Geometric-Algebra LMS Adaptive Filter and its Application to Rotation Estimation

TL;DR

This paper introduces a novel Geometric Algebra-based LMS adaptive filter designed for rotation estimation, demonstrating its effectiveness in 3D point-cloud registration and potential for higher-dimensional applications.

Contribution

The paper develops the first GA-LMS adaptive filter using Geometric Calculus, enabling recursive rotor estimation in any dimension with reduced computational complexity.

Findings

Effective rotation estimation in 3D point-cloud registration

Potential for reduced computational cost in 3D registration algorithms

Applicability to higher-dimensional rotor estimation

Abstract

This paper exploits Geometric (Clifford) Algebra (GA) theory in order to devise and introduce a new adaptive filtering strategy. From a least-squares cost function, the gradient is calculated following results from Geometric Calculus (GC), the extension of GA to handle differential and integral calculus. The novel GA least-mean-squares (GA-LMS) adaptive filter, which inherits properties from standard adaptive filters and from GA, is developed to recursively estimate a rotor (multivector), a hypercomplex quantity able to describe rotations in any dimension. The adaptive filter (AF) performance is assessed via a 3D point-clouds registration problem, which contains a rotation estimation step. Calculating the AF computational complexity suggests that it can contribute to reduce the cost of a full-blown 3D registration algorithm, especially when the number of points to be processed grows. Moreover, the employed GA/GC framework allows for easily applying the resulting filter to estimating rotors in higher dimensions.