Pairwise Rigid Registration Based on Riemannian Geometry and Lie Structures of Orientation Tensors

• Published in: Journal of mathematical imaging and vision Vol. 63; no. 7; pp. 894 - 916
• Main Authors: de Almeida, Liliane Rodrigues, Giraldi, Gilson Antonio, Vieira, Marcelo Bernardes, Miranda Jr, Gastão Florêncio
• Format: Journal Article
• Language: English
• Published: New York Springer US 2021; Springer Nature B.V
• Subjects: Algorithms; Applications of Mathematics; Computer Science; Embedding; Geometry; Image Processing and Computer Vision; Iterative methods; Lie groups; Mathematical analysis; Mathematical Methods in Physics; Matrix algebra; Orientation; Registration; Riemann manifold; Shape factor; Signal,Image and Speech Processing; Tensors; Rigid registration; Orientation tensors; CTSF; Iterative closest point; Lie algebras; Lie groups
• ISSN: 0924-9907; 1573-7683
• DOI: 10.1007/s10851-021-01037-z

Pairwise rigid registration can be developed by comparing local geometry encoded by intrinsic second-order orientation tensors which allows to model the registration problem using the associated Riemannian geometry or related group structures. Everything starts by representing those tensor fields as multivariate normal models that permit us to manipulate Gaussians in two ways: using the Riemannian manifold elements, that can be embedded into matrix spaces with geometric structures, or through Lie group/algebra techniques. In this paper we discuss some points behind these approaches in the context of rigid registration problems. Firstly, they are not equivalent since, in general, there is no isometry linking them. Secondly, embedding methodologies are not invariant with respect to rigid motion. We discuss these points using two variants of the Iterative Closest Point that use the comparative tensor shape factor (CTSF) to match orientation tensors. We replace the CTSF to different criteria computed through geodesic distance and algebraic embeddings and compare the registration algorithms showing that the latter is more efficient for registration of point clouds.