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.

If an approximation of is denoted by , then the residual norms are denoted byFor the sake of simplicity, let us denote Algorithm  2 from [15] by SR1 and Algorithm  2 from [23] by BP.

From Table 1, it is observable that the SR1 method is marginally efficient in terms of accuracy, but it is time consuming, especially for larger dimensions. There is a theoretical explanation for these results. The number of multiplications and divisions included in the SR1 algorithm is about in the case and in the case . Therefore, as it is stated in [15], the rank-one updates algorithm works efficiently in cases where it holds or . In our case, , where represents the width of the blurring process. Since, in the general case, is a relatively small integer, the conditions and are satisfied. This fact causes relative inefficiency of the SR1 method.

Our second attempt is to apply Algorithm SR1 to the matrix , from (10), in order to generate its inverse. Then the Moore-Penrose inverse of comes from the block partitioning method (called BP), as it was described in [23]. This approach is shortly denoted by HSR1 + BP.

It is observable from Table 2 that the method HSR1 + BP is marginally efficient in terms of accuracy, but it is time consuming; see, for example, the case .

From the previous analysis of data included in Tables 1 and 2, we conclude that both algorithms SR1 and HSR1 + BP are not efficient approaches for computing with respect to the BP method in terms of computational speed, especially for large matrices. But both approaches show marginal efficiency in terms of the accuracy.

4. A Combination of SR1 Updates and S-M Formula

Two additional algorithms are defined in the current section. The first one uses the SR1 updates to compute the matrix in the first step and the S-M formula to compute in the subsequent step. The second algorithm replaces the usage of the SR1 updates from the first step by a direct construction of the matrix . The matrix can be generated efficiently in view of the specific structure of the input matrix .

4.1. SR1 Updates in Conjunction with the S-M Formula

As usual, the th column of is denoted by . Then, since is of full row rank, it is advantageous to define a proper SR1 recursive computational method for generating the Moore-Penrose inverse of on the basis of the relationFor that purpose, it is necessary to define the matrix sequenceLemma 1 reveals some of basic properties of the sequence . Following the notation from [20], the notation means that a column is linearly dependent on the previous columns and indicates that is linearly independent of .

Lemma 1. The following statements are valid:(i), for each .(ii), for each .(iii), for each , and , for each .(iv)The following concluding relation is valid:

Proof. (i) This part of the proof is implied by the fact that the columns of are linearly independent.
(ii) According to the part (i), immediately follows , for each . Later, it is easy to verify which implies(iii) This part of the proof can be verified by using (i) and (ii), together with the fact that are all matrices, for each .
(iv) According to part (iii), matrices , , are invertible. Since is invertible, then it holds which completes the proof.

Remark 2. According to Lemma 1 the matrices are of the order and it holds , for each , while , for each . Since , the invertibility of is guaranteed, so that we are able to apply the S-M formula which relates with , for each .
According to Lemma 1, matrices , , are invertible. Therefore, it is possible to define the following matrix sequence:Identity (15) immediately implies

Lemma 3. It holds , for each .

Proof. The proof of lemma follows from the fact that is regular, for each .

Now, we turn our attention to (20). Since the matrices , are regular, it is possible to find a relation between and by applying the Sherman-Morrison formula (2). This relation is given in Proposition 4.

Proposition 4. Let the matrix be defined as in (3). Assume that the matrix sequence is defined in (14). The inverse of is equal tofor each .

In order to present the finite recursive algorithm for computing the Moore-Penrose inverse of with the help of the actual SR1 update formulas (20) and (21), we first define . For each , it is necessary to compute . This requires calculated values of the vectors , and the matrix .

Theorem 5. Let the matrix be defined as in (3), the matrix sequence defined in (14), and the matrix sequence defined in (20). Then the following recursive relations are true:

Proof. Note that from it follows that . Based on the use of (21), the recursive sequences are defined for as in the following:Then the statement (22) can be easily verified using (24).
Also, matrices , are defined as So, (23) holds.

According to Theorem 5, we present Algorithm 1 for computing the Moore-Penrose inverse of the Toeplitz matrix . The algorithm is based on the SR1 modifications to compute and subsequently the S-M formula to compute . For this purpose, we use the abbreviation HSR1 + S-M to denote Algorithm 1.

A Matlab implementation of Algorithm 1 is placed in Appendix.

4.2. An Improvement of the Hybrid Combination in Algorithm 1

By taking into account the fact that the Toeplitz matrix has a specific structure, the matrix can be generated in an efficient way. In this way, an improvement of step  (1) of Algorithm 1 is introduced. The effectiveness of this method will be justified in the numerical experiments presented in Section 4.3.

According to the proper structure of the matrix and step  (1) of Algorithm 1, the general structure of the matrix is defined asLet be one-dimensional vector; then the symmetric Toeplitz matrix generated by is denoted by :Also, for two appropriate vectors , we denote by the (symmetric) Hankel matrix whose first column is and whose last row is .

Theorem 6. The matrix can be expressed as the difference of a specific banded symmetric Toeplitz matrix and a partitioned matrix of the formwhere the block is defined by , for an appropriate Hankel matrix .

Proof. Let and be vectors defined byrespectively. Then we define the following matrices: A simple verification shows that .

Next we present an example, in low dimensions, in order to illustrate Theorem 6.

Example 7. Consider the Toeplitz matrix , defined in (3) for ,  , and : According to (14),