Showing papers on "Circulant matrix published in 1983"

A fast Kalman filter for images degraded by both blur and noise

Abstract: In this paper a fast Kalman filter is derived for the nearly optimal recursive restoration of images degraded in a deterministic way by blur and in a stochastic way by additive white noise. Straightforwardly implemented optimal restoration schemes for two-dimensional images degraded by both blur and noise create dimensionality problems which, in turn, lead to large storage and computational requirements. When the band-Toeplitz structure of the model matrices and of the distortion matrices in the matrix-vector formulations of the original image and of the noisy blurred observation are approximated by circulant matrices, these matrices can be diagonalized by means of the FFT. Consequently, a parallel set of N dynamical models suitable for the derivation of N low-order vector Kalman filters in the transform domain is obtained. In this way, the number of computations is reduced from the order of O(N4) to that of O(N^{2} \log_{2} N) for N × N images.

The inverse of a block-circulant matrix

Abstract: The inverse A^{-1} of a block-circulant matrix (BCM) A is given in a closed form, by using the fact that a BCM is a combination of permutation matrices, whose eigenvalues and eigenvectors are found with the help of the complex roots of unity. Special results are also given when A is block symmetric or symmetric.

A general orthogonalization technique with applications to time series analysis and signal processing

Abstract: A new orthogonalization technique is presented for computing the QR factorization of a general n X p matrix of full rank p (n 2 p). The method is based on the use of projections to solve increasingly larger subproblems recursively and has an O(np2) operation count for general matrices. The technique is readily adaptable to solving linear least-squares problems. If the initial matrix has a circulant structure the algorithm simplifies significantly and gives the so-called lattice algorithm for solving linear prediction problems. From this point of view it is seen that the lattice algorithm is really an efficient way of solving specially structured least-squares problems by orthogonalization as opposed to solving the normal equations by fast Toeplitz algorithms.

TOEPLITZ package users' guide

Abstract: The TOEPLITZ package is a collection of Fortran subroutines for the numerical solution of systems of linear equations with coefficient matrices of Toeplitz or circulant form. This report provides a description of the algorithms and software in the package and includes program listings.

The TOEPLITZ Package Users' Guide

Abstract: The TOEPLITZ package is a collection of Fortran subroutines for the numerical solution of systems of linear equations with coefficient matrices of Toeplitz or circulant form. This report provides a description of the algorithms and software in the package and includes program listings.

A fast algorithm for solving Toeplitz system of equations

Abstract: A fast algorithm for the solution of Toeplitz system of equations is presented. The algorithm requires order N (\log N)^{2} computations where N is the number of equations. For banded Toeplitz matrices the order of computations is reduced to only N \log N + m (\log m)^2 where 2m is the maximum number of nonzero principal subdiagonals of the Toeplitz matrix. The algorithm is in essence a fast implementation of the Trench algorithm in reverse. Thus the algorithm involves imbedding of the given matrix in a cyclic matrix and a fast HD (Half Divisor) algorithm to compute the first row of the inverse matrix. The desired solution is then obtained directly from the first row by applying Fast Fourier Transform techniques in order N \log N computations. Finally, the extension of the algorithm to block Toeplitz matrices is also presented.

Discrete-time multivariable criterion for limit cycle predictions

Abstract: Circulant matrices and their properties provide a new interpretation of limit-cycle phenomena in discrete-time nonlinear feedback systems. The present work extends the results of a recent paper in order to derive a multivariable circle type of criterion with a suitable Nyquist interpretation.