Trifocal Tensors in Three-View Geometry
Abstract
Matrices and tensors are ubiquitous throughout computer vision. The trifocal tensor caT is a 3× 3× 3 tensor that plays a vital role in three-view geometry. However, this tensor, though displayed in a third-order form, is manipulated and operated primarily in a matrix style in standard literature. This article establishes the conformal tensor algebra by replacing ad-hoc matrix collections with well-defined multilinear operators. We treat all three views symmetrically by covariant subscript indexing (Tijk). To make precise what is new beyond notation: the classical point, line, and mixed correspondence constraints are here re-derived, but \emph{uniformly}, each as a single contraction in one shared algebra; the results that are genuinely new include the exact rank characterization of the trifocal tensor --- all mode-k unfoldings caT[k] in RR3 × 9 satisfy rank(caT[k])=3 for non-degenerate setups, and rank deficiency of any unfolding certifies degeneracy of the camera configuration, which yields a near-zero-cost degeneracy alarm and a validation criterion for estimated tensors; the outer-product representation caT = A⊤× bfb4 − B⊤×2 bfa4; and direct epipole extraction via double contractive traces. The coordinate-free multilinear operators seamlessly map to tensor auto-differentiation frameworks (e.g., PyTorch, TensorFlow); a concrete PyTorch experiment demonstrates that the induced $24$-parameter bilinear parametrization acts as a structural prior that eliminates the accuracy drift exhibited by an unstructured $27$-entry autodiff refinement.
Related
- The Geometry of Lee-Yang Tensors
- Gauge-compatible tensors on statistical manifolds: splitting and submanifold geometry
- Finite Weil Jets, Identity Transfer, and the Exponential Geometry of Proper Orthogonal Tensors
- Deligne--Faltings Logarithmic Derived Geometry with a View towards Logarithmic GLSM
- Three-Dimensional Nonlinear LOS-Angle Shaping Guidance with Prescribed Initial Acceleration and Field-of-View Constraints
BibTeX
@misc{xu2026trifocal,
title = {{Trifocal Tensors in Three-View Geometry}},
author = {Yiran Xu and Changqing Xu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.24314},
eprint = {2609.24314},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.24314},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Trifocal Tensors in Three-View Geometry
AU - Yiran Xu
AU - Changqing Xu
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.24314
UR - https://arxiv.org/abs/2609.24314
AB - Matrices and tensors are ubiquitous throughout computer vision. The trifocal tensor caT is a 3× 3× 3 tensor that plays a vital role in three-view geometry. However, this tensor, though displayed in a third-order form, is manipulated and operated primarily in a matrix style in standard literature. This article establishes the conformal tensor algebra by replacing ad-hoc matrix collections with well-defined multilinear operators. We treat all three views symmetrically by covariant subscript indexing (T_ijk). To make precise what is new beyond notation: the classical point, line, and mixed correspondence constraints are here re-derived, but \emph{uniformly}, each as a single contraction in one shared algebra; the results that are genuinely new include the exact rank characterization of the trifocal tensor --- all mode-k unfoldings caT[k] in RR^3 × 9 satisfy rank(caT[k])=3 for non-degenerate setups, and rank deficiency of any unfolding certifies degeneracy of the camera configuration, which yields a near-zero-cost degeneracy alarm and a validation criterion for estimated tensors; the outer-product representation caT = A^⊤× bfb_4 − B^⊤×_2 bfa_4; and direct epipole extraction via double contractive traces. The coordinate-free multilinear operators seamlessly map to tensor auto-differentiation frameworks (e.g., PyTorch, TensorFlow); a concrete PyTorch experiment demonstrates that the induced $24$-parameter bilinear parametrization acts as a structural prior that eliminates the accuracy drift exhibited by an unstructured $27$-entry autodiff refinement.
ER -