Correspondence Free Multivector Cloud Registration using Conformal Geometric Algebra

TL;DR

This paper introduces a novel, correspondence-free method for multivector cloud registration using conformal geometric algebra, enabling registration through orthogonal transformations without direct access to input multivectors, and demonstrating robustness to noise.

Contribution

It presents the first theoretical framework for correspondence-free multivector cloud registration in conformal geometric algebra, utilizing eigenvalue problems for registration without explicit correspondences.

Findings

Effective registration without direct multivector access.

Rotation and translation equivariance in the registration process.

Robustness to high noise levels in datasets.

Abstract

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 https://github.com/Numerical-Geometric-Algebra/RegistrationGA . This work was submitted to the International Journal of Computer Vision and is currently under review.