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<p>Dual quaternions are pivotal for modeling rigid-body motions in robotics, aerospace, and computer graphics, with solving generalized Sylvester dual quaternion matrix equations central to kinematic analysis and trajectory planning. Existing methods often have slow convergence or limited applicability to coupled forms, restricting use. This paper addresses a class of such equations. We establish a necessary and sufficient condition for solution existence via real representation of quaternion matrices, vectorization, and Kronecker products. Two novel algorithms are proposed: dual quaternion generalized conjugate direction and conjugate gradient least squares algorithms, both converging in finite iterations under no rounding errors for theoretical efficiency. Numerical simulations demonstrate the algorithms outperform existing methods in convergence speed and apply to practical engineering like robotic kinematics. Furthermore, experiments on simultaneous restoration of two hand tremor-induced mixed-direction blurred color images validate their effectiveness in handling complex real-world degradation, expanding their application scope to image processing and highlighting strong practical potential.</p>

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Efficient Finite-Iteration Algorithms for Generalized Coupled Sylvester Dual Quaternion Matrix Equations with Application to Color Image Simultaneous Restoration

  • 期刊论文

  • 发表日期：

    2026年4月9日

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  • Xiaomin Cai,
  • Yifen Ke,
  • Chengliang Li,
  • Changfeng Ma

摘要

Dual quaternions are pivotal for modeling rigid-body motions in robotics, aerospace, and computer graphics, with solving generalized Sylvester dual quaternion matrix equations central to kinematic analysis and trajectory planning. Existing methods often have slow convergence or limited applicability to coupled forms, restricting use. This paper addresses a class of such equations. We establish a necessary and sufficient condition for solution existence via real representation of quaternion matrices, vectorization, and Kronecker products. Two novel algorithms are proposed: dual quaternion generalized conjugate direction and conjugate gradient least squares algorithms, both converging in finite iterations under no rounding errors for theoretical efficiency. Numerical simulations demonstrate the algorithms outperform existing methods in convergence speed and apply to practical engineering like robotic kinematics. Furthermore, experiments on simultaneous restoration of two hand tremor-induced mixed-direction blurred color images validate their effectiveness in handling complex real-world degradation, expanding their application scope to image processing and highlighting strong practical potential.