To appear in: SIAM J. MATRIX ANAL. APPL. MEANS AND AVERAGING IN THE GROUP OF ROTATIONS∗
To appear in: SIAM J. MATRIX ANAL. APPL. MEANS AND AVERAGING IN THE GROUP OF ROTATIONS∗
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DOI：
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发表时间：
2002
影响因子：
5.3
通讯作者：
Maher Moakher
中科院分区：
文献类型：
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作者：
Maher Moakher
在本文中，我们给出了精确的定义不同的，适当不变的概念，平均或平均旋转。每个均值都与SO（3）中的一个度量相关联。从弗罗贝纽斯内积中导出的度量产生了一个平均旋转，该平均旋转由最接近给定旋转矩阵的通常算术平均值的特殊正交矩阵给出。与SO（3）上的内在度量相关的平均旋转是给定旋转矩阵的黎曼质心。证明了黎曼平均旋转与正数的几何平均和正厄米特算子的几何平均具有许多共同的性质。我们给出了一些例子与封闭形式的解决方案，这两个概念的意思。
In this paper we give precise definitions of different, properly invariant notions of mean or average rotation. Each mean is associated with a metric in SO(3). The metric induced from the Frobenius inner product gives rise to a mean rotation that is given by the closest special orthogonal matrix to the usual arithmetic mean of the given rotation matrices. The mean rotation associated with the intrinsic metric on SO(3) is the Riemannian center of mass of the given rotation matrices. We show that the Riemannian mean rotation shares many common features with the geometric mean of positive numbers and the geometric mean of positive Hermitian operators. We give some examples with closed-form solutions of both notions of mean.