We present, for the first time, a novel theoretical approach to address the
problem of correspondence free multivector cloud registration in conformal
geometric algebra. Such formalism achieves several favorable properties.
Primarily, it forms an orthogonal automorphism that extends beyond the typical
vector space to the entire conformal geometric algebra while respecting the
multivector grading. Concretely, the registration can be viewed as an
orthogonal transformation (\it i.e., scale, translation, rotation) belonging to
$SO(4,1)$ - group of special orthogonal transformations in conformal geometric
algebra. We will show that such formalism is able to: $(i)$ perform the
registration without directly accessing the input multivectors. Instead, we use
primitives or geometric objects provided by the conformal model - the
multivectors, $(ii)$ the geometric objects are obtained by solving a
multilinear eigenvalue problem to find sets of eigenmultivectors. In this way,
we can explicitly avoid solving the correspondences in the registration
process. Most importantly, this offers rotation and translation equivariant
properties between the input multivectors and the eigenmultivectors.
Experimental evaluation is conducted in datasets commonly used in point cloud
registration, to testify the usefulness of the approach with emphasis to
ambiguities arising from high levels of noise. The code is available at
this https URL . This work was
submitted to the International Journal of Computer Vision and is currently
under review.

我们首次提出了一种新的理论方法，用于解决共形几何代数中不需要对应关系的多矢量云配准问题。该方法通过构建一个超出典型向量空间的正交自同构，并保持多矢量分级，实现了注册。我们将展示该方法能够在不直接访问输入多矢量的情况下进行注册，而是使用由共形模型提供的原始几何对象，即多矢量，通过解决多线性特征值问题来获取几何对象。这样，我们可以明确地避免在注册过程中解决对应关系，并同时实现输入多矢量和特征多矢量之间的旋转等变性质。通过在常用的点云注册数据集上进行实验评估，我们证明了该方法的实用性，特别是在高水平噪声引起的不确定性方面。