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.

本文介绍了一种新的自适应滤波策略，它利用几何（Clifford）代数理论进行开发，提出了一种递归估算叶轮（多向量）的 GA 最小均方（GA-LMS）自适应滤波器，并通过 3D 点云注册问题对其性能进行评估。