  We present, for the first time, a novel theoretical approach to address the
problem of correspondence free multivector cloud registration in conformal
geometric algebra. Such formalism achieves several favorable properties.
Primarily, it forms an orthogonal automorphism that extends beyond the typical
vector space to the entire conformal geometric algebra while respecting the
multivector grading. Concretely, the registration can be viewed as an
orthogonal transformation (\it i.e., scale, translation, rotation) belonging to
$SO(4,1)$ - group of special orthogonal transformations in conformal geometric
algebra. We will show that such formalism is able to: $(i)$ perform the
registration without directly accessing the input multivectors. Instead, we use
primitives or geometric objects provided by the conformal model - the
multivectors, $(ii)$ the geometric objects are obtained by solving a
multilinear eigenvalue problem to find sets of eigenmultivectors. In this way,
we can explicitly avoid solving the correspondences in the registration
process. Most importantly, this offers rotation and translation equivariant
properties between the input multivectors and the eigenmultivectors.
Experimental evaluation is conducted in datasets commonly used in point cloud
registration, to testify the usefulness of the approach with emphasis to
ambiguities arising from high levels of noise. The code is available at
https://github.com/Numerical-Geometric-Algebra/RegistrationGA . This work was
submitted to the International Journal of Computer Vision and is currently
under review.
