Trifocal Tensors in Three-View Geometry

Source abstract

Matrices and tensors are ubiquitous throughout computer vision. The trifocal tensor $\caT$ is a $3\times 3\times 3$ 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 $\RR^{3 \times 9}$ 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 $\caT = A^{\top}\times \bfb_{4} - B^{\top}\times_{2} \bfa_{4}$; 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.