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.
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