Mathematical Advancements in Geometric Algebra for 3D Registration Problems

Abstract

The present PhD thesis investigates the reformulation of geometric registration algorithms using Geometric Algebra (GA) to solve transformation problems traditionally handled with Linear Algebra (LA). Geometric transformations play a key role in fields like robotics, computer vision and 3D modelling, as they allow for crucial tasks such as object recognition, scene reconstruction and motion planning. Unlike LA which relies on matrices and quaternions, GA employs multivectors to encode both rotational and translational transformations in a unified framework. A key contribution of this thesis is the development of Characteristic Multivectors (CMs), a versatile method for recovering geometric transformations while preserving invariant properties across transformations. CMs operate in any dimension or signature and can extend beyond orthonormal frames to handle both noisy and clean data. This flexibility makes them effective for addressing complex geometric problems in higher-dimensional and non-Euclidean spaces, and provides a general and efficient approach that not only solves existing problems but also opens up new possibilities for a wide range of practical applications. The thesis begins by applying 3D Euclidean CMs to the Absolute Orientation problem in 3D registration with known correspondences, demonstrating that the proposed solution offers a viable and efficient alternative to traditional LA methods. It then extends this approach to the Iterative Closest Point (ICP) algorithm, resulting in a new Geometric Algebra-based variant, ICP-GA, which adheres to the standard ICP methodology while solving the 3D registration problem with unknown correspondences. The ICP-GA variant provides comparable accuracy to the standard ICP algorithm proposed by Besl and McKay. Based on the flexibility of CMs, the thesis then introduces 4D Spherical CMs, based on the 1DUp method within Conformal Geometric Algebra to create an optimized variant, CM-ICP, which adheres further to the ICP methodology and efficiently handles transformations in non-Euclidean spaces. This extension successfully addresses limitations present in traditional LA and GA methods and demonstrates superior performance over the standard ICP algorithm in handling complex transformations. Finally, the thesis extends the 1DUp framework to tackle the 3D line registration problem in 4D spherical space, where transformations are recovered through a new approach based on geometric representations of lines. Given the limited number of existing 3D line registration algorithms, this method offers a significant advancement by generalizing traditional point-based techniques. It provides more robust and precise solutions for applications where line registration is critical, such as robotics, medical imaging and 3D modelling. In summary, the present thesis demonstrates how GA provides substantial improvements over traditional LA methods for 3D geometric registration. By offering a unified framework for transformations and the ability to generalize across dimensions, GA enables more efficient and accurate approaches to both point- and line-based registration. These contributions address existing limitations and pave the way for new and versatile applications across various fields.