Weighted Toeplitz Regularized Least Squares Computation for Image Restoration

In this paper, we study mainly on a class of column upper-minus-lower (CUML) Toeplitz matrices without standard Toeplitz structure, which are " similar" to the Toeplitz matrices. Their (−1, −1)-cyclic displacements coincide with cyclic displacement of some standard Toeplitz matrices. We obtain the formula on representation for the inverses of CUML Toeplitz matrices in the form of sums of products of (−1, 1)-circulants and (1, −1)-circulants factor by constructing the corresponding displacement of the matrices. In addition, based on the relation between CUML Toeplitz matrices and CUML Hankel matrices, the inverse formula of CUML Hankel matrices can also be obtained.

“…Matrices have important applications in pure and applied mathematics [3,24,29]. Many scholars study their properties as well as the representation of their inverse formula.…”

Section: Introductionmentioning

confidence: 99%

Skew cyclic displacements and decompositions of inverse matrix for an innovative structure matrix

Jiang¹,

Hong²,

Fu³

2017

“…Toeplitz matrix has become a satisfactory tool in restoration of signals and images [6,24,25]. Toeplitz inversion formulae involving circulant matrices have also been presented in [1,20,21].…”

Section: Introductionmentioning

confidence: 99%

A new Toeplitz inversion formula, stability analysis and the value

Zheng¹,

Fu²,

Shon³

2017

In this paper, Toeplitz and Hankel inversion formulae are presented by the idea of skew cyclic displacement. A new Toeplitz inversion formula can be denoted as a sum of products of skew circulant matrices and upper triangular Toeplitz matrices. A new Hankel inversion formula can be denoted as a sum of products of skew left circulant matrices and upper triangular Toeplitz matrices. The stability of their inverse formulae are discussed and their algorithms are given respectively. How the analogue of our formulae lead to a more efficient way to solve the Toeplitz and Hankel linear system of equations are proposed.

“…Toeplitz matrix has become a satisfactory tool in restoration of signals and images [5,26,27]. In [4,7], the authors introduced a generalization of Toeplitz matrices, called conjugate-Toeplitz (CT) matrices, and showed that certain properties of Toeplitz matrices could be extended to CT matrices.…”

Section: Introductionmentioning

confidence: 99%

Gohberg-Semencul type formula and application for the inverse of a conjugate-Toeplitz matrix involving imaginary circulant matrices

Jiang¹,

Hong²

2017

Gohberg-Semencul type inverse formula of conjugate-Toeplitz (CT) is obtained by constructing a kind of imaginary cyclic displacement transform. The stability of decomposition formula of inverse is investigated, and its algorithm is also given. Numerical example is provided to verify the feasibility of the inverse formula. How the analogue of our formula leads to a more efficient way to solve the conjugate-Toeplitz linear system of equations is proposed. The corresponding inverse, stability, and algorithm of conjugate-Hankel (CH) matrix are also considered.