Linear Fundamental Matrix Estimation from 7 or 5 Points

Repository for our CVPR 2026 paper: Linear Fundamental Matrix Estimation from 7 or 5 Points

This repository contains a C++ implementation of the V-Umlaut fundamental-matrix solver, without a RANSAC pipeline, using synthetic experiments to understand how different noise models affect the solver's results.

The solver's linear formula is derived with Macaulay2 in derivation/v_umlaut.m2, and implemented in Solver.cpp.

To use the V-Umlaut solver as a drop-in minimal fundamental matrix solver inside PoseLib's RANSAC pipeline, see the v-umlaut branch in our PoseLib fork.

Our real-data experiments in the paper follow the evaluation pipeline of this repository for the paper Practical Solutions to the Relative Pose of Three Calibrated Cameras by Tzamos et al., CVPR 2025. We will share our fork of this pipeline that replicates the real-data experiment figures in our paper soon.

Synthetic Experiments

These experiments reproduce Section 4 / Figure 4 of our paper: the relative-pose accuracy of 5pF-V-Umlaut against the 5pE (5-point essential) and 7pF (7-point fundamental) solvers, under two pixel-noise models, on two camera/scene configurations, without RANSAC. The synthetic data generation and the two noise models are described in detail in Section 4 of our paper.

The plotting code that replicates Figure 4 will be shared soon. For now, the binary reproduces individual data points (below).

Dependencies

• A C++17 compiler and CMake ≥ 3.10
• Eigen 3

The two baseline solvers used for comparison, Nister's 5-point relative-pose solver (5pE) and the 7-point fundamental-matrix solver (7pF), are minimally extracted from their PoseLib implementations and included in this repository under external/poselib/. PoseLib itself is therefore not a dependency.

Build

# Linux / macOS
cmake -S . -B build -DCMAKE_BUILD_TYPE=Release && cmake --build build -j
#   -> build/v_umlaut

# Windows
cmake -S . -B build -G "Visual Studio 17 2022" -A x64 -DEigen3_DIR=<prefix>/share/eigen3/cmake
cmake --build build --config Release
#   -> build/Release/v_umlaut.exe

Reproducing a Figure 4 data point

Each run performs many trials and prints the median rotation/translation error for all three solvers. One data point in Figure 4 corresponds to a single (scene, noise-model, σ) setting. For the Standard scene at σ = 1 px over 10,000 trials:

# Figure 4a — independent-point noise
./v_umlaut general 10000 --independent-point-noise 1.0

# Figure 4b — dependent-point noise
./v_umlaut general 10000 --dependent-point-noise 1.0

Swap general → wami for the WAMI configuration, and vary σ (e.g. 0.5, 1.5, 2.0) to move along the x-axis of Figure 4. The three methods are printed as PoseLib-5pt (5pE), PoseLib-7pt (7pF), and V-Umlaut (5pF-V-Umlaut).

Citation

@InProceedings{V-Umlaut_2026_CVPR,
    author    = {Kucukpinar, Taci Ata and Mogollon, Juan and Fraser, Joshua and Duff, Timothy and Palaniappan, Kannappan},
    title     = {Linear Fundamental Matrix Estimation from 7 or 5 Points},
    booktitle = {Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR)},
    month     = {June},
    year      = {2026},
    pages     = {21464-21473}
}