【70周年院庆学术论坛】

第五届四元数矩阵计算及其应用国际会议的日程安排

会议报告摘要

香港理工大学：祁力群
题目：Unit Dual Quaternion Directed Graphs, Formation Control and General Weighted Directed Graphs
摘要：We study the multi-agent formation control problem in a directed graph. The relative configurations are expressed by unit dual quaternions (UDQs). We call such a weighted directed graph a unit dual quaternion directed graph (UDQDG). We show that a desired relative configuration scheme is reasonable or balanced in a UDQDG if and only if there is a diagonal matrix with UDQ diagonal elements such that the dual quaternion Laplacian is similar to the unweighted Laplacian of the underlying directed graph. A direct method and a unit gain graph method are proposed to solve the balance problem of general unit weighted directed graphs. We then study the balance problem of general non-unit weighted directed graphs. Numerical experiments for UDQDG are reported.

香港浸会大学：Michael K. Ng
题目：Hermitian Quaternion Toeplitz Matrices
摘要：In this talk, we discuss recent results about quaternion Toeplitz matrices. Both theoretical and numerical results will be studied.

上海大学：王卿文
题目：A generalized Sylvester dual quaternion matrix equation with applications
摘要：Dual quaternions have important applications in fields such as information control, robotics, and hand-eye calibration. At the same time, matrix equations play a crucial role in system control, particularly the generalized Sylvester matrix equation AX-EXF=CY+D, which has extensive applications in higher-order linear systems. However, research on this matrix equation in the context of dual quaternions has not yet been discovered. Therefore, this paper aims to fill this research gap by establishing the necessary and sufficient conditions for the solvability of this generalized Sylvester matrix equation over dual quaternions and providing a general solution when it is consistent. As an application, we design a color image encryption and decryption scheme based on this generalized Sylvester matrix equation. Experimental results demonstrate the high feasibility and effectiveness of the proposed scheme.

澳门大学：高潔欣(Kit Ian KOU)
题目：Digital Hypercomplex Analytic Signal: A Novel Framework for Multidimensional Signal Processing
摘要：The concept of the analytic signal has been a cornerstone in signal processing, providing a powerful tool for analyzing real-valued signals by extracting instantaneous amplitude, phase, and frequency information. However, extending this framework to multidimensional and hypercomplex domains remains a challenging yet promising area of research. This paper introduces the Digital Hypercomplex Analytic Signal (DHAS), a novel approach that generalizes the analytic signal to hypercomplex numbers, such as quaternions and Clifford numbers, enabling the analysis of multidimensional signals in a unified manner. By leveraging hypercomplex Fourier transforms and Cauchy-Riemann-like conditions, the proposed framework preserves the geometric and algebraic properties of hypercomplex numbers while providing a robust tool for applications in image processing, robotics, and communications. We present theoretical foundations, computational algorithms, and experimental results demonstrating the efficacy of DHAS in extracting meaningful features from multidimensional data. This work bridges the gap between classical signal processing and hypercomplex algebra, opening new avenues for advanced signal analysis in higher-dimensional spaces.

杭州电子科技大学：凌晨
题目：A metric function for dual quaternion matrices and related least-squares problems
摘要：Solving dual quaternion equations is an important issue in many fields such as scientific computing and engineering applications. In this paper, we first introduce a new metric function for dual quaternion matrices. Then, we reformulate dual quaternion overdetermined equations as a least squares problem, which is further converted into a bi-level optimization problem. Numerically, we propose two implementable proximal point algorithms for finding approximate solutions of dual quaternion overdetermined equations. The relevant convergence theorems have also been established. Preliminary simulation results on synthetic and color image datasets demonstrate the effectiveness of the proposed algorithms.

National Research University Higher School of Economics, Russia：Dmitry Shirokov
题目：On calculation of spin group elements in terms of Clifford algebras, quaternions, and split-quaternions
摘要：We present a method for calculation of spin groups elements for known pseudo-orthogonal group elements with respect to the corresponding two-sheeted coverings. We present our results using the Clifford algebra formalism in the case of arbitrary dimension and signature, and then in some special cases explicitly using matrices, quaternions, and split-quaternions. The different formalisms are convenient for different possible applications in physics, engineering, and computer science.

俄罗斯东北联邦大学，临沂大学：姜同松
题目：A Novel Strategy for Color Image Denoising Using Quaternion SVD and Its Application to Bearing Weak Fault Diagnosis
摘要：Color short-time Fourier transform (STFT) image is commonly used in bearing fault diagnosis to highlight fault features. However, without additional denoising, fault information in STFT image is often obscured by non-fault related signal components and significant noise. While the quaternion model has shown effectiveness in color image denoising, existing quaternion color image denoising methods are not as applicable to STFT images in weak fault scenarios because they fail to extract fault features amid interference from non-fault related components and strong noise. In this talk, we introduce a novel bearing weak fault diagnosis method that utilizes a quaternion color image denoising scheme to analysis STFT image for the first time. This method comprises two stages: first, a new and fast quaternion singular value decomposition (QSVD) algorithm which is based on the real representation of a quaternion matrix is proposed for computing the singular components (SCs) of a pure quaternion matrix which represents an STFT image; second, an indicator, which is defined as fault energy ratio (FER), is introduced to select fault related SCs and mitigate the interference of non-fault related components and noise by leveraging fault characteristics in frequency domain. Results from both simulated analyses and real experimental signals demonstrate that the new and fast QSVD algorithm is more efficient than several existing QSVD algorithms, and the proposed bearing fault diagnosis method is effective in signal denoising even when fault features are weak.

江苏师范大学：贾志刚
题目：CSTV-QBO: A New Cross-Space Total Variation Regularization Model for Color Image Restoration with Quaternion Blur Operator
摘要：The cross-channel deblurring problem in color image processing is difficult to solve due to the complex coupling and structural blurring of color pixels. Until now, there are few efficient algorithms that can reduce color artifacts in deblurring process. To solve this challenging problem, we present a novel cross-space total variation (CSTV) regularization model for color image deblurring by introducing a quaternion blur operator and a cross-color space regularization functional. The existence and uniqueness of the solution are proved and a new L-curve method is proposed to find a balance of regularization terms on different color spaces. The Euler-Lagrange equation is derived to show that CSTV has taken into account the coupling of all color channels and the local smoothing within each color channel. A quaternion operator splitting method is firstly proposed to enhance the ability of color artifacts reduction of the CSTV regularization model. This strategy also applies to the well-known color deblurring models.

复旦大学：邵美悦
题目：The complex embedding technique and algorithm development for quaternion eigenvalue problems
摘要：Embedding is a powerful tool to study quaternions, mainly for theoretical analysis. In the past decade, the real embedding technique has been used to develop structure-preserving algorithms for quaternion (right) eigenvalue problems. We show that much more can be achieved using the complex embedding technique. We use the complex embedding technique to solve several building blocks in quaternion matrix computation.  This results in several new eigensolvers for quaternion matrices.

北京航空航天大学：崔春风
题目：Eigenvalues of Dual Quaternion Matrices with Applications to Formation Control and SLAM
摘要：In this presentation, I will report our recent work on eigenvalues of dual quaternion matrices and their applications in formation control and Simultaneous Localization and Mapping (SLAM). We proposed a control law based on the Laplacian matrix of unit dual quaternion directed graphs (UDQDGs), extending the work of [Olfati-Saber and Murray, 2004] to UDQDGs. Furthermore, we proved the global asymptotic convergence with R-linear convergence rate. Furthermore, I shall report the proximal linearized Riemannian alternating direction method of multipliers method (PieADMM) for solving SLAM problems and its convergence results.