pith. sign in

arxiv: 1907.10415 · v1 · pith:UMM3PDMEnew · submitted 2019-07-24 · 🧮 math.AG

The Rank of Trifocal Grassmann Tensors

Marina Bertolini , Gian Mario Besana , Gilberto Bini , Cristina Turrini This is my paper

Pith reviewed 2026-05-24 16:52 UTC · model grok-4.3

classification 🧮 math.AG
keywords Grassmann tensorstrifocal tensorstensor rankprojection matricesmulti-view geometryalgebraic geometryscene reconstructioncanonical forms
0
0 comments X

The pith

A canonical form for combined projection matrices yields a closed formula for the rank of trifocal Grassmann tensors when centers are generic.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines trifocal Grassmann tensors that arise from three projections of a k-dimensional projective space onto lower-dimensional view spaces, a setup drawn from multi-view scene reconstruction. It constructs a canonical form for the stacked projection matrices that encode the three centers and their images. Under the assumption that the three centers satisfy a natural generality condition, this canonical form directly produces an explicit formula for the rank of the resulting tensor. The same technique recovers an earlier result on the rank of bifocal Grassmann tensors and tracks how ranks behave for sequences of tensors that approach degenerate center configurations.

Core claim

When the centers of the three projections lie in general position, the canonical form of the combined projection matrices supplies a closed formula for the rank of the associated trifocal Grassmann tensor; the same reduction also reproduces the known rank formula for the two-projection case.

What carries the argument

The canonical form of the combined projection matrices, which reduces the rank computation to an algebraic count once the centers satisfy the generality assumption.

If this is right

  • The rank of any trifocal Grassmann tensor with generic centers is given by an explicit algebraic expression obtained from the canonical matrices.
  • The identical reduction produces the rank formula for bifocal Grassmann tensors, matching results obtained by other methods.
  • Sequences of Grassmann tensors that converge to those arising from degenerate center configurations realize a spectrum of possible rank jumps.
  • The canonical-form technique applies uniformly across different choices of the dimension k and the dimensions of the view spaces.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The explicit rank formula may be used to test whether a given tensor arises from a realizable triple of projections in computer-vision pipelines.
  • Degenerate-limit examples could serve as test cases for numerical stability of rank-computation algorithms when centers approach collinearity or coplanarity.
  • The same matrix-reduction idea might extend to tensors associated with four or more views, though the paper does not carry out that extension.

Load-bearing premise

The centers of projections satisfy a natural generality assumption.

What would settle it

Compute the rank of an explicit trifocal Grassmann tensor for a concrete choice of generic centers in low dimension (for example k=2) and check whether it equals the number given by the closed formula derived from the canonical form.

read the original abstract

Grassmann tensors arise from classical problems of scene reconstruction in computer vision. Trifocal Grassmann tensors, related to three projections from a projective space of dimension k onto view-spaces of varying dimensions are studied in this work. A canonical form for the combined projection matrices is obtained. When the centers of projections satisfy a natural generality assumption, such canonical form gives a closed formula for the rank of the trifocal Grassmann tensors. The same approach is also applied to the case of two projections, confirming a previous result obtained with different methods in [6]. The rank of sequences of tensors converging to tensors associated with degenerate configurations of projection centers is also considered, giving concrete examples of a wide spectrum of phenomena that can happen.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper studies trifocal Grassmann tensors from three projections in projective space of dimension k. It derives a canonical form for the combined projection matrices and shows that, under a natural generality assumption on the projection centers, this form yields a closed formula for the tensor rank. The same canonical-form method recovers a known bifocal result from [6] and is used to analyze rank behavior for sequences of tensors approaching degenerate projection-center configurations.

Significance. The explicit closed-form rank expression under a standard generality assumption is a concrete advance for multi-view geometry in algebraic geometry and computer vision. Recovering the bifocal case with the same technique and providing concrete examples of limiting rank phenomena strengthen the contribution. The canonical-form approach is methodologically clean and could support further explicit computations.

minor comments (3)
  1. [Abstract] Abstract: the phrase 'such canonical form gives a closed formula' is slightly vague; state the formula or cite the precise theorem number where it appears.
  2. The generality assumption on projection centers is invoked repeatedly; consider isolating it as a numbered hypothesis or definition early in the text for easy reference.
  3. Notation for the view-space dimensions and the combined projection matrix could be summarized in a short table or diagram to aid readability across sections.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision for our manuscript on the rank of trifocal Grassmann tensors. The report contains no specific major comments or requested changes.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper obtains an explicit canonical form for the combined projection matrices of three (or two) projections and, under a stated generality assumption on the centers, directly extracts a closed rank formula for the associated Grassmann tensors. No equation or step is shown to reduce by construction to a fitted parameter, a self-definition, or a prior result whose only justification is a self-citation chain. The bifocal confirmation is explicitly noted to use different methods, and the trifocal derivation is presented as independent algebraic work. The derivation chain therefore remains self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The sole load-bearing premise visible in the abstract is the generality assumption on projection centers that enables both the canonical form and the closed rank formula.

