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E.D. Kolaczyk (Department of Mathematics \& Statistics, Boston University), R.D. Nowak (ECE Department, Michigan State University)
We present a new approach for producing deconvolved spectra and images, based on a novel non-parametric, multi-scale statistical model designed explicitly for Poisson limited data. The framework within which we work is completely general, requiring only that the user specify the manner in which the data were ``blurred'' (for example, through an instrument-specific PSF). Therefore, we anticipate our method serving in problems involving data at any of a variety of energies, especially at the x-ray and gamma-ray levels, for deconvolution problems arising in a variety of missions -- particularly when there is not yet a good analytical model for the source(s).
The underlying statistical framework explicitly models the process of (dis)aggregating counts across multiple resolutions. The result is a multi-scale (though not wavelet-based) representation of the source object to be recovered through the deconvolution. Furthermore, this framework is built completely within the context of the original Poisson data likelihood, so that we proceed without the use of statistical approximations (such as \chi2 approximations) or data transformations. Adopting a Bayesian paradigm, a flexible prior probability structure is used to regularize the set of possible solutions to what is formally a statistical inverse problem. This prior is both intuitive and interpretable, in that it models the degree to which counts are allowed to be (dis)aggregated at each location-scale combination.
Estimates of the ``deblurred'' source object are obtained in our procedure using standard Bayesian statistical techniques (i.e., based on the mode of the posterior distribution of the object given the data). Despite the generality of this method and the potentially complex structures that may be modeled, these estimates may be produced using an efficient iterative algorithm (i.e., the expectation-maximization (EM) algorithm), wherein iterates at each stage are yielded by closed-form solutions to simple algebraic equations. In particular, this allows us to perform our computations without recourse to Monte Carlo simulation methods. We present examples in this talk using both simulated and actual data.
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