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Triangulation of Points Constrained to a Plane

This paper derives a formula for the number of complex critical points in planar triangulation to quantify its algebraic complexity and demonstrates through experiments that incorporating planar constraints improves reconstruction speed and accuracy compared to unconstrained methods.

Original authors: Petr Hrubý, Elima Shehu

Published 2026-05-01

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Petr Hrubý, Elima Shehu

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to figure out where a specific object is in a room, but you can only see it through a series of windows (cameras) from different angles. This is the basic job of triangulation: using multiple 2D pictures to reconstruct a 3D location.

Usually, computers assume objects can be floating anywhere in the 3D space, like a balloon drifting in a void. This makes the math very messy and difficult because there are too many possibilities.

However, this paper points out a simple truth about our world: most things we see are flat. Walls, floors, ceilings, and tables are all planes. When a point (like a corner on a wall) is stuck to a flat surface, the math changes completely.

Here is what the authors did, explained simply:

1. The "Flat World" Shortcut

The authors realized that if you know a point is on a flat plane (like a wall), you don't need to guess its height, depth, and width independently. You only need to figure out its position on that 2D surface.

Think of it like this:

The Old Way (Unconstrained): Trying to find a lost cat in a giant, empty 3D warehouse. You have to search up, down, left, right, forward, and backward. It takes a long time and you might get lost.
The New Way (Constrained): Knowing the cat is hiding on a specific flat roof. Now you only need to search left and right on that roof. The search space is much smaller and easier to solve.

2. Counting the "Possible Answers"

In math, when you try to solve a problem with noisy data (blurry photos), there are often many "critical points"—mathematical solutions that look like the answer, but only one is the true answer.

The authors did some heavy algebra (using a branch of math called algebraic geometry) to count exactly how many of these "possible answers" exist for a flat surface.

They found a specific formula: $4.5m^2 - 6.5m + 3$ (where $m$ is the number of cameras).
Why this matters: This number tells them how "hard" the problem is. Because the number is lower for flat surfaces than for floating points, it proves mathematically that solving for flat surfaces is inherently less complex and faster.

3. Building Better Tools (Solvers)

Knowing the problem is easier, they built new "solvers" (algorithms) specifically designed to exploit this flatness.

They created tools for 2 cameras and 3 cameras.
They tested these tools on two types of data:
1. Fake Data: Computer-generated images where they knew the exact answer.
2. Real Data: Actual photos taken of a room with a laser scanner (which gives the perfect "ground truth").

4. The Results: Faster and Sharper

The experiments showed that their new "flat-aware" tools were superior to the standard tools that ignore the flatness:

Accuracy: They found the 3D location of points much more precisely. The standard tools often made mistakes because they tried to force flat points into a 3D "cloud" that didn't fit.
Speed: Because the math was simplified, the calculations finished faster.
Stability: The new tools were less likely to crash or give nonsense results when the photos were a bit blurry or noisy.

The Bottom Line

The paper argues that by acknowledging that "walls are flat," we can stop doing unnecessary math. By treating points as "anchored" to a plane, we get a clearer, faster, and more accurate picture of the 3D world, especially in indoor environments like offices or homes.

They have made their code public so others can use these "flat-surface" tools to improve their own 3D reconstruction projects.

1. Problem Statement

The paper addresses the triangulation problem in Structure-from-Motion (SfM) pipelines, specifically focusing on 3D points constrained to a common plane (coplanar points).

Context: In many real-world scenarios (indoor environments, urban facades), visual features lie on planar surfaces. Standard triangulation methods treat points as generic elements in $\mathbb{P}^3$ , ignoring this geometric constraint.
Issue: Treating coplanar points as generic 3D points introduces unnecessary degrees of freedom, leading to higher algebraic complexity, slower solvers, and reduced accuracy when reconstructing from noisy image data.
Goal: To derive the intrinsic algebraic complexity of planar triangulation, develop efficient solvers, and demonstrate that explicitly enforcing planar constraints improves reconstruction accuracy and stability compared to unconstrained methods.

2. Methodology

The authors employ applied algebraic geometry to model the problem, specifically utilizing multiview varieties and Euclidean Distance (ED) degrees.

A. Mathematical Formulation

Planar-Anchored Multiview Variety ( $M^\Pi_C$ ): The authors define a specific algebraic variety representing all possible image correspondences of points lying on a fixed plane $\Pi$ $Π$ observed by $m$ $m$ cameras.
- Unlike the standard multiview variety (which has dimension 3), the planar-anchored variety is isomorphic to $\mathbb{P}^2$ (dimension 2).
- The image correspondences are characterized by planar homographies ( $H_{ji}$ ) between views, derived from the camera matrices and the plane equation.
Euclidean Distance Degree (EDdeg): This invariant measures the number of complex critical points of the optimization problem: finding the point on the variety closest to noisy image data. It serves as a proxy for the algebraic complexity of the solver.

