Summed squared distance error reduction by simultaneous multiprojections and applications
Applied Mathematics and Computation
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Signal synthesis and reconstruction is considered when the signal is to be determined by N constraint sets, Ci. The solution sought is required to minimize a weighted quadratic cost functional $\hat J$. Emphasis is on cases in which the intersection of the sets Ci is empty. Our proposed procedure employs a suitably weighted simultaneous projection iteration method. It is shown that the iterates generated by the algorithm converge weakly to a global minimizer of $\hat J$ provided the set of fixed points of the algorithm is nonempty. If the problem is consistent $(C_o:=\bigcap C_i \neq \emptyset),$ weak convergence is to an element in Co. However, it is indicated that large classes of inconsistent problems, which could not be treated by existing methods, admit a solution as well.