Multiscale Analysis of Photon-Limited Astronomical Signals and Images

Rebecca Willett
Rice University
Electrical and Computer Engineering

The nonparametric multiscale methods presented here, unlike traditional 
wavelet-based methods, are well suited to photon-limited astronomical 
applications and capable of efficiently solving Poisson inverse problems 
such as deblurring and superresolution image reconstruction, a computational
process used to reconstruct high- resolution images from several blurred and 
noisy low-resolution images. The recursive partitioning scheme underlying 
these methods is based on multiscale likelihood factorizations of the Poisson 
data model. These partitions allow the construction of multiscale signal 
decompositions based on polynomials in one dimension and multiscale image 
decompositions based on platelets in two dimensions. Platelets are localized 
functions at various positions, scales and orientations that can produce highly 
accurate, piecewise linear approximations to images consisting of smooth regions 
separated by smooth boundaries.  

Statistical risk analysis establishes the theoretical near-optimality of these 
methods for certain interesting classes of signals and images. Because platelet-based 
maximum penalized likelihood methods for image analysis are both tractable and 
computationally efficient, existing image reconstruction methods based on 
expectation- maximization type algorithms can be easily enhanced with platelet- based 
techniques. These methods are useful in several important applications, including Gamma 
Ray Burst intensity estimation, astronomical image deblurring, and superresolution 
reconstruction.  

Simulations demonstrate the effectiveness of the proposed methods, including their ability
to distinguish between tightly grouped stars with a small set of observations.

This is joint work with Robert Nowak.