Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Minimal Solver for Relative Pose Estimation with Unknown Focal Length from Two Affine Correspondences

Zhenbao Yu, Shirong Ye, Ronghe Jin, Shunkun Liang, Zibin Liu, Huiyun Zhang, Banglei Guan

TL;DR

The paper presents a minimal solver for jointly estimating the relative pose and a still-unknown focal length from two affine correspondences, leveraging a known vertical direction from an IMU to reduce rotation to $3$-DOF. The core idea is to derive four polynomial constraints in the rotation parameter $s$ and focal length $f$ from two ACs, and solve them via a polynomial eigenvalue approach that yields $s$ and then $f$, with translation recovered from the nullspace. Extensive experiments on synthetic data and real-world datasets (KITTI, smartphone, and vehicular platforms) show that the proposed OUR-2AC-3DOF-f method offers superior numerical stability and accuracy, outperforming state-of-the-art baselines by substantial margins and often with faster runtime. The approach is particularly appealing for real-time, IMU-enabled vision systems in harsh, cluttered environments where minimal sampling and robustness to noise are crucial.

Abstract

In this paper, we aim to estimate the relative pose and focal length between two views with known intrinsic parameters except for an unknown focal length from two affine correspondences (ACs). Cameras are commonly used in combination with inertial measurement units (IMUs) in applications such as self-driving cars, smartphones, and unmanned aerial vehicles. The vertical direction of camera views can be obtained by IMU measurements. The relative pose between two cameras is reduced from 5DOF to 3DOF. We propose a new solver to estimate the 3DOF relative pose and focal length. First, we establish constraint equations from two affine correspondences when the vertical direction is known. Then, based on the properties of the equation system with nontrivial solutions, four equations can be derived. These four equations only involve two parameters: the focal length and the relative rotation angle. Finally, the polynomial eigenvalue method is utilized to solve the problem of focal length and relative rotation angle. The proposed solver is evaluated using synthetic and real-world datasets. The results show that our solver performs better than the existing state-of-the-art solvers.

A Minimal Solver for Relative Pose Estimation with Unknown Focal Length from Two Affine Correspondences

TL;DR

The paper presents a minimal solver for jointly estimating the relative pose and a still-unknown focal length from two affine correspondences, leveraging a known vertical direction from an IMU to reduce rotation to -DOF. The core idea is to derive four polynomial constraints in the rotation parameter and focal length from two ACs, and solve them via a polynomial eigenvalue approach that yields and then , with translation recovered from the nullspace. Extensive experiments on synthetic data and real-world datasets (KITTI, smartphone, and vehicular platforms) show that the proposed OUR-2AC-3DOF-f method offers superior numerical stability and accuracy, outperforming state-of-the-art baselines by substantial margins and often with faster runtime. The approach is particularly appealing for real-time, IMU-enabled vision systems in harsh, cluttered environments where minimal sampling and robustness to noise are crucial.

Abstract

In this paper, we aim to estimate the relative pose and focal length between two views with known intrinsic parameters except for an unknown focal length from two affine correspondences (ACs). Cameras are commonly used in combination with inertial measurement units (IMUs) in applications such as self-driving cars, smartphones, and unmanned aerial vehicles. The vertical direction of camera views can be obtained by IMU measurements. The relative pose between two cameras is reduced from 5DOF to 3DOF. We propose a new solver to estimate the 3DOF relative pose and focal length. First, we establish constraint equations from two affine correspondences when the vertical direction is known. Then, based on the properties of the equation system with nontrivial solutions, four equations can be derived. These four equations only involve two parameters: the focal length and the relative rotation angle. Finally, the polynomial eigenvalue method is utilized to solve the problem of focal length and relative rotation angle. The proposed solver is evaluated using synthetic and real-world datasets. The results show that our solver performs better than the existing state-of-the-art solvers.
Paper Structure (16 sections, 28 equations, 6 figures, 5 tables)

This paper contains 16 sections, 28 equations, 6 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Related work
  3. Geometric Constraint
  4. Relationship of Fundamental Matrix and Local Affinities
  5. Relative Pose Estimation With Known Vertical Direction and Unknown Focal Length
  6. Problem formulation
  7. Polynomial eigenvalue solver
  8. Experiments
  9. Experiments on synthetic data
  10. Number stability
  11. Noise Resilience
  12. principal point noise
  13. Experiments on KITTI dataset
  14. Experiments on smartphone dataset
  15. Experiments on vehicular mobile platform dataset
  16. ...and 1 more sections

Figures (6)

  • Figure 1: Probability density functions over focal length, rotation error and translation error noise free for 1000 runs under random motion. (a) Focal length error, (b) Rotation error, (b) Translation error.
  • Figure 2: Focal length, Rotation and translation error for four methods under random motion. First column: Focal length error (%); Second column: Rotation error (degree); Third column: translation error (degree): (a)(b)(c) adding image noise with perfect IMU data. (d)(e)(f) adding pitch angle noise and fix the image noise as 1.0 pixel. (g)(h)(i) adding roll angle noise and fix the image noise as 1.0 pixel.
  • Figure 3: Focal length, Rotation and translation error for four methods under planar motion. First column: Focal length error (%); Second column: Rotation error (degree); Third column: translation error (degree): (a)(b)(c) adding image noise with perfect IMU data. (d)(e)(f) adding pitch angle noise and fix the image noise as 1.0 pixel. (g)(h)(i) adding roll angle noise and fix the image noise as 1.0 pixel
  • Figure 4: Focal length, Rotation and translation error for four methods under sideways motion. First column: Focal length error (%); Second column: Rotation error (degree); Third column: translation error (degree): (a)(b)(c) adding image noise with perfect IMU data. (d)(e)(f) adding pitch angle noise and fix the image noise as 1.0 pixel. (g)(h)(i) adding roll angle noise and fix the image noise as 1.0 pixel
  • Figure 5: Focal length, rotation and translation error for four methods under forward motion. First column: Focal length error (%); Second column: Rotation error (degree); Third column: translation error (degree): (a)(b)(c) adding image noise with perfect IMU data. (d)(e)(f) adding pitch angle noise and fix the image noise as 1.0 pixel. (g)(h)(i) adding roll angle noise and fix the image noise as 1.0 pixel
  • ...and 1 more figures