QuadSync: Quadrifocal Tensor Synchronization via Tucker Decomposition

Daniel Miao; Gilad Lerman; Joe Kileel

cs.CVFeb 26, 2026

QuadSync: Quadrifocal Tensor Synchronization via Tucker Decomposition

Authors: Daniel Miao, Gilad Lerman, Joe Kileel

Organizations: School of Mathematics, University of Minnesota · Department of Mathematics and Oden Institute for Computational Engineering and Sciences, University of Texas at Austin

Abstract

In structure from motion, quadrifocal tensors capture more information than their pairwise counterparts (essential matrices), yet they have often been thought of as impractical and only of theoretical interest. In this work, we challenge such beliefs by providing a new framework to recover $n$ cameras from the corresponding collection of quadrifocal tensors. We form the block quadrifocal tensor and show that it admits a Tucker decomposition whose factor matrices are the stacked camera matrices, and which thus has a multilinear rank of (4,~4,~4,~4) independent of $n$ . We develop the first synchronization algorithm for quadrifocal tensors, using Tucker decomposition, alternating direction method of multipliers, and iteratively reweighted least squares. We further establish relationships between the block quadrifocal, trifocal, and bifocal tensors, and introduce an algorithm that jointly synchronizes these three entities. Numerical experiments demonstrate the effectiveness of our methods on modern datasets, indicating the potential and importance of using higher-order information in synchronization.

Figures & tables

Figure 1: Mean location error for ETH3D datasets

Figure 2: Mean location error for EPFL datasets

Figure 3: QuadSync retrieved camera poses on near-collinear views from plant_scene_1 dataset from ETH3D SLAM.

Figure 4: Randomized updates in QuadSync tested on ETH3D ‘relief’ dataset.

Figure 5: Ground truth location of 10 cameras. $\times$

perc. (%)	noise(%)	$\bar{e_{t}}$	$\hat{e_{t}}$	$\bar{e_{r}}$	$\hat{e_{r}}$
100	0.00	0.00	0.00	0.00	0.00
100	1	0.04	0.04	0.23	0.18
100	5	0.24	0.22	2.67	2.52
80	0	0.00	0.00	0.00	0.00
80	1	0.04	0.03	0.37	0.37
80	5	0.36	0.36	3.07	2.68

Table 1: Results for synthetic experiments with collinear cameras.

\bar{e_{t}}

: mean location error.

\hat{e_{t}}

: median location error.

\bar{e_{r}}

: mean rotation error.

\hat{e_{r}}

: median rotation error.

Figure 6: Retrieved poses (colored, where each cluster has a distinct color) vs. ground truth poses (black) for CastleP30 with handpicked clusters.

size	noise	$\bar{e_{t}}$	$\hat{e_{t}}$	$\bar{e_{r}}$	$\hat{e_{r}}$	time(s)
Full (30 cams)	0	0	0	0	0	1666.3
Cluster1 (10 cams)	0	0	0	0	0	27.3
Cluster2 (15 cams)	0	0	0	0	0	92.6
Cluster3 (15 cams)	0	0	0	0	0	97.7
Aligned (30 cams)	0	0	0	0	0	150.2
Full (30 cams)	1	0.11	0.10	0.08	0.09	1944.8

Table 2: Results for synthetic distributed synchronization.

\bar{e_{t}}

: mean location error.

\hat{e_{t}}

: median location error.

\bar{e_{r}}

: mean rotation error.

\hat{e_{r}}

: median rotation error. Noise is added as percentage onto the cameras. In the noisy cases, only the quadrifocal tensors were synchronized.

Dataset	Completion %	Joint Opt. (s)	TrifocalSync (s)
courtyard (18/38)	33.10	111.2	137.9
electro (12/39)	78.70	30.6	45.13
facade (46/76)	61.98	4226.2	1633.0
kicker (20/30)	31.84	157.3	174.3
meadow (6/14)	42.13	11.7	9.1
office (13/21)	39.92	34.2	53.3

Table 3: Completion rates and runtimes of different methods

Dataset	QuadSync	Joint Opt.	Trifocal Sync	LUD	NRFM	MPLS BATA	MPLS CS
courtyard (18/38)	0.0477	0.0489	0.1753	0.0403	0.1104	0.0711	0.0234
electro (12/39)	0.0199	0.0200	0.0197	0.0605	0.0257	0.0617	0.0604
facade (46/76)	3.0294	0.0248	0.0222	0.2497	0.2068	0.0984	0.0158
kicker (20/30)	0.0078	0.0078	0.0204	0.0366	0.0246	0.1934	0.0178
meadow (6/14)	0.0401	0.1071	1.4341	0.6408	0.2207	1.8229	0.0691
office (13/21)	0.0039	0.0037	0.0151	0.0262	0.0187	0.0166	0.0086

Table 4: Mean location error by method

Dataset	QuadSync	Joint Opt.	Trifocal Sync	LUD	NRFM	MPLS BATA	MPLS CS
courtyard	0.0307	0.0324	0.0947	0.0308	0.0668	0.0598	0.0194
electro	0.0151	0.0152	0.0133	0.0256	0.0081	0.0221	0.0181
facade	2.5853	0.0135	0.0140	0.0108	0.0124	0.0253	0.0128
kicker	0.0068	0.0063	0.0173	0.0090	0.0110	0.0788	0.0101
meadow	0.0390	0.0922	1.1006	0.2374	0.1285	0.7635	0.0358
office	0.0028	0.0027	0.0090	0.0035	0.0047	0.0043	0.0033

Table 5: Median location error by method

Dataset	QuadSync	Joint Opt.	Trifocal Sync	LUD	NRFM	MPLS
courtyard	0.3059	0.3295	1.2398	0.1886	0.1886	0.1056
electro	0.1022	0.1020	0.1208	0.0722	0.0722	0.0603
facade	2.1185	0.1131	0.1239	0.0442	0.0442	0.0506
kicker	0.2063	0.2028	0.2986	0.1162	0.1162	0.2697
meadow	0.2856	0.6100	12.4079	0.1403	0.1403	0.1861
office	0.1653	0.1449	0.5749	0.0674	0.0674	0.0547

Table 6: Mean rotation error by method

Dataset	QuadSync	Joint Opt.	Trifocal Sync	LUD	NRFM	MPLS
courtyard	0.2144	0.2283	1.2651	0.1909	0.1909	0.1234
electro	0.0883	0.0888	0.1186	0.0630	0.0630	0.0638
facade	0.9737	0.0731	0.0812	0.0449	0.0449	0.0457
kicker	0.1735	0.1701	0.2877	0.1146	0.1146	0.1650
meadow	0.2002	0.5049	7.3067	0.0902	0.0902	0.1337
office	0.0994	0.1070	0.2807	0.0717	0.0717	0.0703

Table 7: Median rotation error by method

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