Mathematical Sciences: Inverse Estimation Problems

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Proposal: DMS 9504485 PI: Frits Ruymgaart Institution: Texas Tech Title: Inverse Estimation Problems Abstract: Inverse estimation is concerned with indirect curve estimation: the curve of interest is to be recovered from observations, subject to random blur, on a transformation of the curve. This class of problems is very broad and formally contains ordinary, direct, curve estimation as a special case. Many interesting examples come from physics, signal processing, statistics, and applied mathematics. The estimation problem can essentially be solved by inverting the transformation involved. Since this inverse is not in general continuous, the problem is typically ill-posed and regularization of the inverse is required. Two important questions are how restriction, due to limitations in time or space, of the transformation relate to the unrestricted transformation, and how recovery of irregular curves can be modified so as to avoid the Gibbs phenomenon. Along with a comprehensive study of deconvolution on locally compact Abelian groups, this research addresses these questions. Also, the study of some examples of practical interest are explored. Inverse estimation arises when an object, e.g., part of the human brain, can only be indirectly observed, and recovery of information about the object is required from the indirect measurements. Indirect measurement techniques, such as those used in medical imagery, are attractive because they are noninvasive. Perfect reconstruction of the image would, in principle, be possible if an unlimited number of uncorrupted measurements were available. In practice, however, one can only obtain finitely many data that, moreover, are corrupted by measurement errors. Here statistical considerations and techniques play a role in the image reconstruction process. Aspects addressed in this research include how the reconstruction depends on the specific way in which the indirect data are collected, and al so how it should be tuned to prior information about the object to be recovered.