Pairwise rigid registration can be developed by comparing local geometry encoded by intrinsic second-order orientation tensors which allows to model the registration problem using the associated Riemannian geometry or related group structures. Everything starts by representing those tensor fields as multivariate normal models that permit us to manipulate Gaussians in two ways: using the Riemannian manifold elements, that can be embedded into matrix spaces with geometric structures, or through Lie group/algebra techniques. In this paper we discuss some points behind these approaches in the context of rigid registration problems. Firstly, they are not equivalent since, in general, there is no isometry linking them. Secondly, embedding methodologies are not invariant with respect to rigid motion. We discuss these points using two variants of the Iterative Closest Point that use the comparative tensor shape factor (CTSF) to match orientation tensors. We replace the CTSF to different criteria computed through geodesic distance and algebraic embeddings and compare the registration algorithms showing that the latter is more efficient for registration of point clouds.

可以通过比较由固有二阶取向张量编码的局部几何来开发成对的刚性配准，这可以使用关联的黎曼几何或相关的组结构对配准问题进行建模。一切都始于将那些张量场表示为多元法线模型，从而使我们可以通过两种方式操纵高斯：使用可以被嵌入具有几何结构的矩阵空间中的黎曼流形元素，或者通过李群/代数技术。在本文中，我们讨论了在刚性注册问题中这些方法背后的一些观点。首先，它们是不等价的，因为通常没有等轴测图链接它们。其次，嵌入方法对于刚性运动不是不变的。我们使用迭代最接近点的两个变体来讨论这些点，这些变体使用比较张量形状因子（CTSF）来匹配方向张量。我们将CTSF替换为通过测地距离和代数嵌入计算出的不同标准，并比较配准算法，结果表明后者对于点云的配准更为有效。