Kronecker product approximations for dense block Toeplitz‐plus‐Hankel matrices

Abstract

AbstractIn this paper, we consider the approximation of dense block Toeplitz‐plus‐Hankel matrices by sums of Kronecker products of Toeplitz‐plus‐Hankel matrices. We present an algorithm for efficiently computing the matrix approximation that requires the factorization of matrices of much smaller dimension than that of the original. The main results are described for block Toeplitz matrices with Toeplitz‐plus‐Hankel blocks, but the algorithms can be readily adjusted for other related structures that arise in image processing applications, such as block Toeplitz with Toeplitz blocks and block Toeplitz‐plus‐Hankel with Toeplitz‐plus‐Hankel blocks. Our work extends the techniques in Kamm and Nagy (SIAM J. Matrix Anal. Appl. 2000; 22:155–172) and Nagy et al. (SIAM J. Matrix Anal. Appl. 2004; 25:829–841), which consider similar matrices, but with the added restriction that the matrices have a banded/block‐banded structure. We illustrate the effectiveness of our algorithm by using the output of the algorithm to construct preconditioners for systems from two different applications: diffuse optical tomography and atmospheric image deblurring. Copyright © 2007 John Wiley & Sons, Ltd.