GLOBAL REGISTRATION OF MULTIPLE POINT CLOUDS USING SEMIDEFINITE PROGRAMMING

Chaudhury, KN and Khoo, Y and Singer, A (2015) GLOBAL REGISTRATION OF MULTIPLE POINT CLOUDS USING SEMIDEFINITE PROGRAMMING. In: SIAM JOURNAL ON OPTIMIZATION, 25 (1). pp. 468-501.

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Abstract

Consider N points in R-d and M local coordinate systems that are related through unknown rigid transforms. For each point, we are given (possibly noisy) measurements of its local coordinates in some of the coordinate systems. Alternatively, for each coordinate system, we observe the coordinates of a subset of the points. The problem of estimating the global coordinates of the N points (up to a rigid transform) from such measurements comes up in distributed approaches to molecular conformation and sensor network localization, and also in computer vision and graphics. The least-squares formulation of this problem, although nonconvex, has a well-known closed-form solution when M = 2 (based on the singular value decomposition (SVD)). However, no closed-form solution is known for M >= 3. In this paper, we demonstrate how the least-squares formulation can be relaxed into a convex program, namely, a semidefinite program (SDP). By setting up connections between the uniqueness of this SDP and results from rigidity theory, we prove conditions for exact and stable recovery for the SDP relaxation. In particular, we prove that the SDP relaxation can guarantee recovery under more adversarial conditions compared to earlier proposed spectral relaxations, and we derive error bounds for the registration error incurred by the SDP relaxation. We also present results of numerical experiments on simulated data to confirm the theoretical findings. We empirically demonstrate that (a) unlike the spectral relaxation, the relaxation gap is mostly zero for the SDP (i.e., we are able to solve the original nonconvex least-squares problem) up to a certain noise threshold, and (b) the SDP performs significantly better than spectral and manifold-optimization methods, particularly at large noise levels.

Item Type:	Journal Article
Publication:	SIAM JOURNAL ON OPTIMIZATION
Publisher:	SIAM PUBLICATIONS
Additional Information:	Copy right for this article belongs to the SIAM PUBLICATIONS, 3600 UNIV CITY SCIENCE CENTER, PHILADELPHIA, PA 19104-2688 USA
Keywords:	global registration; rigid transforms; rigidity theory; spectral relaxation; spectral gap; convex relaxation; semidefinite program (SDP); exact recovery; noise stability
Department/Centre:	Division of Electrical Sciences > Electrical Engineering
Date Deposited:	06 May 2015 05:43
Last Modified:	06 May 2015 05:43
URI:	http://eprints.iisc.ac.in/id/eprint/51505

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