Synchronization over Cartan motion groups via contraction

TL;DR

This paper introduces a novel method for synchronization over Cartan motion groups using group contraction, simplifying the problem by reducing it to a unitary representation, with demonstrated advantages over existing methods.

Contribution

The paper develops a contraction-based approach for synchronization over Cartan motion groups, including the special Euclidean group, with detailed analysis and numerical validation.

Findings

Effective reduction of non-compact synchronization problems to unitary representations

Numerical results show advantages over state-of-the-art methods

Applicable to rigid motion groups like SE(3)

Abstract

Group contraction is an algebraic map that relates two classes of Lie groups by a limiting process. We utilize this notion for the compactification of the class of Cartan motion groups. The compactification process is then applied to reduce a non-compact synchronization problem to a problem where the solution can be obtained by means of a unitary, faithful representation. We describe this method of synchronization via contraction in detail and analyze several important aspects of this application. One important special case of Cartan motion groups is the group of rigid motions, also called the special Euclidean group. We thoroughly discuss the synchronization over this group and show numerically the advantages of our approach compared to some current state-of-the-art synchronization methods on both synthetic and real data.