Plücker Coordinates for Ray Camera Models

• Plücker coordinates are an algebraic representation for 3D rays, defined by a pair of vectors with an inherent orthogonality constraint ensuring physical validity.
• They are used in camera calibration, pose estimation, and 3D reconstruction by enabling efficient, division-free computations and robust correction methods.
• Recent advances integrate Plücker representations with learning-based attention mechanisms, enhancing geometric consistency and real-time processing in modern vision systems.

Plücker coordinates are an algebraic representation for lines in projective 3-space, characterized by a pair of vectors whose orthogonality is central to ensuring they correspond to valid physical rays. Their adoption in ray-based camera representations underlies many advances in computer vision, geometric computation, camera calibration, and feed-forward 3D reconstruction. This article details the theory, computation, and practical use of Plücker coordinates in ray-based camera models, focusing on underlying principles, algorithmic advances, projective formulations, efficiency, and recent learning-based applications.

1. Mathematical Foundations of Plücker Coordinates in Ray-Based Camera Models

A line (or ray) in 3D can be represented via Plücker coordinates as a pair $(\mathbf{d}, \mathbf{m})$, where $\mathbf{d} \in \mathbb{R}^3$ is a direction vector and $\mathbf{m} = \mathbf{o} \times \mathbf{d}$ encodes the moment, with $\mathbf{o}$ any point on the line. The representation is homogeneous and invariant to translation along the direction. For a line between two points $\mathbf{p}_1, \mathbf{p}_2$, one canonical choice is:

$$\mathbf{d} = \mathbf{p}_2 - \mathbf{p}_1, \quad \mathbf{m} = \mathbf{p}_1 \times (\mathbf{p}_2 - \mathbf{p}_1)$$

To be a valid line, the 6D vector must satisfy the Klein quadric constraint:

$\mathbf{d}^\top \mathbf{m} = 0$

This orthogonality is necessary and sufficient for correspondence to a physical line in projective space (Cardoso et al., 2016).

In projective geometry, lines may also be encoded as elements of the Grassmannian $G(2, 4)$ or, for n-flats, as isotropic subspaces of a symplectic space, with explicit Plücker coordinates providing both Euclidean and projective formulations (Carrillo-Pacheco et al., 2016, Skala, 2022).

2. Correction and Validation of Plücker Representations

In practical vision problems, noise leads to 6D vectors deviating from the Klein quadric. The Plücker correction problem seeks, for arbitrary vectors $\mathbf{a}, \mathbf{b}$, the closest $(\mathbf{x}_\ast, \mathbf{y}_\ast)$ with $\mathbf{x}_\ast^\top \mathbf{y}_\ast = 0$ that minimize:

$$\min_{\mathbf{x}^\top \mathbf{y} = 0} \|\mathbf{a} - \mathbf{x}\|^2 + \|\mathbf{b} - \mathbf{y}\|^2$$

A closed-form, global solution is given by (Cardoso et al., 2016):

$$\begin{aligned} p &= \mathbf{a}^\top \mathbf{b} \ q &= \|\mathbf{a}\|^2 + \|\mathbf{b}\|^2 \ \alpha &= \frac{2p}{q + \sqrt{q^2 - 4p^2}} \ \mathbf{x}_\ast &= \frac{1}{1-\alpha^2} (\mathbf{a} - \alpha \mathbf{b}) \ \mathbf{y}_\ast &= \frac{1}{1-\alpha^2} (\mathbf{b} - \alpha \mathbf{a}) \ \end{aligned}$$

This solution is both optimal and computationally efficient, with speed-ups exceeding 6x over SVD-based approaches, enabling real-time application in high-resolution camera models and large-scale geometric processing.

3. Projective Geometry, Duality, and Computational Methods

Plücker coordinates naturally arise in projective geometry, particularly when representing joins and meets via determinants and extended cross products (Skala, 2022). Homogeneous coordinates facilitate unified representation of finite and infinite points, lines, and planes, supporting robust symbolic and numerical computation.

Key formulations include:

• Line through two points: $p = x_1 \times x_2$
• Intersection of two lines: $x = p_1 \times p_2$
• Plane through three points: $p = x_1 \times x_2 \times x_3$
• Intersection of three planes: $x = p_1 \times p_2 \times p_3$

These computations are division-free, stable under floating-point arithmetic, and well-suited for parallel vectorized evaluation, particularly on GPUs.

Such projective approaches underpin ray-based intersection, barycentric coordinate determination, fitting, and calibration algorithms. Division-free line intersection tests based on Plücker coordinates are established in the geometric and graphics literature as robust, branchless, and SIMD-amenable (Skala, 2022).

4. Ray-Based Camera Representations: Calibration, Pose Estimation, and Reconstruction

Calibration and Pose from Lines

Plücker coordinates underpin several ray-based camera calibration and pose algorithms (Zhang et al., 11 Mar 2025, Přibyl et al., 2016, Zhang et al., 22 Feb 2024). Lines from LiDAR and image data are represented in Plücker form, permitting constraint-based registration:

• Line transformation: For rotation $R$ and translation $P$:

$$\begin{aligned} {}^{C}\mathbf{n}_{l} &= R {}^{L}\mathbf{n}_{l} + \lfloor P \times \rfloor R {}^{L}\mathbf{v}_{l} \ {}^{C}\mathbf{v}_{l} &= R {}^{L}\mathbf{v}_{l} \end{aligned}$$

• Co-perpendicular constraint: ${}^{C}\mathbf{n}_l' \cdot {}^{C}\mathbf{v}_l = 0$
• Co-parallel constraint: ${}^{C}\mathbf{n}_l' \times {}^{C}\mathbf{n}_l = 0$

Calibration decouples rotation (nonlinear) and translation (linear least squares) for greater robustness and interpretability (Zhang et al., 11 Mar 2025).

