Solving quaternion kinematical differential equations(QKDE) is one of the most significant problems in the automation, navigation, aerospace and aeronautics literatures. Most existing approaches for this problem neith...Solving quaternion kinematical differential equations(QKDE) is one of the most significant problems in the automation, navigation, aerospace and aeronautics literatures. Most existing approaches for this problem neither preserve the norm of quaternions nor avoid errors accumulated in the sense of long term time. We present explicit symplectic geometric algorithms to deal with the quaternion kinematical differential equation by modelling its time-invariant and time-varying versions with Hamiltonian systems and adopting a three-step strategy. Firstly,a generalized Euler's formula and Cayley-Euler formula are proved and used to construct symplectic single-step transition operators via the centered implicit Euler scheme for autonomous Hamiltonian system. Secondly, the symplecticity, orthogonality and invertibility of the symplectic transition operators are proved rigorously. Finally, the explicit symplectic geometric algorithm for the time-varying quaternion kinematical differential equation, i.e., a non-autonomous and non-linear Hamiltonian system essentially, is designed with the theorems proved. Our novel algorithms have simple structures, linear time complexity and constant space complexity of computation. The correctness and efficiencies of the proposed algorithms are verified and validated via numerical simulations.展开更多
How to combine color and multiscale information is a fundamental question for computer vision, and quite a few color diffusion techniques have been presented. Most of these proposed techniques do not consider the dire...How to combine color and multiscale information is a fundamental question for computer vision, and quite a few color diffusion techniques have been presented. Most of these proposed techniques do not consider the direct interactions between color channel pairs. In this paper, a new method of color diffusion considering these effects is presented, which is based on quaternion diffusion （QD） equation. In addition to showing the solution to linear QD and its analysis, one form of nonlinear QD is discussed. Compared with other color diffusion techniques, considering the interactions between channel pairs, QD has the following advantages： 1） staircasing effect is avoided; 2） as diffusion tensor, the image derivative is regularized without requiring additional convolution; 3） less time is needed. Experimental results demonstrate the effectiveness of linear and nonlinear QD applied to natural color images for denoising by both visual and quantitative evaluations.展开更多
基金supported by the Fundamental Research Funds for the Central Universities of China(ZXH2012H005)supported in part by the National Natural Science Foundation of China(61201085,51402356,51506216)+1 种基金the Joint Fund of National Natural Science Foundation of China and Civil Aviation Administration of China(U1633101)the Joint Fund of the Natural Science Foundation of Tianjin(15JCQNJC42800)
文摘Solving quaternion kinematical differential equations(QKDE) is one of the most significant problems in the automation, navigation, aerospace and aeronautics literatures. Most existing approaches for this problem neither preserve the norm of quaternions nor avoid errors accumulated in the sense of long term time. We present explicit symplectic geometric algorithms to deal with the quaternion kinematical differential equation by modelling its time-invariant and time-varying versions with Hamiltonian systems and adopting a three-step strategy. Firstly,a generalized Euler's formula and Cayley-Euler formula are proved and used to construct symplectic single-step transition operators via the centered implicit Euler scheme for autonomous Hamiltonian system. Secondly, the symplecticity, orthogonality and invertibility of the symplectic transition operators are proved rigorously. Finally, the explicit symplectic geometric algorithm for the time-varying quaternion kinematical differential equation, i.e., a non-autonomous and non-linear Hamiltonian system essentially, is designed with the theorems proved. Our novel algorithms have simple structures, linear time complexity and constant space complexity of computation. The correctness and efficiencies of the proposed algorithms are verified and validated via numerical simulations.
文摘How to combine color and multiscale information is a fundamental question for computer vision, and quite a few color diffusion techniques have been presented. Most of these proposed techniques do not consider the direct interactions between color channel pairs. In this paper, a new method of color diffusion considering these effects is presented, which is based on quaternion diffusion （QD） equation. In addition to showing the solution to linear QD and its analysis, one form of nonlinear QD is discussed. Compared with other color diffusion techniques, considering the interactions between channel pairs, QD has the following advantages： 1） staircasing effect is avoided; 2） as diffusion tensor, the image derivative is regularized without requiring additional convolution; 3） less time is needed. Experimental results demonstrate the effectiveness of linear and nonlinear QD applied to natural color images for denoising by both visual and quantitative evaluations.
