高阶连续的单位四元数插值曲线

给出了k阶连续的单位四元数插值曲线的一种构造算法．构造的曲线不仅插值给定的朝向序列,而且插值给定的角速度序列．单位四元数空间S^3与三维空间R^3之间的映射是该算法的基础：S^3与R^3之间的映射公式保证了曲线的插值朝向序列性质,S^3到R^3的指数映射导数公式保证了单位四元数曲线的插值角速度序列性质．     

一类C~2连续的单位四元数插值样条曲线

为了能够快速生成光滑的单位四元数插值样条曲线,实时控制三维物体关键帧动画,提出一种基于四次多项式的单位四元数插值样条曲线,并证明了它的插值性和C~2连续性.首先选择适当的四次多项式调配函数,使其生成的样条曲线能够插值给定的型值点列并达到C~2连续;然后通过累和形式的基函数作指数、代表关键帧方向的单位四元数为底的指数函数的连乘形式把欧氏空间中的样条曲线推广到单位四元数空间S~3中.文中构造的曲线能够自动通过给定的关键帧朝向序列,避免经典的B样条单位四元数曲线从型值点反求控制顶点过程中求解非线性方程组带来的时间开销,提高了计算效率.实验结果表明,该方案可以有效地应用于三维关键帧动画.     

彩色图像缩放的四元数插值算法

将彩色图像像素点的3个颜色分量当作一个纯四元数来处理,在充分考虑了图像3个颜色分量的内在相关性、图像本身的非线性机制的基础上,构造了基于球面线性四元数插值方法和代数三角混合四元数插值样条方法的彩色图像插值算法.实验结果表明,四元数方法用于彩色图像插值时,所处理的图像清晰度和色彩亮度均较传统方法有较大改进,边缘细节上也更丰富.     

C3连续的单位四元数插值样条曲线

目的 构造一类C³连续的单位四元数插值样条曲线,证明它的插值性和连续性,并把它应用于刚体关键帧动画设计中。方法 利用R³空间中插值样条曲线的5次多项式调配函数的累和形式构造了S³空间中单位四元数插值样条曲线,它不仅能精确通过一系列给定的方向,而且能生成C³连续的朝向曲线。结果 与Nielson的单位四元数均匀B样条插值曲线的迭代构造方法相比,所提方法避免了为获取四元数B样条曲线控制顶点对非线性方程组迭代求解的过程,提高了运算效率;与单位四元数代数三角混合插值样条曲线的构造方法（Su方法）相比,所提方法只用到多项式基,运算速度更快。本例中创建关键帧动画所需的时间与Nielson方法和Su方法相比平均下降了73%和33%。而且,相比前两种方法,所提方法产生的四元数曲线连续性更高,由C²连续提高到C³连续,这意味着动画中刚体的朝向变化更加自然。结论 仿真结果表明,本文方法对刚体关键帧动画设计是有效的,对实时性和流畅性要求高的动画设计场合尤为适用。     

从运动捕获数据中提取关键帧

在四元数表示下导出了两个旋转之间差异的一种简单形式，将人体各关节上总的旋转变化作为帧间距，设计了一种从运动捕获数据中提取关键帧的高效算法，通过向量的线性插值和四元数的球面线性插值重建了原始动画。实验表明，该算法具有很好的数据压缩效果，而且提取出的关键帧在视觉上对原始动画具有概括力。     

四元数在图形学中的应用

本文以复数为引导介绍四元数的代数模型,有效减低了学习难度。基于四元数的几何模型,进一步介绍了四元数在计算机图形学中的方向插值和旋转应用,为从事实时图形绘制的技术人员提供了理论基础。     

在实时仿真中应用拉格朗日四点插值算法的可行性分析

在动力学连续系统实对仿真的函数生成中,通常采用的插值算法是线性插值。其它更高次的插值方法由于计算公式复杂而较少被采用。本文指出,为节省函数表所占用的计算机内存空间,在实时仿真中采用拉格朗日四点插值是可行的。并且给出了该插值公式的实时形式。文中提出了n—i 混合插值算法,分析了它的运算量和速度。文中还分析并举例说明了拉格朗日四点插值算法节省内存空间的情况。     

基于Hermite样条曲线的关键帧插值角色动画

针对角色关键帧动画中的运动平滑性问题,以基于运动捕获数据的角色动画为基础,提出以Hermite样条曲线为基本算法结合四元数球面线性插值算法的方法,对虚拟人体骨骼运动的动画关键帧进行插值,实现虚拟人体角色的动作平滑过渡。经实验证明,本方法实现简单高效,获得的关键帧动画平滑流畅,可广泛用于人体类角色动画的研究。     

解析法图像重建中的插值技术研究

为了给图像重建中插值方法的选择提供依据,研究了线性插值和样条插值对重建图像的质量影响.分析了常用的插值技术,从一维插值的角度,从理论和实验两个方面分析了线性插值和样条插值的性能,将两种方法应用到滤波反投影法图像重建中,分析了两种方法在不同的投影采样间隔的情况下对重建图像的质量影响.结果发现在一维信号插值时,样条插值精度高,但在图像重建时,采用更精确的样条插值反而使得重建图像的质量下降.最后指出图像重建时应该采用能更好的抑制振荡噪声的线性插值方法.     

