{
  "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.24314",
  "author_total": 2,
  "authors": [
    {
      "name": "Yiran Xu"
    },
    {
      "name": "Changqing Xu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f83k2g4maww65.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.24314",
  "fulltext_url": null,
  "id": "f83k2g4maww65",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.24314",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AG"
    ],
    [
      "openalex",
      "2605|Tensor decomposition and applications"
    ]
  ],
  "openalex": "W7213998755",
  "pdf_url": "https://arxiv.org/pdf/2609.24314",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f83k2g4maww65.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.24314",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.24314"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Trifocal Tensors in Three-View Geometry",
  "title_text": "Trifocal Tensors in Three-View Geometry",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/f83k2g4maww65",
  "venue": "arXiv",
  "versions": []
}