems on Skellam shrinkage, one showing the near-optimality of shrinkage in the Bayesian setting and the other providing for unbiased risk estimation in a frequentist context. These results serve to yield new estimators in the Haar transform domain, including an unbiased risk estimate for shrinkage of Haar-Fisz variance-stabilized data, along with accompanying low-complexity algorithms for inference. We conclude with a simulation study demonstrating the efficacy of our Skellam shrinkage estimators both for the standard univariate wavelet test functions as well as a variety of test images taken from the image processing literature, confirming that they offer some performance improvements over existing alternatives.
ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2011.2165933