基于样条模型插值的方法在科学可视化上的应用

插值方法和插值基函数的选择是可视化技术的一个关键问题。在规则数据场中，本文提出了一个基于样条模型插值来重建三维结构的方法。该模型以样条基函数为背景，其函数系数则由周围采样点数据重复迭代产生。实验表明该方法的逼近效果明显优于传统的线性插值拟合得出的图像质量。     

Quaternion rational surfaces: Rational surfaces generated from the quaternion product of two rational space curves

A quaternion rational surface is a surface generated from two rational space curves by quaternion multiplication. The goal of this paper is to demonstrate how to apply syzygies to analyze quaternion rational surfaces. We show that we can easily construct three special syzygies for a quaternion rational surface from a μ-basis for one of the generating rational space curves. The implicit equation of any quaternion rational surface can be computed from these three special syzygies and inversion formulas for the non-singular points on quaternion rational surfaces can be constructed. Quaternion rational ruled surfaces are generated from the quaternion product of a straight line and a rational space curve. We investigate special μ-bases for quaternion rational ruled surfaces and use these special μ-bases to provide implicitization and inversion formulas for quaternion rational ruled surfaces. Finally, we show how to determine if a real rational surface is also a quaternion rational surface.     

彩色图像四元数频域奇异值分解水印算法

将四元数傅里叶变换与四元数奇异值分解技术相结合并引入到对彩色图像的水印处理,提出一种基于四元数频域奇异值分解的彩色图像盲水印算法.首先对彩色载体图像进行分块并采用四元数傅里叶变换(quaternion Fourier transform,QFT)得到其频域矩阵,然后对频域矩阵中的单位小块进行四元数奇异值分解(quaternion singular value decomposition,QSVD),得到实系数奇异值,使用奇偶量化调制法将水印信号嵌入到单位小块的最大奇异值中.仿真实验结果表明,嵌入的水印分布在空域图像各彩色分量中,在不可见性以及鲁棒性的比较中优于传统的彩色图像亮度域以及独立多通道处理方法.     

Quaternion wavelet phase based stereo matching for uncalibrated images

This paper proposes a quaternion wavelet phase based stereo matching (QWPSM) scheme for uncalibrated image pairs. In this scheme, we estimate the disparity by directly establishing correspondences between quaternionic phase structures of two quaternion wavelet filtered (QWF) images. Firstly, linear-phase quaternion wavelet filters (LPQWFs) are constructed from real biorthogonal wavelet bases. Then, quaternion phases are extracted under each scale through quaternion wavelet filtering of the multiscale transformed image pyramids. The disparity estimation is formed as a minimization process of a local energy weighted cost function, and propagated from coarse to fine scales. Costs can adaptively alleviate the negative effects of phase singularities, which are the main causes of mismatches in phase-based stereo matching. Multiscale matching strategy is used to avoid phase wrapping and improve convergence speed. Experimental results are promising in various image pairs.     

Motion trajectory of human arms based on the dual quaternion with motion tracker

In this paper, we present an algorithm for human motion capture of the real-time motion trajectory of human arms based on wireless inertial 3D motion trackers. It aims to improve the accuracy of inertial motion captures and quickly reconstruct some human movements. To evaluate the performance of the proposed dual quaternion algorithm, we present the prototype design. The wireless inertial measurement system and Kinect device are introduced simultaneously in capturing human motion. The dual quaternion algorithm incorporates features of the quaternion rotation and translation. So the singular points of Euler angles can be avoided. Dual quaternion algorithm and DCM(direction cosine matrix) are used to reconstruct human arm movements respectively. Compared with the computing speed in Matlab, the speed of the dual quaternion is faster than it of DCM. To the end, we propose a 3D ADAMS human robotic model for simulating the motion trajectory using dual quaternion algorithm. The results show that the dual quaternion can achieve capabilities of a positive DCM solving, which completed between body segments rotating and translating the coordinate system transformation. Also it can effectively drive in real-time a human model to animate movement, and provide a good algorithm.     

