Segre-Determinantal Loci and the Image Variety for Three Flatland Cameras

Abstract: Motivated by applications of algebraic geometry to reconstruction problems in computer vision, we initiate a study of the equations of degeneracy loci associated with linearly dependent points on Segre varieties. When these points are constrained to lie on a common hyperplane, we prove that the vanishing ideals of these loci are prime, Cohen-Macaulay, and generated by the natural maximal minors, and that these minors form a universal Gröbner basis.

• The paper establishes that the vanishing ideals of Segre-determinantal loci are prime, Cohen-Macaulay, and universally generated by maximal minors.
• It proves that the maximal minors form a universal Gröbner basis, leveraging combinatorial and matroidal methods to simplify computations in multiview geometry.
• It demonstrates that for three flatland cameras, the image variety coincides with the Segre-determinantal locus, providing robust criteria for 2D scene reconstruction.

Segre-Determinantal Loci and the Image Variety for Three Flatland Cameras

Introduction and Motivation

This paper develops a comprehensive algebraic-geometric characterization of degeneracy loci defined by linear dependence constraints on Segre varieties, with a core motivation in multiview geometry, particularly in the context of scene reconstruction from multiple projective camera views in flatland. The authors focus on instances where observed point correspondences from “flatland cameras” (namely, projections from $\mathbb{P}^2$ to $\mathbb{P}^1$) raise complex degeneracy conditions corresponding to the linear dependence of their Segre embeddings. The results generalize the study of image varieties and determinantal loci, providing rigorous commutative and computational algebraic foundations for solvability and implicitization in multiview geometry, especially in settings involving three flatland cameras.

Segre-Determinantal Loci: Definitions and Core Results

The Segre-determinantal loci $\Seg_{m_\bullet, n}$ are defined as the closed subvarieties of $(\mathbb{P}^{m_\bullet})^{\times n}$ whose $n$ Segre-embedded points are linearly dependent, i.e., they lie on a common hyperplane in $\mathbb{P}^{M(m_\bullet)}$. The authors give a precise parameter count and show that, for $n > M(m_\bullet)$, these loci are irreducible of codimension $n - M(m_\bullet)$.

A strong algebraic result is established: the vanishing ideals of these Segre-determinantal loci are prime, Cohen-Macaulay, and are generated by the maximal minors of the underlying Segre matrix. Further, the maximal minors form a universal Gröbner basis for these ideals. This universality implies that these minors provide Gröbner bases for all monomial orders, significantly simplifying computational approaches.

The authors rigorously prove these structural results using incidence variety arguments, dimension counts, and the Eagon-Northcott theorem. The Jacobian criterion is explicitly verified in the flatland ($m_\bullet = (1,1,1)$) case.

Universal Gröbner Bases: Combinatorial and Matroidal Perspectives

The proof that the set of maximal minors forms a universal Gröbner basis leverages recent advances in the combinatorics of determinantal ideals and algebraic matroids. Using a criterion by Huang and Larson (Huang et al., 2024), the authors establish that the “spread complex” associated to the minors aligns with the algebraic matroid of the variety, ensuring universality.

Explicit combinatorial descriptions of facets and bases are given, exploiting the tensor structure of the Segre embedding and providing clear pathways for studying projections and algebraic independence relevant in computer vision.

Flatland Camera Image Varieties: Reconstruction and Equational Characterization

Turning to the imaging problem, the paper formalizes the image variety of three flatland cameras as the Zariski closure of points in $(\mathbb{P}^1)^{3n}$ consistent with a scene reconstruction. The main algebraic result is:

For all $\mathbb{P}^1$0, the image variety of three flatland cameras coincides scheme-theoretically with the Segre-determinantal locus $\mathbb{P}^1$1. The ideal is generated by $\mathbb{P}^1$2 minors of the Segre matrix whose rows are the embedded point correspondences.

This establishes necessary and sufficient conditions for feasibility of the 2D reconstruction from three projections to $\mathbb{P}^1$3. In particular, the affirmative solution holds generically for $\mathbb{P}^1$4 point triples; for $\mathbb{P}^1$5 the locus is a hypersurface, and the defining Segre determinant aligns with recent work in determinantal geometry (Pratt, 14 May 2025).

A thorough dimension analysis confirms the expected dimension formula and dominant parameterization in the flatland trifocal tensor case.

Generalizations and Connections to Multiview Geometry

The paper analyzes the extension of these results to higher-dimensional multiview settings, generalizing both the camera models and the Segre-determinantal loci. For classical pinhole cameras ($\mathbb{P}^1$6), the Segre-determinantal loci encapsulate but do not exhaust the image varieties, necessitating more elaborate constraints (e.g., fundamental matrices, trifocal tensors), many of whose ideals remain open problems in computational algebraic geometry.

Explicit examples for the two and three pinhole camera cases demonstrate that for sufficiently many point correspondences, the Segre minors alone do not provide a complete set of equations for the image variety; additional geometrically meaningful determinants (e.g., the 8-point or 7-point constraints) are necessary.

Implications and Theoretical Developments

This work delivers a robust algebraic foundation for analyzing and computing image varieties, degeneracy loci, and implicitization problems in multiview geometry—areas critical in algebraic vision and computational geometry. By delivering explicit, universal Gröbner bases and solidifying Cohen-Macaulay and primeness properties, the results facilitate effective symbolic and numerical computations for solving reconstruction and feasibility problems in computer vision.

Furthermore, the combinatorial and matroidal approaches developed here suggest potential for further generalizations in tensor completion, rigidity and matrix completion problems, expanding the interaction between commutative algebra, algebraic geometry, and combinatorics.

Conclusion

This paper presents a detailed structure theory for Segre-determinantal loci, with universal Gröbner bases and Cohen-Macaulay properties, and definitively links these loci to image varieties for three flatland cameras. The work provides effective criteria and explicit constructions for feasibility and implicitization problems foundational in algebraic vision, while uncovering deeper connections between determinantal geometry, algebraic matroids, and computational approaches to multiview geometry. The theoretical framework and explicit results presented invite further exploration of advanced imaging models, higher codimension degeneracy loci, and their role in the algorithmic and geometric foundations of computer vision and related fields (2606.31965).