Nonparametric Estimation of Intensity Maps Using Haar Wavelets and Poisson Noise Characteristics

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Abstract
We present a method for the nonparametric (model-free) estimation of an intensity map underlying two-dimensional count data with Poisson noise characteristics. Specifically, we extend the so-called TIPSH (translation invariant Poisson smoothing using Haar wavelets) methodology, which was formulated for denoising Poisson time series data in the context of γ-ray bursts. In addition to the obvious extension of TIPSH to the two-dimensional context of image data, the primary contribution of this paper is in showing the extreme generality under which exact thresholding may be done when using the Haar transform. This generality allows for the efficient evaluation of arbitrarily complex models for observed data within a multiscale framework. This latter characteristic has played a critical role in a recent analysis of EGRET data which provided strong evidence for anomalous large-scale emission in high-energy γ-rays (the so-called γ-ray halo). Here we concentrate on describing the details of the new thresholding scheme as well as exploring the results of a number of simulation studies.