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FracGM: A Fast Fractional Programming Technique for Geman-McClure Robust Estimator

Bang-Shien Chen, Yu-Kai Lin, Jian-Yu Chen, Chih-Wei Huang, Jann-Long Chern, Ching-Cherng Sun

TL;DR

This work presents a fast algorithm for Geman-McClure robust estimation, FracGM, leveraging fractional programming techniques and demonstrates the proposed FracGM solver with Wahba's rotation problem and 3-D point-cloud registration along with relaxation pre-processing and projection post-processing.

Abstract

Robust estimation is essential in computer vision, robotics, and navigation, aiming to minimize the impact of outlier measurements for improved accuracy. We present a fast algorithm for Geman-McClure robust estimation, FracGM, leveraging fractional programming techniques. This solver reformulates the original non-convex fractional problem to a convex dual problem and a linear equation system, iteratively solving them in an alternating optimization pattern. Compared to graduated non-convexity approaches, this strategy exhibits a faster convergence rate and better outlier rejection capability. In addition, the global optimality of the proposed solver can be guaranteed under given conditions. We demonstrate the proposed FracGM solver with Wahba's rotation problem and 3-D point-cloud registration along with relaxation pre-processing and projection post-processing. Compared to state-of-the-art algorithms, when the outlier rates increase from 20% to 80%, FracGM shows 53% and 88% lower rotation and translation increases. In real-world scenarios, FracGM achieves better results in 13 out of 18 outcomes, while having a 19.43% improvement in the computation time.

FracGM: A Fast Fractional Programming Technique for Geman-McClure Robust Estimator

TL;DR

This work presents a fast algorithm for Geman-McClure robust estimation, FracGM, leveraging fractional programming techniques and demonstrates the proposed FracGM solver with Wahba's rotation problem and 3-D point-cloud registration along with relaxation pre-processing and projection post-processing.

Abstract

Robust estimation is essential in computer vision, robotics, and navigation, aiming to minimize the impact of outlier measurements for improved accuracy. We present a fast algorithm for Geman-McClure robust estimation, FracGM, leveraging fractional programming techniques. This solver reformulates the original non-convex fractional problem to a convex dual problem and a linear equation system, iteratively solving them in an alternating optimization pattern. Compared to graduated non-convexity approaches, this strategy exhibits a faster convergence rate and better outlier rejection capability. In addition, the global optimality of the proposed solver can be guaranteed under given conditions. We demonstrate the proposed FracGM solver with Wahba's rotation problem and 3-D point-cloud registration along with relaxation pre-processing and projection post-processing. Compared to state-of-the-art algorithms, when the outlier rates increase from 20% to 80%, FracGM shows 53% and 88% lower rotation and translation increases. In real-world scenarios, FracGM achieves better results in 13 out of 18 outcomes, while having a 19.43% improvement in the computation time.
Paper Structure (16 sections, 3 theorems, 18 equations, 5 figures, 2 tables, 2 algorithms)

This paper contains 16 sections, 3 theorems, 18 equations, 5 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Geman-McClure Robust Solvers
  4. Robust Estimation for Point Cloud Registration
  5. Fractional Programming for Geman-McClure Robust Estimator
  6. Sum-of-Ratio Programming
  7. Solving Auxiliary Variables
  8. Global Optimality of FracGM
  9. Spatial Perception with FracGM
  10. Robust Rotation Solver with FracGM
  11. Robust Registration Solver with FracGM
  12. Experiments and Applications
  13. Rotation Estimation with Synthetic Dataset
  14. Point Cloud Registration with Synthetic Dataset
  15. Point Cloud Registration with 3DMatch Dataset
  16. ...and 1 more sections

Key Result

Lemma 1

If $(\bar{\boldsymbol{x}},\bar{\boldsymbol{\beta}})$ is a solution of Problem eq_primal, then there exists $\bar{\boldsymbol{\mu}}\in\mathbb{R}^N$ such that $\bar{\boldsymbol{x}}$ is a solution of the following problem for $\boldsymbol{\mu}=\bar{\boldsymbol{\mu}}$ and $\boldsymbol{\beta}=\bar{\bolds where $\boldsymbol{\mu}_i>0$ and $c^2>\boldsymbol{\beta}_i$. Furthermore, $\bar{\boldsymbol{x}}$ al

Figures (5)

  • Figure 2: 3-D registration of the Stanford Bunny dataset, with $N=100$ points of the source set in blue and target set in red with 80% outliers, while the light-colored points are only for visualization purpose. Our proposed FracGM for rotation estimation and point cloud registration is more accurate than state-of-the-art methods.
  • Figure 3: Rotation error and Performance profile comparison with (1) SVD H87, (2) RCQP BG17, (3) GNC-TLS YAT20, (4) TEASER YSC21, (5) GNC-GM YAT20, and (6) FracGM. Note that TEASER cannot solve large size problems, thus it is not shown in the extreme outlier cases.
  • Figure 4: Convergence comparison between state-of-the-art iterative methods.
  • Figure 5: Performance comparison between state-of-the-art robust registration solvers in the synthetic dataset. FracGM is comparable to or better than other methods in terms of rotation and translation errors.
  • Figure 6: Performance comparison between TEASER++ YSC21 and FracGM registration solver with different noise bounds for the Stanford Bunny point cloud. Our solver is more insensitive to noise configuration with respect to error models, and sustains more accurate results than TLS-based approach.

Theorems & Definitions (5)

  • Lemma 1: Variant of J12
  • proof
  • Theorem 2
  • proof
  • Proposition 3