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https://hdl.handle.net/2142/19116
Description
Title
Constrained signal reconstruction
Author(s)
Potter, Lee Carson
Issue Date
1990
Doctoral Committee Chair(s)
Arun, K.S.
Department of Study
Electrical and Computer Engineering
Discipline
Computer Engineering
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Engineering, Electronics and Electrical
Language
eng
Abstract
The research reported in this dissertation addresses the reconstruction of signals and images from linear measurements subject to convex constraints. The objectives are to describe the existence and uniqueness of solutions, to characterize reconstructions, and to develop algorithms for efficiently computing reconstructions. A prototypical inverse problem of this type is the extrapolation of a positive semidefinite sequence, which is equivalent to the covariance extension and trigonometric moment problems. Classical results are extended to incorporate the additional convex constraints imposed by spectral support limits and bounding functions. An order N$\sp2$ matrix test is given for the extendibility of a partial covariance sequence subject to a spectral support constraint, and recursive reconstruction algorithms employing the Levinson algorithm follow from the constructive proof.
For the general problem of arbitrary linear measurements and arbitrary convex constraints, the signal recovery problem is formulated in Hilbert space as the determination of the signal closest to a nominal signal and lying in the intersection of the convex constraint sets and a linear variety determined by the measurements. A finite parameterization for this solution is established, and conditions for the existence of the parameter vector are derived. Together, the form of the solution and the constraint qualification constitute a novel optimization result and provide an easily applied reconstruction framework. The parameters are shown to satisfy a nonlinear system of equations in IR$\sp{\rm N}$, or equivalently, to be the fixed point of a nonlinear operator. Iterative algorithms are developed for determining the parameters and are shown to produce a minimum norm least-squares solution when data are rendered nonextendible by measurement noise. Convergence proofs rely on properties of firmly nonexpansive operators since the nonlinear operator is neither a contraction nor has compact domain. The applicability and flexibility of the results are demonstrated on two practical problems.
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