Mathematical Problems in Engineering

Volume 2016 (2016), Article ID 9065438, 16 pages

http://dx.doi.org/10.1155/2016/9065438

## Computing the Pseudoinverse of Specific Toeplitz Matrices Using Rank-One Updates

^{1}University of Niš, Faculty of Sciences and Mathematics, Višegradska 33, 18000 Niš, Serbia^{2}Department of Economics, Division of Mathematics and Informatics, National and Kapodistrian University of Athens, Sofokleous 1 Street, 10559 Athens, Greece^{3}Faculty of Computer Science, Goce Delčev University, Goce Delčev 89, 2000 Štip, Macedonia

Received 29 February 2016; Accepted 3 July 2016

Academic Editor: Masoud Hajarian

Copyright © 2016 Predrag S. Stanimirović et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Application of the pure rank-one update algorithm as well as a combination of rank-one updates and the Sherman-Morrison formula in computing the Moore-Penrose inverse of the particular Toeplitz matrix is investigated in the present paper. Such Toeplitz matrices appear in the image restoration process and in many scientific areas that use the convolution. Four different approaches are developed, implemented, and tested on a number of numerical experiments.

#### 1. Introduction

Let and denote the set of all complex matrices and the set of all complex matrices of , respectively. The identity matrix of an appropriate order is denoted by . The conjugate transpose, the range, the rank, and the null space of are denoted by , , , and , respectively.

Representation and computation of various generalized inverses are closely related to the following Penrose equations: The set of all matrices obeying the conditions contained in a subset is denoted by . Any matrix from is called -inverse of and is denoted by . By we denote the set of all -inverses of with . For any matrix there exists a single element in the set , called the Moore-Penrose inverse of and denoted by .

A rank-one modification of a matrix is the matrix , which is created from and two vectors and . The Sherman-Morrison formula (S-M shortly) gives the basic relationship between the inverses and (for more details see, e.g., [1]):The identity (2) provides a numerically efficient way to compute the inverse of the rank-one update . The S-M formula is important in many different fields of numerical computation; see for example [2–7].

On the other hand, Toeplitz matrices arise in a number of various theoretical investigations and applications. A number of iterative processes for finding generalized inverses of an arbitrary Toeplitz matrix by modifying Newton’s method have been developed so far. The main results were stated in [8–13]. Adaptations of the iterative processes to the Toeplitz structure are based on the usage of the displacement operator as well as the concept of displacement representation and -displacement rank of matrices.

A variety of methods for computing the Moore-Penrose inverse of a rank-one modified matrix have been developed so far. Main results were derived in [14–18]. Relationships between various generalized inverses of an arbitrary matrix and corresponding generalized inverses of its rank-one modifications were investigated in [19]. The leading idea in [15, 16] was successive computation of the symmetric rank-one (SR1) updates of a given matrix , where denotes the th row of and . The authors of the paper [15] introduced a computational procedure for the Moore-Penrose inverse of a symmetric rank-one perturbed matrix. Using this method, the authors of [16] proposed a finite method for computing the minimum-norm least-squares solution of the linear system .

The results derived in [20] reveal that both the SR1 updates techniques and the S-M recursive rule are useful tools in the computation of various matrix products involving the Moore-Penrose inverse of certain symmetric matrices. Particularly, the algorithms introduced in [20] are numerically efficient in computation of and inverses.

In the present paper, we investigate the possibilities to apply the SR1 update procedure and the S-M formula in the computation of the Moore-Penrose inverse of specific Toeplitz matrices that appear in the image restoration process. Our main motivation arises from the convenience to apply the SR1 update and the S-M procedure in removing the blur which is always present in digital images. Firstly, both the SR1 update and the S-M formula are based on the usage of columns (or rows) of the input matrix. On the other hand, the matrices which appear in the mathematical model of blur in computer-generated images possess very specific structure which can be used to accelerate SR1 and S-M procedures. Namely, entries in Toeplitz matrices are constant along main diagonal parallels and, moreover, possess a significant proportion of zero elements.

The paper is organized as follows. Some basic notations and necessary facts are restated in Section 2. Also, some additional motivation is presented in the same section. Usage of the pure SR1 update algorithm, proposed in [15], in the computation of the Moore-Penrose inverse of a kind of Toeplitz matrices is considered in Section 3. A hybrid combination of the SR1 and the S-M recursive rules is defined in Section 4. An improvement of the SR1 procedure, which is derived on the basis of the specific structure of the underlying Toeplitz matrix, is presented in the same section. An application of introduced methods in image restoration is presented in Section 5.

#### 2. Preliminaries and Motivation

Toeplitz matrices or diagonally constant matrices are matrices having constant diagonal entries. Toeplitz matrices which are applicable in the image restoration process contain nonzero main diagonal parallels above the main diagonal, where defines the blurring process. In what follows, let us consider the Toeplitz matrix of such form:The assumption is active.

To clarify notation, Toeplitz matrices of the general form (3) will be denoted shortly by We investigate the use of the SR1 update method, as described in [15, Algorithm 2], during the numerical computation of the Moore-Penrose inverse of Toeplitz matrices satisfying the pattern. Also, we examine different improvements of the original method. The improvements are based on appropriate adaptations of the SR1 method and the S-M formula to the characteristic structure of underlying matrices of type . The method of SR1 updates is based on the expression which computes the Moore-Penrose inverse of the first columns of the initial matrix using the Moore-Penrose inverse of its first columns. In detail, the SR1 method from [15] starts from the well-known representation of the Moore-Penrose. If the th row of is denoted by , then Chen and Ji in [15] defined the matrix sequences and , asClearly, is the rank-one modification of and

Recall that the Moore-Penrose inverse of a general rank-one modified matrix , where is an arbitrary matrix and , are arbitrary vectors, is obtained in [14, Theorem 3.1.3]. The general theorem from [14] suggests six different cases that one has to follow in order to establish a relation between and . In [15], the authors proved that the real number , corresponding to the term in (2), satisfies . Later using this result in conjunction with the fact that is a positive semidefinite matrix, the six cases of Theorem 3.1.3 from [14] can be reduced to the two-case problem. This reduction simplifies the SR1 updates formulas.

Let us denote the first columns of by Also, the last columns of are denoted by . Then the matrix is given in the block form where the square block is the nonsingular band Toeplitz matrix and collects the last columns of .

An application of Greville’s partitioning method from [21] and the block partitioning method (BP method, shortly) from [22] in computing the Moore-Penrose inverse of the matrix is presented in [23]. According to Algorithms 1 and 2 and Lemma 3 from [23], it is clear that the specific structure of matrix enables computation simply by computing the inverse of the nonsingular block .

In this paper, we investigate some alternative methods for computing using the SR1 updates and the S-M formula. Our intention is to decrease computational complexity as much as possible using a specific structure of Toeplitz matrices of the general form .

#### 3. Computing the Pseudoinverse of a Toeplitz Matrix by Rank-One Updates

Our first attempt consists in applying the unmodified SR1 method from [15] in order to compute the pseudoinverse of the matrix that belongs to the class . Obtained results are compared with corresponding results derived by applying Algorithm 2 from [23] (the BP method). The results of this comparison are presented in Table 1, where , , and are the parameters which define the Gaussian blur modeled by the Toeplitz matrix . In the rest of the paper, it is assumed that the matrix is of the order , where and represents the width of the blurring function.