QuadSync: Quadrifocal Tensor Synchronization via Tucker Decomposition
This work addresses the challenge of utilizing quadrifocal tensors for camera recovery in structure from motion, potentially benefiting researchers and practitioners in computer vision by enabling the use of higher-order information.
This paper introduces a new framework to recover 'n' cameras from a collection of quadrifocal tensors, which are typically considered impractical. The authors demonstrate that a block quadrifocal tensor admits a Tucker decomposition with a multilinear rank of (4, 4, 4, 4) and develop the first synchronization algorithm for these tensors.
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.