Skellam shrinkage: Wavelet-based intensity estimation for inhomogeneous Poisson data

TL;DR

This paper introduces Skellam shrinkage methods for wavelet-based intensity estimation of inhomogeneous Poisson data, providing new estimators with theoretical guarantees and demonstrating superior performance through simulations.

Contribution

It develops novel Skellam shrinkage estimators for Poisson data in the wavelet domain, including Bayesian and frequentist optimality results and practical algorithms.

Findings

Skellam distribution naturally models wavelet coefficients of Poisson data.

Proposed estimators outperform existing methods in simulations.

Algorithms are computationally efficient and effective for image denoising.

Abstract

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 substantial performance improvements over existing alternatives.