Ed. Kolaczyk et Dd. Dixon, Nonparametric estimation of intensity maps using Haar wavelets and Poissonnoise characteristics, ASTROPHYS J, 534(1), 2000, pp. 490-505
We present a method for the nonparametric (model-free) estimation of an int
ensity map underlying two-dimensional count data with Poisson noise charact
eristics. Specifically, we extend the so-called TIPSH (translation invarian
t Poisson smoothing using Haar wavelets) methodology, which was formulated
for denoising Poisson time series data in the context of gamma-ray bursts.
In addition to the obvious extension of TIPSH to the two-dimensional contex
t of image data, the primary contribution of this paper is in showing the e
xtreme generality under which exact thresholding may be done when using the
Haar transform. This generality allows for the efficient evaluation of arb
itrarily complex models for observed data within a multiscale framework. Th
is latter characteristic has played a critical role in a recent analysis of
EGRET data which provided strong evidence for anomalous large-scale emissi
on in high-energy gamma-rays (the so-called gamma-ray halo). Here we concen
trate on describing the derails of the new thresholding scheme as well as e
xploring the results of a number of simulation studies.