Parameter Estimation from Uncertain Data in Geometric Algebra

Abstract

We show how standard parameter estimation methods can be applied to Geometric Algebra in order to fit geometric entities and operators to uncertain data. These methods are applied to three selected problems. One of which is the perspective pose estimation problem. We show experiments with synthetic data and compare the results of our algorithm with standard approaches.In general, our aim is to find multivectors that satisfy a particular constraint, which depends on a set of uncertain measurements. The specific problem and the type of multivector, representing a geometric entity or a geometric operator, determine the constraint. We consider the case of point measurements in Euclidian 3D-space, where the respective uncertainties are given by covariance matrices. We want to find a best fitting circle or line together with their uncertainty. This problem can be expressed in a linear manner, when it is embedded in the corresponding conformal space. In this space, it is also possible to evaluate screw motions and their uncertainty, in very much the same way.The parameter estimation method we use is a least-squares adjustment method, which is based on the so-called Gauss-Helmert model, also known as mixed model with constraints. For this linear model, we benefit from the implicit linearization when expressing our constraints in conformal space. The multivector representation of the entities we are interested in also allows their uncertainty to be expressed by covariance matrices. As a by-product, this method provides such covariance matrices.

Least Square for Grassmann-Cayley Agelbra in Homogeneous Coordinates

“…Some least square applications in geometric algebra has been investigated by Gebken et al [5], applied to line and circle fitting in the conformal space of Euclidean 3D-space, but to our knowledge, nothing has been done concerning the least square formulation of the wedge product.…”

Properties and Applications of the Simplified Generalized Perpendicular Bisector

“…However, rotation reconstruction methods from point correspondences are used in many application domains such as Computer Vision or body movement analysis. Different approaches can be found in the literature [4,17,9]. Their drawbacks are the impossibility to extend them in dimension higher than two or the subtle geometrical model and computations they need.…”

Automatic Multivector Differentiation and Optimization

“…An important problem in these applications is the estimation of geometric objects and transformations from noisy data. Most estimation techniques based on geometric algebra employ singular value decomposition or other linear least squares methods, see [4,5,20,21,31,39]. Valkenburg and Alwesh [48] employ nonlinear optimization in a calibration method of multiple stationary 3D points as part of an optical positioning system using the conformal model of geometric algebra.…”