Calculating the Rotor Between Conformal Objects

Abstract: 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

REFORM: Rotor Estimation From Object Resampling and Matching

Abstract: 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 magnitude-based 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