用四元数状态切换无迹卡尔曼滤波器估计的飞行器姿态

在较大初始姿态误差角下, 针对捷联惯导/CCD星敏感器(strap-intertial navigation system/CCD star sensor, SINS/CCD)姿态估计系统扩展卡尔曼滤波(extended Kalman filter, EKF)算法精度下降的问题, 提出了基于四元数的状态切换无迹卡尔曼滤波算法. 通过状态实时切换降低了全维无迹卡尔曼滤波(unscented Kalman filter, UKF)的维数, 减小了计算复杂度, 提高了系统的实时性. 文中采用基于特征向量求解的代价函数法计算四元数均值避免了UKF算法中四元数规范化的限制; 利用乘性误差四元数表示姿态更新点与估计点之间的距离, 解决了四元数协方差阵奇异性问题. 仿真实验结果表明: 与EKF相比, 该算法在精度上有较大提高; 与全维UKF算法和修正罗德里格斯参数UKF算法相比, 该算法精度相当但估计时间均有不同程度的减少.     

基于四元数衍生无迹卡尔曼滤波的二段式多旋翼无人机姿态估计算法

对传统多旋翼无人机姿态估计算法难以兼顾高精度、强实时性以及抗干扰能力差的问题,首先基于一种计算量较小的衍生无迹卡尔曼滤波算法,在量测更新中,将加速度数据和磁力计数据分为两个阶段进行姿态四元数校正处理,然后从旋转四元数的本质出发,推测出四元数各元素分别包含着不同的姿态角信息,最后将校正四元数分别乘上为降低校正过程中的相互干扰所设计的系数,提出一种基于四元数衍生无迹卡尔曼滤波的二段式多旋翼无人机姿态估计算法.通过使用PIXHAWK飞控数据,与传统姿态估计算法进行仿真实验对比,实验表明,本文提出算法与传统使用扩展卡尔曼滤波(EKF)或无迹卡尔曼滤波(UKF)的姿态估计算法相比,在实时性、解算精度和抗干扰能力方面有较大提升.     

Hidden Markov tree model of images using quaternion wavelet transform

The quaternion wavelet transform is regarded as a new multi-scale tool for signal and image processing, which can effectively capture local shifts and image texture information. The marginal and joint distributions of the quaternion wavelet transform coefficients are measured by the histogram. The mutual information is utilized to measure the dependence between the coefficients. The authors have drawn the conclusion that the quaternion coefficients can be modeled by a Gaussian Mixture model conditioned to the magnitudes of generalized coefficients, with intensive analysis of the statistical properties of the decomposition coefficients. In this paper a new hidden Markov tree model utilizing quaternion wavelet transforms is proposed based on the authors’ findings. In order to demonstrate its effectiveness, the new statistical model was applied to image de-noising. The experimental results show that the proposed statistical model exhibits better performance than other related image de-noising algorithms that are also based on hidden Markov tree models.     

Rigid body dynamics with a scalable body,quaternions and perfect constraints

In this paper, we present a formulation of the quaternion constraint for rigid body rotations in the form of a standard perfect bilateral mechanical constraint, for which the associated Lagrangian multiplier has the meaning of a constraint force. First, the equations of motion of a scalable body are derived. A scalable body has three translational, three rotational, and one uniform scaling degree of freedom. As generalized coordinates, an unconstrained quaternion and a displacement vector are used. To the scalable body, a perfect bilateral constraint is added, restricting the quaternion to unit length and making the body rigid. This way a quaternion based differential algebraic equation (DAE) formulation for the dynamics of a rigid body is obtained, where the 7×7 mass matrix is regular and the unit length restriction of the quaternion is enforced by a mechanical constraint. Finally, the equations of motion in the form of a DAE are linked to the Newton–Euler equations of motion of a rigid body. The rigid body DAE formulation is useful for the construction of (energy) consistent integrators.     

四元数扩维无迹卡尔曼滤波算法及其在大失准角快速传递对准中的应用

针对快速传递对准中量测失准角为大角度的情况,在非线性欧拉角误差模型基础上,推导了一种基于乘性四元数的等效快速传递对准模型.为解决四元数在无迹卡尔曼滤波（UKF）算法中的应用问题,提出了一种基于四元数的状态扩维无迹卡尔曼滤波（Q--AUKF）算法.该算法将系统噪声增广到状态向量中,解决了乘性四元数噪声无法进行向量意义下四则运算的问题.针对四元数加权均值规范化的限制,采用平均四元数算法保证其正交规范化要求.最后将其应用到快速传递对准中的仿真实验结果表明,在量测误差角为大角度的情况下,该算法具有更高的估计精度与收敛速度.