In this paper we propose an approach to approximate a truncated singular value decomposition of a large structured matrix. By first decomposing the matrix into a sum of Kronecker products, our approach can be used to approximate a large number of singular values and vectors more efficiently than other well known schemes, such as randomized matrix algorithms or iterative algorithms based on Golub-Kahan bidiagonalization. We provide theoretical results and numerical experiments to demonstrate the accuracy of our approximation and show how the approximation can be used to solve large scale ill-posed inverse problems, either as an approximate filtering method, or as a preconditioner to accelerate iterative algorithms.
Since the publication by Kamm and Nagy @cite_4 , the algorithm has been extended in three general ways. First, the original 2D algorithm was modified to enable use on 3D problems @cite_26 . Second, Kilmer and Nagy extended the work from banded matrices to dense matrices @cite_31 . And third, the choices of boundary conditions have expanded to reflexive @cite_16 , anti-reflective @cite_24 , and whole-sample symmetric reflective boundary conditions @cite_14 , among others. All prior work uses the simple @math computations proposed in @cite_1 and @cite_4 .
@cite_26 @cite_4 @cite_14 @cite_1 @cite_24 @cite_31 @cite_16
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We derive Kronecker product approximations, with the help of tensor decompositions, to construct approximations of severely ill-conditioned matrices that arise in three-dimensional (3-D) image processing applications. We use the Kronecker product approximations to derive preconditioners for iterative regularization techniques; the resulting preconditioned algorithms allow us to restore 3-D images in a computationally efficient manner. Through examples in microscopy and medical imaging, we show that the Kronecker approximation preconditioners provide a powerful tool that can be used to improve efficiency of iterative image restoration algorithms. This paper considers the problem of finding n × n matrices Ak and Bk that minimize @math , where @math denotes Kronecker product and T is a banded n × n block Toeplitz matrix with banded n × n Toeplitz blocks. It is shown that the optimal Ak and Bk are banded Toeplitz matrices, and an efficient algorithm for computing the approximation is provided. An image restoration problem from the Hubble Space Telescope (HST) is used to illustrate the effectiveness of an approximate SVD preconditioner constructed from the Kronecker product decomposition. Reflexive boundary conditions (BCs) assume that the array values outside the viewable region are given by a symmetry of the array values inside. The reflection guarantees the continuity of the image. In fact, there are usually two choices for the symmetry: symmetry around the meshpoint and symmetry around the midpoint. The first is called whole-sample symmetry in signal and image processing, the second is half-sample. Many researchers have developed some fast algorithms for the problems of image restoration with the half-sample symmetric BCs over the years. However, little attention has been given to the whole-sample symmetric BCs. In this paper, we consider the use of the whole-sample symmetric boundary conditions in image restoration. The blurring matrices constructed from the point spread functions (PSFs) for the BCs have block Toeplitz-plus-PseudoHankel with Toeplitz-plus-PseudoHankel blocks structures. Recently, regardless of symmetric properties of the PSFs, a technique of Kronecker product approximations was successfully applied to restore images with the zero BCs, half-sample symmetric BCs and anti-reflexive BCs, respectively. All these results extend quite naturally to the whole-sample symmetric BCs, since the resulting matrices have similar structures. It is interesting to note that when the size of the true PSF is small, the computational complexity of the algorithm obtained for the Kronecker product approximation of the resulting matrix in this paper is very small. It is clear that in this case all calculations in the algorithm are implemented only at the upper left corner submatrices of the big matrices. Finally, detailed experimental results reporting the performance of the proposed algorithm are presented. Abstract Image restoration applications often result in ill-posed least squares problems involving large, structured matrices. One approach used extensively is to restore the image in the frequency domain, thus providing fast algorithms using fast Fourier transforms (FFTs). This is equivalent to using a circulant approximation to a given matrix. Iterative methods may also be used effectively by exploiting the structure of the matrix. While iterative schemes are more expensive than FFT-based methods, it has been demonstrated that they are capable of providing better restorations. As an alternative, we propose an approximate singular value decomposition (SVD), which can be used in a variety of applications. Used as a direct method, the computed restorations are comparable to iterative methods but are computationally less expensive. In addition, the approximate SVD may be used with the generalized cross validation method to choose regularization parameters. It is also demonstrated that the approximate SVD can be an effective preconditioner for iterative methods. In 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 (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.