| This thesis is concerned with the problem of restoring a signal from noisy measurements subject to a variety of constraints on the signal and the observation noise. Such problems can be constructed as straightforward constrained optimization problems in the following manner. A cost functional is used to assign a cost to every signal which satisfied the signal and the noise constraints. Then the signal with minimum cost is selected as the best estimate of the signal. Equivalently, these problems can be mapped to alternative problems, known as duals, whose solution can be used to obtain the signal to be estimated using a predetermined parametric model of the signal. In other words, the signal to be restored has a parametric representation, and its parameters are the solution of the dual problem. This thesis is an attempt to provide a general framework for the previous work done in signal restoration using the duals and to explore new perspectives regarding the applicability of the approach. The contributions of the thesis include an explicit and relatively simple formula to construct a relatively large class of dual problems and signal models, a sufficient condition under which the dual approach is applicable, a class of fast algorithms for least-squares signal restoration whose computational complexity is similar to solving several sets of Toeplitz equations, and two criteria for selecting a cost functional which use a prior estimate of the signal to be restored in a consistent manner. |
