One-Dimensional Line Multiview Varieties

• One-dimensional line multiview varieties are defined as the Zariski closure of lines in projective space mapped to tuples of lines in projective planes via full-rank cameras.
• The construction employs Plücker coordinates and Grassmannian geometry to capture constraints, determine dimensionality, and ensure generically injective correspondence.
• Applications include robust line triangulation and calibration in multi-view systems, with methods involving Gröbner bases, multidegree analysis, and Euclidean distance degree computations.

A one-dimensional line multiview variety, also known as the line $n$-view variety or unanchored line multiview variety, is the Zariski closure of the image of the map that projects one-dimensional linear subspaces (lines) in projective space $\PP^m$—most classically, in %%%%2%%%%—to tuples of their images as lines in $n$ projective planes $\PP^2$, induced by $n$ full-rank projective cameras. These varieties encode the fundamental constraints on multi-image line correspondences in computational vision and algebraic geometry, generalizing classical relations for point correspondences. They form a central object in the intersection of algebraic geometry, geometric computer vision, and tensor-based methods in multiview geometry.

1. Mathematical Definition and Canonical Construction

Let $C_1,\dots,C_n$ be $n$ full-rank $3\times(m+1)$ projective camera matrices acting as linear maps from $\PP^m$ to $\PP^2$, each with a well-defined center $c_i=\ker C_i$. A projective line $L\subset \PP^m$ can be represented as the span of two independent homogeneous vectors $X_0,X_1\in \CC^{m+1}$, or equivalently as a point on the Grassmannian $\Gr(2,m+1)$. The camera $C_i$ maps the line $L=\langle X_0,X_1\rangle$ to the image line $C_i L = \langle C_i X_0, C_i X_1 \rangle \subset \PP^2$ provided the line does not meet the camera center.

The multiview map is

$\varphi: \Gr(2,m+1) \dashrightarrow (\PP^2)^n,\quad L \mapsto (C_1 L, \dots, C_n L).$

In Plücker coordinates, this is given by

$\varphi_i((p_{IJ}))_\alpha = \sum_{I<J} (\text{minor of %%%%18%%%% omitting row %%%%19%%%%})\cdot p_{IJ},$

where $p_{IJ}(L) = \det[X_I;X_J]$ and $\alpha=0,1,2$. The one-dimensional line multiview variety $MV_1(C_1,\dots,C_n)$ is defined as the Zariski closure $\overline{\varphi(\Gr(2,m+1))} \subset (\PP^2)^n$ (Rydell, 2023).

2. Dimension, Injectivity, and Geometric Properties

For generic cameras (i.e., centers pairwise in general position), and in the classical case $m=3$, the dimension of the one-dimensional line multiview variety is

$\dim MV_1 = 2\cdot \min(m-1, n).$

For example, for $\PP^3$ and $n \geq 2$, $\dim MV_1 = 2\min(2,n)$, yielding dimension $4$ as soon as $n \ge 2$ (Rydell, 2023, Duff et al., 2024). The critical threshold for generic injectivity (i.e., triangulability of a world line from its $n$ image lines) is $n \geq m-1$. Below this, the inverse image has positive dimension; above, the map is generically injective.

Furthermore, for $n\geq m-1$ with generic centers, $MV_1$ is isomorphic to the blowup of $P^m$ along the base locus of the joint projection $(C_1,\dots,C_n)$, and for $m=3$, $n\geq 2$, every tuple of image lines corresponds to a unique world line ($L = \bigcap_i H_i$), with $H_i$ the back-projected planes. Thus, the inverse correspondence amounts to solving linear equations for the Plücker coordinates of $L$.

3. Algebraic Equations and Ideals

Suppose $C_1,\dots,C_m$ are generic $3\times 4$ (rank $3$) camera matrices for lines in $\PP^3$. The associated line multiview variety in $(\PP^2)^m$ is set-theoretically cut out by the vanishing of all $3\times 3$ minors of the $4\times m$ measurement (backprojection) matrix

$$M(\ell) = [C_1^T \ell_1 \mid C_2^T \ell_2 \mid \dots \mid C_m^T \ell_m],$$

where $\ell_i\in \PP^2$ represent image lines as dual vectors. In symbols,

$\mathcal{L}_C = \{\ell\in (\PP^2)^m \mid \rank M(\ell) \leq 2\},$

I($\mathcal{L}_C$) is the ideal generated by the $3\times 3$ minors of $M(\ell)$ (Breiding et al., 2023). Each $3\times 3$ minor asserts that three back-projected planes $H_i,H_j,H_k$ meet in a line in $\PP^3$, and the condition imposed over all cameras enforces concurrency through a common line.

