@MISC{Chu_lowrank, author = {Moody T. Chu and Robert J. Plemmons}, title = {Low Rank Circulant Approximation}, year = {} }

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Abstract

Partially due to the fact that the empirical data collected in practice by devices with finite bandwidth often neither maintain the specified structure nor induce a certain desired rank, structured low rank approximation arises in many important applications. Approximation by structured low rank matrices has particular significance in signal and image processing. This paper discusses a procedure for low rank approximation involving circulant structure. The low rank circulant real approximation problem is not as straightforward as the usual truncated singular value decomposition since a conjugate-even set of eigenvalues must be maintained to guarantee a real-valued approximation. The fast Fourier transform together with a sorting scheme are employed to compute the nearest real-valued circulant approximation with a specific rank to a given target matrix. Extensions of this work to block circulant with circulant blocks is possible which could lead to an efficient way of preconditioning image post-processing computations.