Correspondence Free Multivector Cloud Registration using Conformal Geometric Algebra

TL;DR

This paper introduces a correspondence-free multivector cloud registration framework in conformal geometric algebra (CGA), formulating registration as an orthogonal transform in $SO(4,1)$ that includes rotation, translation, and scale without direct access to input multivectors. It builds a theoretical pipeline around eigenmultivector extraction from multilinear covariance operators and a multivector-coefficients approach to estimate the rigid transformation, proving rotation/translation equivariance and eigenvalue invariance. The CGA-EVD and CGA-COEF algorithms are validated on Stanford 3D scans, showing robustness to noise and competitive performance against PCA-based and learning-based baselines, with qualitative visualization of CGA primitives illustrating the registration. The work offers a novel geometric-algebra perspective on registration, enabling correspondence-free alignment and suggesting future regionalization of eigenmultivectors to handle low-overlap scenarios, augmented by open-source code.

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.

Figures (10)

• Figure 1: Benchmark results for rotation angle of $\theta=5^\circ$ and translation magnitude $\|\bm{t}\| = 0.01$ of the VGA-EVD, CGA-EVD, ICPbesl1992method, PASTA-3Dmarchi2023sharp, DCPwang2019deep, Go-ICPyang2015go, TEASER++yang2020teaser methods for the Bunny, Armadillo and Dragon datasets. Error in the rotation (top row), and translation (bottom row).
• Figure 2: The Stanford Armadillo dataset for increasing levels of Gaussian Noise, i.e.$\sigma\in\{0, 0.001, 0.005, 0.01, 0.02\}$
• Figure 3: The Stanford Armadillo dataset and its respective primitives: (a) eigenvectors, (b) eigenbivectors, (c) translation invariant vectors extracted from the eigenbivectors.
• Figure 4: The Stanford Bunny dataset and its respective primitives: (a) eigenvectors, (b) eigenbivectors, (c) translation invariant vectors extracted from the eigenbivectors.
• Figure 5: Multiple non-registered point clouds of the Stanford dataset with added Gaussian noise, $\sigma=0.01$, and their respective circle primitives.

Theorems & Definitions (76)

• Definition 1
• Definition 2
• Definition 3
• Theorem C.1
• proof
• Theorem C.2
• Theorem C.3
• proof
• Theorem C.4: Decomposition of bivectors and skew symmetric transformations
• proof : Proof of Theorem \ref{['theo:skew:decomp']}