Tensor-Based Synchronization and the Low-Rankness of the Block Trifocal Tensor

TL;DR

This paper reveals a low-rank structure in the block trifocal tensor that enables more accurate camera pose synchronization in three-view geometry, surpassing traditional pairwise methods.

Contribution

It introduces an explicit Tucker factorization of the block trifocal tensor, establishing a low multilinear rank that facilitates a novel synchronization algorithm.

Findings

The Tucker rank is (6,4,4) regardless of camera number under certain conditions.

The proposed algorithm outperforms state-of-the-art methods in location accuracy.

Exploiting higher-order interactions improves synchronization performance.

Abstract

The block tensor of trifocal tensors provides crucial geometric information on the three-view geometry of a scene. The underlying synchronization problem seeks to recover camera poses (locations and orientations up to a global transformation) from the block trifocal tensor. We establish an explicit Tucker factorization of this tensor, revealing a low multilinear rank of $(6,4,4)$ independent of the number of cameras under appropriate scaling conditions. We prove that this rank constraint provides sufficient information for camera recovery in the noiseless case. The constraint motivates a synchronization algorithm based on the higher-order singular value decomposition of the block trifocal tensor. Experimental comparisons with state-of-the-art global synchronization methods on real datasets demonstrate the potential of this algorithm for significantly improving location estimation accuracy. Overall this work suggests that higher-order interactions in synchronization problems can be exploited to improve performance, beyond the usual pairwise-based approaches.