Pose estimation (DLT method):

Given image/3D line correspondences, Plücker parameterization allows formulation of the line projection matrix via linear least squares. With at least 9 lines, the method is exact in noise-free data; greater redundancy yields higher noise robustness. Algebraic outlier rejection (AOR) replaces RANSAC for efficiency in large correspondence sets (Přibyl et al., 2016).

Few-view 3D Reconstruction and Ray Diffusion

Recent models employ Plücker coordinates as the backbone of ray-based distributed representations for camera pose (ray bundles) and structure. Cameras are modeled as collections of Plücker rays per patch/pixel, facilitating tight coupling with spatial features and efficient transformer-based inference; both regression and denoising diffusion approaches can be formulated on ray bundles (Zhang et al., 22 Feb 2024). This framework supports richer uncertainty modeling, generalizes beyond perspective cameras, and outperforms pose regression via global parameter vectors.

Table: Ray-centric camera representations

Approach	Camera Representation	Core Operation
PLK-Calib (Zhang et al., 11 Mar 2025)	Matched Plücker line pairs	Decoupled rot/translation
DLT from lines (Přibyl et al., 2016)	Plücker-encoded line sets	Linear matrix estimation
Ray diffusion (Zhang et al., 22 Feb 2024)	Bundle of Plücker rays per patch	Regression & diffusion
PlückeRF (Bahrami et al., 4 Jun 2025)	Feature-augmented Plücker rays (grid/image)	Attention via line geometry

5. Attention Mechanisms and Geometric Data Association in Learned Models

Learned ray-based representations exploit line-to-line distances in Plücker space to bias cross-attention and self-attention (Bahrami et al., 4 Jun 2025). The analytic line-to-line distance is:

$$d(\mathbf{l}_1, \mathbf{l}_2) = \begin{cases} \frac{|\mathbf{d}_1^T \mathbf{m}_2 + \mathbf{d}_2^T \mathbf{m}_1|}{\|\mathbf{d}_1 \times \mathbf{d}_2\|_2} & \text{if } \mathbf{d}_1 \times \mathbf{d}_2 \neq 0 \ \|\mathbf{d}_1 \times (\mathbf{m}_1 - (\mathbf{d}_1^T \mathbf{d}_2) \mathbf{m}_2)\|_2 & \text{if } \mathbf{d}_1 \times \mathbf{d}_2 = 0 \end{cases}$$

In transformer attention:

$$\text{softmax}\left(\frac{\mathbf{K}\mathbf{Q}^T}{\sqrt{d_I} - \gamma d(\mathbf{l}_q, \mathbf{l}_k)}\right)$$

This enables preferential information sharing between rays (or pixels/features) that are geometrically proximate or candidate-intersecting, substantially improving geometric consistency, data fusion from multiple views, and reconstruction fidelity, particularly in the few-view regime.

Empirical results indicate systematic improvements in PSNR, SSIM, and perceptual metrics over triplane and point-based alternatives (Bahrami et al., 4 Jun 2025). Ablation studies show the necessity of explicit Plücker-based attention bias for optimal performance.

6. Algebraic Geometry, Grassmannians, and Physical Validity

General ray-based camera geometry is encoded by subspaces of a symplectic vector space, corresponding to the Lagrangian-Grassmannian $L(n,2n)$ (Carrillo-Pacheco et al., 2016). The set of physically admissible rays obeys both quadratic Plücker relations (for the Grassmannian $G(n,2n)$) and explicit additional linear constraints—Plücker linear relations—enforcing maximal isotropicity:

$$H_{\mathbf{a}} := \sum_{i=1}^n X_{i,\mathbf{a},2n-i+1} = 0,\quad \forall \mathbf{a} \in I(n-2, 2n)$$

This algebraic characterization enables rigorous testing and enforcement of physicality in ray or n-flat parameterizations. These constraints generalize to arbitrary field characteristics and support applications in coding theory, projective camera models, and structured error correction in computational imaging.

7. Computational Impact, Parallelism, and Efficiency

The closed-form Plücker correction and projective formulations support large-scale, real-time geometric processing. Algorithms rely chiefly on vector-vector or matrix operations (extended cross products, determinant forms) with minimal branching or division, making them naturally efficient and robust on modern hardware (GPUs, SIMD CPUs) (Skala, 2022).

Division-free computation and symbolic manipulation enhance numerical stability, prevent overflow/underflow, and scale effectively to millions of rays or pixels per frame in camera-based applications. This is particularly impactful for ray-based calibration, reconstruction, SLAM, and graphics where efficiency at scale is mandatory.

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Plücker coordinates provide a foundational, algebraically rigorous, and computationally efficient method for representing and manipulating rays and lines in camera models. Their use as the primitive in ray-based camera representations supports advanced geometric algorithms, efficient parallel computation, robust calibration, and state-of-the-art learning-based 3D reconstruction, with direct implications for the speed, fidelity, and scalability of modern computer vision systems.