Motor Estimation using Heterogeneous Sets of Objects in Conformal Geometric Algebra

Hadfield

Adv. Appl. Clifford Algebras

2019

In this paper we will address the problem of recovering covariant transformations between objects-specifically; lines, planes, circles, spheres and point pairs. Using the covariant language of conformal geometric algebra (CGA), we will derive such transformations in a very simple manner. In CGA, rotations, translations, dilations and inversions can be written as a single rotor, which is itself an element of the algebra. We will show that the rotor which takes a line to a line (or plane to a plane etc) can easily be formed and we will investigate the nature of the rotors formed in this way. If we can recover the rotor between one object and another of the same type, a useable metric which tells us how close one line (plane etc) is to another, can be a function of how close this rotor is to the identity. Using these ideas, we find that we can define metrics for a number of common problems, specifically recovering the transformation between sets of noisy objects.

Section: Related Workmentioning

confidence: 99%

Calculating the Rotor Between Conformal Objects

Hadfield

Adv. Appl. Clifford Algebras

2019

“…For cases in which our query model is a small displacement (where displacement here will refer to rotation and translation) from the reference model, we would expect that simply assigning each object in the query model to its closest object in the reference model would give us a good number of correct matches. Several authors have proposed cost functions between objects [12,13], and while many of these are extremely effective for extracting motors between circles and other round elements, they tend to fail to extract the transformation between parallel lines and planes. To counteract this problem we choose the cost function described in [6] (the properties of this cost function are further explored in [6]).…”

Section: Proximity-based Matchingmentioning

confidence: 99%

“…Our reference might be, for example, a CAD model, and our query model might represent the output of fitting primitives to LIDAR data or structurefrom-motion point clouds. Many authors have tackled the problem of rotor estimation between groups of pre-matched geometric objects [5,6,12,13] and others have applied conformal geometric algebra to 3D registration of point and sphere clouds [2,9]. In this paper we tackle the problem of registration and rotor estimation for primitives of any grade.…”

Section: Introductionmentioning

confidence: 99%

REFORM: Rotor Estimation From Object Resampling and Matching

Hadfield

Adv. Appl. Clifford Algebras

Ramage

et al. 2019

In this paper we tackle the problem of correspondence and rotor estimation between models composed of geometric primitives of different types. We frame this problem as searching for the rotor that takes a query model to a reference model. The situations that we consider are those in which our query model: contains additional primitives not present in the reference; is missing primitives that are present in the reference. We will also look at cases in which there are a large number of primitives per model. These are all common issues facing any SLAM-type (simultaneous localisation and mapping) systems. To overcome these problems we introduce an inter-object rotor magnitudebased matching function and a subsampled iterative rotor estimation and matching algorithm. We title the finished algorithm: Rotor Estimation From Object Resampling and Matching-REFORM. REFORM builds on ideas from the RANSAC (RAndom SAmple Consensus) [7] and ICP (Iterative Closest Point) [3, 11] algorithms and extends these to multivector correspondence. It is easily parallelisable and designed for good convergence performance with models of real objects.

Proceedings of the Computer Graphics International Conference

“…which is solved using Levenberg-Marquardt algorithm which is has been developed to work for conformal points and lines. e algorithm is presented in [21].…”

Section: Error Functionsmentioning

confidence: 99%

Initial alignment of point clouds using motors

Kleppe

Tingelstad

Egeland

2017

Self Cite

is paper presents an approach for initial alignment or coarse registration of a partial 3D point cloud of objects. e method is based on computing the centroid of the points in the point cloud, and a line derived from the surface normals. is approach uses conformal geometric algebra and non-linear least squares optimization to achieve the results. e method performs well in experiments, and it is shown that it performs more accurately the more points are sampled.