# Beyond Gr"obner Bases: Basis Selection for Minimal Solvers

arXiv: https://arxiv.org/abs/1803.04360 · HTML: https://mlsanity.com/paper/1803.04360/

## Key facts

- Cited by 25 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Mar 2018.

## Built on by (most cited first, 25 of 25)

- [The 8-Point Algorithm as an Inductive Bias for Relative Pose Prediction by ViTs](https://mlsanity.com/paper/2208.08988/) (2022)
- [Learning to Solve Hard Minimal Problems](https://mlsanity.com/paper/2112.03424/) (2021)
- [ARCS: Accurate Rotation and Correspondence Search](https://mlsanity.com/paper/2203.14493/) (2022)
- [FAR: Flexible, Accurate and Robust 6DoF Relative Camera Pose Estimation](https://mlsanity.com/paper/2403.03221/) (2024)
- [Estimation Contracts for Outlier-Robust Geometric Perception](https://mlsanity.com/paper/2208.10521/) (2022)
- [Snapshot of Algebraic Vision](https://mlsanity.com/paper/2210.11443/) (2022)
- [GPU-Based Homotopy Continuation for Minimal Problems in Computer Vision](https://mlsanity.com/paper/2112.03444/) (2021)
- [Optimizing Elimination Templates by Greedy Parameter Search](https://mlsanity.com/paper/2203.14901/) (2022)
- [Order-One Rolling Shutter Cameras](https://mlsanity.com/paper/2403.11295/) (2024)
- [Trust Your IMU: Consequences of Ignoring the IMU Drift](https://mlsanity.com/paper/2103.08286/) (2021)
- [Partially calibrated semi-generalized pose from hybrid point correspondences](https://mlsanity.com/paper/2209.15072/) (2022)
- [Sparse resultant based minimal solvers in computer vision and their connection with the action matrix](https://mlsanity.com/paper/2301.06443/) (2023)
- [Automatic Solver Generator for Systems of Laurent Polynomial Equations](https://mlsanity.com/paper/2307.00320/) (2023)
- [Algebra and Geometry of Camera Resectioning](https://mlsanity.com/paper/2309.04028/) (2023)
- [Trifocal Tensor and Relative Pose Estimation with Known Vertical Direction](https://mlsanity.com/paper/2512.19110/) (2025)
- [On Relative Pose Recovery for Multi-Camera Systems](https://mlsanity.com/paper/2102.11996/) (2021)
- [MinBackProp -- Backpropagating through Minimal Solvers](https://mlsanity.com/paper/2404.17993/) (2024)
- [Simulator HC: Regression-based Online Simulation of Starting Problem-Solution Pairs for Homotopy Continuation in Geometric Vision](https://mlsanity.com/paper/2411.03745/) (2024)
- [A Minimal Solver for Relative Pose Estimation with Unknown Focal Length from Two Affine Correspondences](https://mlsanity.com/paper/2512.22833/) (2025)
- [Online Stability Improvement of Groebner Basis Solvers using Deep Learning](https://mlsanity.com/paper/2401.09328/) (2024)
- [Three-view Focal Length Recovery From Homographies](https://mlsanity.com/paper/2501.07499/) (2025)
- [PLMP -- Point-Line Minimal Problems for Projective SfM](https://mlsanity.com/paper/2503.04351/) (2025)
- [Radially Distorted Homographies, Revisited](https://mlsanity.com/paper/2508.21190/) (2025)
- [Learning Fast Monomial Orders for Gr"obner Basis Computations](https://mlsanity.com/paper/2602.02972/) (2026)
- [Solving Minimal Problems Without Matrix Inversion Using FFT-Based Interpolation](https://mlsanity.com/paper/2605.06572/) (2026)

## Abstract (arXiv)

Many computer vision applications require robust estimation of the underlying geometry, in terms of camera motion and 3D structure of the scene. These robust methods often rely on running minimal solvers in a RANSAC framework. In this paper we show how we can make polynomial solvers based on the action matrix method faster, by careful selection of the monomial bases. These monomial bases have traditionally been based on a Gr"obner basis for the polynomial ideal. Here we describe how we can enumerate all such bases in an efficient way. We also show that going beyond Gr"obner bases leads to more efficient solvers in many cases. We present a novel basis sampling scheme that we evaluate on a number of problems.