TY - JOUR
T1 - SE(3) Synchronization by eigenvectors of dual quaternion matrices
AU - Hadi, Ido
AU - Bendory, Tamir
AU - Sharon, Nir
N1 - Publisher Copyright:
© 2024 The Author(s). Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.
PY - 2024/9/1
Y1 - 2024/9/1
N2 - In synchronization problems, the goal is to estimate elements of a group from noisy measurements of their ratios. A popular estimation method for synchronization is the spectral method. It extracts the group elements from eigenvectors of a block matrix formed from the measurements. The eigenvectors must be projected, or 'rounded', onto the group. The rounding procedures are constructed ad hoc and increasingly so when applied to synchronization problems over non-compact groups. In this paper, we develop a spectral approach to synchronization over the non-compact group, the group of rigid motions of. We based our method on embedding into the algebra of dual quaternions, which has deep algebraic connections with the group. These connections suggest a natural rounding procedure considerably more straightforward than the current state of the art for spectral synchronization, which uses a matrix embedding of. We show by numerical experiments that our approach yields comparable results with the current state of the art in synchronization via the spectral method. Thus, our approach reaps the benefits of the dual quaternion embedding of while yielding estimators of similar quality.
AB - In synchronization problems, the goal is to estimate elements of a group from noisy measurements of their ratios. A popular estimation method for synchronization is the spectral method. It extracts the group elements from eigenvectors of a block matrix formed from the measurements. The eigenvectors must be projected, or 'rounded', onto the group. The rounding procedures are constructed ad hoc and increasingly so when applied to synchronization problems over non-compact groups. In this paper, we develop a spectral approach to synchronization over the non-compact group, the group of rigid motions of. We based our method on embedding into the algebra of dual quaternions, which has deep algebraic connections with the group. These connections suggest a natural rounding procedure considerably more straightforward than the current state of the art for spectral synchronization, which uses a matrix embedding of. We show by numerical experiments that our approach yields comparable results with the current state of the art in synchronization via the spectral method. Thus, our approach reaps the benefits of the dual quaternion embedding of while yielding estimators of similar quality.
KW - applied non-commutative algebra
KW - dual quaternions
KW - group synchronization
KW - spectral algorithms
UR - http://www.scopus.com/inward/record.url?scp=85198718369&partnerID=8YFLogxK
U2 - 10.1093/imaiai/iaae014
DO - 10.1093/imaiai/iaae014
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AN - SCOPUS:85198718369
SN - 2049-8772
VL - 13
JO - Information and Inference
JF - Information and Inference
IS - 3
M1 - iaae014
ER -