Author

# Jianli Zhao

Bio: Jianli Zhao is an academic researcher from Liaocheng University. The author has contributed to research in topics: Mathematics & Quaternion. The author has an hindex of 6, co-authored 6 publications receiving 86 citations.

##### Papers

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TL;DR: A fast structure-preserving algorithm for computing the singular value decomposition of quaternion matrices by applying orthogonal JRS -symplectic matrices is proposed.

43 citations

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TL;DR: By applying particular structure of the real representations of quaternion matrices and the Moore-Penrose generalized inverse, the expressions of the minimal norm least squares solution, the pure imaginary least square solution, and the real least squares Solution for the quaternions matrix equation A X = B are derived.

25 citations

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TL;DR: From necessary and sufficient conditions for the submatrices to be zero and other special forms, respectively, some Hermitian properties for least squares solution to matrix equation AXB = C are derived.

15 citations

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TL;DR: A new imageWatermarking scheme based on the real SVD and Arnold scrambling to embed a color watermarking image into a color host image and can extract the watermark from the watermarked host image.

Abstract: We propose a new image watermarking scheme based on the real SVD and Arnold scrambling to embed a color watermarking image into a color host image. Before embedding watermark, the color watermark image with size of is scrambled by Arnold transformation to obtain a meaningless image . Then, the color host image with size of is divided into nonoverlapping pixel blocks. In each pixel block , we form a real matrix with the red, green, and blue components of and perform the SVD of . We then replace the three smallest singular values of by the red, green, and blue values of with scaling factor, to form a new watermarked host image . With the reserve procedure, we can extract the watermark from the watermarked host image. In the process of the algorithm, we only need to perform real number algebra operations, which have very low computational complexity and are more effective than the one using the quaternion SVD of color image.

13 citations

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TL;DR: Some closed-form formulas for calculating maximal and minimal ranks and inertias of P-X with respect to X are given and necessary and sufficient conditions for X>P(>=P,

Abstract: In this paper, we give some closed-form formulas for calculating maximal and minimal ranks and inertias of P-X with respect to X, where P@?C"H^n is given, X is common Hermitian least squares solutions to matrix equations A"1XA"1^*=B"1 and A"2XA"2^*=B"2. As application, we derive necessary and sufficient conditions for X>P(>=P,

10 citations

##### Cited by

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TL;DR: The properties of the Kronecker product related to the mixed matrix products, the vector operator, and the vec-permutation matrix are studied and several theorems and their proofs are given.

Abstract: This paper studies the properties of the Kronecker product related to the mixed matrix products, the vector operator, and the vec-permutation matrix and gives several theorems and their proofs. In addition, we establish the relations between the singular values of two matrices and their Kronecker product and the relations between the determinant, the trace, the rank, and the polynomial matrix of the Kronecker products.

110 citations

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TL;DR: Some necessary and sufficient solvability conditions for the mixed Sylvester matrix equations are given, and parameterize general solution when it is solvable, and the maximal and minimal ranks of the general solution are investigated.

88 citations

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TL;DR: The solutions of the reformulated distributed optimization problems are equivalent to least squares solutions to original matrix equations; the exponential convergence of the algorithms to a least squares solution to the matrix equation for any initial condition is proved.

Abstract: This paper investigates the distributed computation of the well-known linear matrix equation in the form of ${{AXB}} = F$ , with the matrices $A$ , $B$ , $X$ , and $F$ of appropriate dimensions, over multiagent networks from an optimization perspective. In this paper, we consider the standard distributed matrix-information structures, where each agent of the considered multiagent network has access to one of the subblock matrices of $A$ , $B$ , and $F$ . To be specific, we first propose different decomposition methods to reformulate the matrix equations in standard structures as distributed constrained optimization problems by introducing substitutional variables; we show that the solutions of the reformulated distributed optimization problems are equivalent to least squares solutions to original matrix equations; and we design distributed continuous-time algorithms for the constrained optimization problems, even by using augmented matrices and a derivative feedback technique. Moreover, we prove the exponential convergence of the algorithms to a least squares solution to the matrix equation for any initial condition.

57 citations

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TL;DR: A robust adaptive beamforming scheme based on two-component electromagnetic (EM) vector-sensor arrays is proposed by extending the well-known worst-case constraint into the quaternion domain, and achieves a better performance than the traditional diagonal loading scheme, in the case of smaller sample sizes and higher SNRs.

50 citations

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TL;DR: Numerical examples illustrate that the performance of the developed Lanczos-based method for low-rank quaternion approximation is better than that of the state-of-the-art methods.

Abstract: In many color image processing and recognition applications, one of the most important targets is to compute the optimal low-rank approximations to color images, which can be reconstructed with a small number of dominant singular value decomposition (SVD) triplets of quaternion matrices. All existing methods are designed to compute all SVD triplets of quaternion matrices at first and then to select the necessary dominant ones for reconstruction. This way costs quite a lot of operational flops and CPU times to compute many superfluous SVD triplets. In this paper, we propose a Lanczos-based method of computing partial (several dominant) SVD triplets of the large-scale quaternion matrices. The partial bidiagonalization of large-scale quaternion matrices is derived by using the Lanczos iteration, and the reorthogonalization and thick-restart techniques are also utilized in the implementation. An algorithm is presented to compute the partial quaternion singular value decomposition. Numerical examples, including principal component analysis, color face recognition, video compression and color image completion, illustrate that the performance of the developed Lanczos-based method for low-rank quaternion approximation is better than that of the state-of-the-art methods.

47 citations