For non-generic configurations (four or more collinear camera centers), one must supplement with additional higher-degree generators (such as quartics) tailored to the collinear subsets (Breiding et al., 2023, Breiding et al., 2022).

4. Invariants: Multidegree and Euclidean Distance Degree

The multidegree of the variety encodes the intersection numbers with products of hyperplanes. For $\mathcal{L}^{3,2}$, the multidegree is determined by the patterns of how hyperplanes are distributed across factors; e.g., intersecting with two hyperplanes in each of two factors gives $1$, or with one in each of four views gives $2$ (Duff et al., 2024).

The Euclidean distance degree (ED degree) quantifies the algebraic complexity of solving the nearest-point problem on $MV_1$ (i.e., minimum squared distance to a tuple of image lines). For a rational curve $Y \subset \PP^N$ of degree $e$ projected via $n$ generic cameras, the affine ED degree is

$\EDdeg(X) = 3en - 2.$

Specializing to the one-dimensional line multiview variety in $\PP^3$ ($e=2$), the formula is $\EDdeg = 6n-2$ (Finkel et al., 20 Dec 2025), resolving prior conjectures; this is uniform for all $n$ and any $h\geq 2$. For the full non-curve line multiview variety $\mathcal{L}^{3,2}$, the ED degree has been conjectured (for $n\ge4$) to follow the quartic in $n$: $\EDdeg(\mathcal{L}_n^{3,2}) = \frac{27}{4}n^4 - 27 n^3 + \frac{121}{4} n^2 - 13 n + 6$ (Duff et al., 2024).

5. Specializations and Low-Dimensional Examples

For pencils of lines (one-parameter families through a point), the image curve in $(\PP^2)^n$ is a smooth rational curve, cut out by $2\times 2$ minors of the $4\times n$ back-projection matrix; its multidegree is $1$ in each factor and its ED degree is $2$, illustrating exceptional geometric and numerical simplicity (Breiding et al., 2022).

In the $n=2$ case with generic cameras in $\PP^3$, the line multiview map is dominant and the variety fills $(\PP^2)^2$. For $n=3$, $\mathcal{L}_3^{3,2}$ is a cubic hypersurface in $(\PP^2)^3$ of dimension $4$, cut out by a single $3\times3$ minor—the so-called trifocal line-line-line determinant (Duff et al., 2024, Breiding et al., 2023).

6. Computation, Gröbner Bases, and Non-Generic Cases

While the $3\times3$ minors generate the ideal for the generic case, they do not form a Gröbner basis for $m\geq3$. For $m\geq5$, the reduced Gröbner basis consists of the union of those for all $5$-camera subproblems. In collinear cases (four or more collinear camera centers), the ideal must be enlarged by determinants of specific quartic matrices, and this extended ideal is saturated and radical (Breiding et al., 2023).

These findings enable practical algorithms for symbolic and numerical line triangulation and calibration in multi-camera computer vision systems. The line Grassmann tensor (e.g., trifocal or quadrifocal tensors) can be constructed directly from the multilinear constraints defining the variety, facilitating recovery of camera parameters from line correspondences (Rydell, 2023).

7. Classification and Equivalence

Recent work classifies line multiview varieties under ED-equivalence, i.e., up to isomorphisms preserving the number and location of ED-critical points. Among fourteen minimal ED-classes in low dimensions, $\mathcal{L}^{3,2}$ forms one such class, distinct from the trivial Plücker-to-Plücker image and various anchored classes (lines through a point, lines meeting one to three disjoint world lines, etc.) (Duff et al., 2024).

The concurrency variety (tuples of lines in $\PP^3$ concurrent at a point), more generally, and the line multiview varieties, are related via elimination and intersection with congruence subvarieties. This framework leads to explicit generators for the multi-image and multiview constraints underlying classical and non-classical projective multi-camera models (Ponce et al., 2016).

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References:

• (Rydell, 2023) Projections of Higher Dimensional Subspaces and Generalized Multiview Varieties
• (Finkel et al., 20 Dec 2025) The Euclidean distance degree of curves: from rational to line multiview varieties
• (Duff et al., 2024) Metric Multiview Geometry -- a Catalogue in Low Dimensions
• (Breiding et al., 2023) Line Multiview Ideals
• (Breiding et al., 2022) Line Multiview Varieties
• (Ponce et al., 2016) Congruences and Concurrent Lines in Multi-View Geometry