发表机构
Georgia State University; Suzhou University of Science and Technology（佐治亚州立大学; 苏州科技大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文为三视图几何中的三焦点张量建立共形张量代数，以统一多线性算子重推导经典约束，并给出精确秩刻画、外积表示及对极点提取方法，其参数化在PyTorch中作为结构先验消除精度漂移。
AI 中文摘要
矩阵和张量在计算机视觉中无处不在。三焦点张量 $\caT$ 是一个 $3\times 3\times 3$ 的张量，在三视图几何中起着至关重要的作用。然而，尽管该张量以三阶形式呈现，但在标准文献中，它主要按矩阵风格进行操纵和运算。本文通过用定义明确的多线性算子取代临时性的矩阵集合，建立了共形张量代数。我们通过协变下标索引（$T_{ijk}$）对称地处理所有三个视图。为了明确超越符号的新颖之处：经典的点、线和混合对应约束在此被重新推导，但以统一的方式，每个约束都是共享代数中的一次单一收缩；真正新颖的结果包括三焦点张量的精确秩刻画——在非退化设置下，$\RR^{3 \times 9}$ 中所有模式-$k$ 展开 $\caT[k]$ 满足 $\rank(\caT[k])=3$，且任何展开的秩亏缺都证明相机配置的退化性，这产生了一个近零成本的退化警报和估计张量的验证标准；外积表示 $\caT = A^{\top}\times \bfb_{4} - B^{\top}\times_{2} \bfa_{4}$；以及通过双重收缩迹直接提取对极点。无坐标多线性算子无缝映射到张量自动微分框架（例如，PyTorch、TensorFlow）；一个具体的 PyTorch 实验表明，所引入的 $24$ 参数双线性参数化作为结构先验，消除了非结构化 $27$ 项自动微分细化所表现出的精度漂移。
英文摘要
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.
Comments80 pages, 2 figures