axioms (1)
  • domain assumption The centers of projections satisfy a natural generality assumption.
    Invoked in the abstract as the condition under which the canonical form yields the closed rank formula.

pith-pipeline@v0.9.0 · 5648 in / 1148 out tokens · 25695 ms · 2026-05-24T16:52:46.614353+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages · 1 internal anchor

  1. [1]

    The ideal of the trifocal variety

    Chris Aholt and Luke Oeding. The ideal of the trifocal variety. Math. Comp. , 83(289):2553–2574, 2014

    work page 2014

  2. [2]

    Constraints for the trifoc al tensor

    Alberto Alzati and Alfonso Tortora. Constraints for the trifoc al tensor. In Tensors in image processing and computer vision , Adv. Pattern Recognit., pages 261–269. Springer, London, 2009

    work page 2009

  3. [3]

    The hitchhiker guide to: Secant va rieties and tensor decomposition

    Alessandra Bernardi, Enrico Carlini, Maria Virginia Catalisano, Aless andro Gimigliano, and Alessandro Oneto. The hitchhiker guide to: Secant va rieties and tensor decomposition. Mathematics, 6(12), 2018

    work page 2018

  4. [4]

    Matrices dropping rank in codimension one and critical loci in computer vision

    Marina Bertolini, GianMario Besana, Roberto Notari, and Cristina T urrini. Matrices dropping rank in codimension one and critical loci in compute r vision. arXiv:1902.00376, 2019. 18 M.BERTOLINI, G.M.BESANA, G.BINI, AND C.TURRINI

    work page internal anchor Pith review Pith/arXiv arXiv 1902

  5. [5]

    Critical loc i for projective reconstruction from multiple views in higher dimension: a c ompre- hensive theoretical approach

    Marina Bertolini, GianMario Besana, and Cristina Turrini. Critical loc i for projective reconstruction from multiple views in higher dimension: a c ompre- hensive theoretical approach. Linear Algebra Appl. , 469:335–363, 2015

    work page 2015

  6. [6]

    Generalize d funda- mental matrices as grassmann tensors

    Marina Bertolini, GianMario Besana, and Cristina Turrini. Generalize d funda- mental matrices as grassmann tensors. Annali di Matematica Pura ed Applicata (1923 -) , 7 2016

    work page 1923

  7. [7]

    The Bordiga surface as critical locus for 3-view reconstructions

    Marina Bertolini, Roberto Notari, and Cristina Turrini. The Bordiga surface as critical locus for 3-view reconstructions. J. Symbolic Comput. , 91:74–97, 2019

    work page 2019

  8. [8]

    Critical configurations for 1 -view in pro- jections from Pk → P2

    Marina Bertolini and Cristina Turrini. Critical configurations for 1 -view in pro- jections from Pk → P2. Journal of Mathematical Imaging and Vision , 27:277– 287, 2007

    work page 2007

  9. [9]

    Tensor rank

    Elias Erdtman and Carl J¨ onsson. Tensor rank. Master’s thesis , Link¨ opings Universitet, Link¨ opings, Sweden, June 2012

    work page 2012

  10. [10]

    Multiple view geometry in computer vision

    Richard Hartley and Andrew Zisserman. Multiple view geometry in computer vision. Cambridge University Press, Cambridge, second edition, 2003. Wit h a foreword by Olivier Faugeras

    work page 2003

  11. [11]

    Hartley and Frederik Schaffalitzky

    Richard I. Hartley and Frederik Schaffalitzky. Reconstruction from projections using grassmann tensors. Int. J. Comput. Vision , 83(3):274–293, July 2009

    work page 2009

  12. [12]

    A common framework for multiple view tensors

    Anders Heyden. A common framework for multiple view tensors. In Computer VisionECCV’98, pages 3–19. Springer Berlin Heidelberg, 1998

    work page 1998

  13. [13]

    W. V. D. Hodge and D. Pedoe. Methods of Algebraic Geometry , volume 1. Cambridge University Press, 1994

    work page 1994

  14. [14]

    J. M. Landsberg. Tensors: geometry and applications , volume 128 of Gradu- ate Studies in Mathematics . American Mathematical Society, Providence, RI, 2012

    work page 2012

  15. [15]

    F. Enriques

    Luke Oeding. The quadrifocal variety. Linear Algebra Appl., 512:306–330, 2017. E-mail address : marina.bertolini@unimi.it Dipartimento di Matematica “F. Enriques”, Universit `a degli Studi di Milano, Via Saldini 50, 20133 Milano, Italy E-mail address : gbesana@depaul.edu College of Computing and Digital Media, DePaul University, 243 South W abash, Chicago...

    work page 2017