B. Theoretical Derivation

Main Theorem: The authors prove that for $m \geq 2$ generic calibrated cameras, the ED degree of the affine planar-anchored point multiview variety is:
$\text{EDdeg}(M^\Pi_C) = \frac{9}{2}m^2 - \frac{13}{2}m + 3$
Proof Strategy: The proof involves calculating the topological Euler characteristic of the variety and its intersections with the "distance hypersurface" and the "hyperplane at infinity." They utilize:
- Bertini's Theorem: To establish the smoothness of generic divisors.
- Whitney Stratification: To handle singularities (nodes) arising from the intersection of divisors at infinity.
- Milnor Fibers: To correct the Euler characteristic for singular points.

C. Solver Development

Based on the theoretical ED degree, the authors construct minimal solvers for $m=2$ and $m=3$ views:

Problem Reduction: The optimization is reduced to finding roots of a polynomial system in two variables ( $a, b$ ) representing the coordinates on the plane.
Implementation:
- Homotopy Continuation (HC): Implemented via HomotopyContinuation.jl.
- Gröbner Basis (GB): Implemented using an automatic generator to create elimination templates.
Performance: The GB solver for 2 views takes ~6.9 $\mu$ s, while the HC solver takes ~172.9 $\mu$ s. For 3 views, GB takes ~82.9 $\mu$ s and HC takes ~1341.6 $\mu$ s.

3. Key Contributions

Algebraic Complexity Formula: Derivation of the closed-form formula for the ED degree of planar triangulation ( $\frac{9}{2}m^2 - \frac{13}{2}m + 3$ ), quantifying the intrinsic difficulty of the problem.
Geometric Characterization: Proof that planar-anchored multiview varieties are isomorphic to $\mathbb{P}^2$ and are smooth, simplifying the algebraic structure compared to the general 3D case.
Efficient Solvers: Development of specialized minimal solvers for 2 and 3 views that exploit the planar constraint, significantly reducing the number of solutions to find (e.g., 8 solutions for $m=2$ vs. higher counts in unconstrained cases).
Empirical Validation: Comprehensive experiments on both synthetic and real-world data demonstrating superior performance over unconstrained methods.

4. Experimental Results

A. Synthetic Data

Setup: Randomly generated cameras, planes, and points with added Gaussian noise ( $\epsilon = 10^{-12}$ ).
Comparison: Planar-constrained triangulation (P.C) vs. Unconstrained triangulation (P.UC).
Findings:
- Accuracy: The constrained method (P.C) yields significantly lower triangulation errors (logarithmic average relative error) across 2, 3, and 4 views.
- Runtime: The constrained method is faster due to the reduced dimensionality of the search space.

B. Real-World Data (Lamar Dataset)

Setup: Used the CAB sequence from the Lamar dataset (indoor scenes with NavVis M6 trolley). Ground truth was obtained via laser scanning.
Pipeline:
1. Detected planes using a monocular depth network and RANSAC (without using GT depth).
2. Triangulated points using three strategies: Unconstrained (P.UC), Constrained (P.C), and Hybrid (P.H).
Findings:
- Accuracy: For $m=3$ , the peak error for the constrained method (P.C) was ~4 cm, compared to ~10 cm for the unconstrained method (P.UC).
- Stability: The constrained methods produced fewer "failed" triangulations (outliers) compared to the unconstrained approach.
- Hybrid Approach: A hybrid strategy (using GB first, falling back to HC or unconstrained if reprojection error is high) proved robust.

5. Significance and Impact

Theoretical Advancement: The paper bridges the gap between algebraic geometry and computer vision by providing a rigorous complexity analysis for a specific, highly relevant geometric constraint (planarity).
Practical Utility: The results suggest that standard SfM pipelines (like COLMAP) could significantly improve reconstruction quality and speed by explicitly detecting and utilizing planar constraints during the triangulation stage, rather than treating all points as generic 3D entities.
Future Directions: The authors propose extending this framework to line triangulation and developing approximate solvers (similar to existing fast 3-view triangulation approximations) to further reduce runtime while maintaining stability.

In summary, this work demonstrates that leveraging the "planar prior" is not just a heuristic but an algebraically grounded necessity for high-precision, efficient 3D reconstruction in structured environments.