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A Subspace-Constrained Tyler's Estimator and its Applications to Structure from Motion

Feng Yu, Teng Zhang, Gilad Lerman

TL;DR

The paper tackles robust subspace recovery in the presence of heavy outliers by proposing the Subspace-Constrained Tyler's Estimator (STE), a method that regularizes Tyler's M-estimator to concentrate energy on a chosen $d$-subspace. STE blends Tyler's M-estimator with a variant of the fast median subspace, enabling reliable subspace recovery with fewer inliers and improved computational efficiency. The authors provide theoretical guarantees under a generalized haystack model, showing STE can succeed with DS-SNR below 1 given good initialization, and they validate the method on Structure from Motion tasks, including robust fundamental matrix estimation and initial camera removal, where STE demonstrates competitive or superior performance to state-of-the-art RSR methods. The results suggest STE is a practical and principled tool for robust 3D reconstruction pipelines, offering improvements in accuracy and speed and opening avenues for further integration and analysis in robust vision tasks.

Abstract

We present the subspace-constrained Tyler's estimator (STE) designed for recovering a low-dimensional subspace within a dataset that may be highly corrupted with outliers. STE is a fusion of the Tyler's M-estimator (TME) and a variant of the fast median subspace. Our theoretical analysis suggests that, under a common inlier-outlier model, STE can effectively recover the underlying subspace, even when it contains a smaller fraction of inliers relative to other methods in the field of robust subspace recovery. We apply STE in the context of Structure from Motion (SfM) in two ways: for robust estimation of the fundamental matrix and for the removal of outlying cameras, enhancing the robustness of the SfM pipeline. Numerical experiments confirm the state-of-the-art performance of our method in these applications. This research makes significant contributions to the field of robust subspace recovery, particularly in the context of computer vision and 3D reconstruction.

A Subspace-Constrained Tyler's Estimator and its Applications to Structure from Motion

TL;DR

The paper tackles robust subspace recovery in the presence of heavy outliers by proposing the Subspace-Constrained Tyler's Estimator (STE), a method that regularizes Tyler's M-estimator to concentrate energy on a chosen -subspace. STE blends Tyler's M-estimator with a variant of the fast median subspace, enabling reliable subspace recovery with fewer inliers and improved computational efficiency. The authors provide theoretical guarantees under a generalized haystack model, showing STE can succeed with DS-SNR below 1 given good initialization, and they validate the method on Structure from Motion tasks, including robust fundamental matrix estimation and initial camera removal, where STE demonstrates competitive or superior performance to state-of-the-art RSR methods. The results suggest STE is a practical and principled tool for robust 3D reconstruction pipelines, offering improvements in accuracy and speed and opening avenues for further integration and analysis in robust vision tasks.

Abstract

We present the subspace-constrained Tyler's estimator (STE) designed for recovering a low-dimensional subspace within a dataset that may be highly corrupted with outliers. STE is a fusion of the Tyler's M-estimator (TME) and a variant of the fast median subspace. Our theoretical analysis suggests that, under a common inlier-outlier model, STE can effectively recover the underlying subspace, even when it contains a smaller fraction of inliers relative to other methods in the field of robust subspace recovery. We apply STE in the context of Structure from Motion (SfM) in two ways: for robust estimation of the fundamental matrix and for the removal of outlying cameras, enhancing the robustness of the SfM pipeline. Numerical experiments confirm the state-of-the-art performance of our method in these applications. This research makes significant contributions to the field of robust subspace recovery, particularly in the context of computer vision and 3D reconstruction.
Paper Structure (21 sections, 2 theorems, 22 equations, 11 figures, 8 tables, 2 algorithms)

This paper contains 21 sections, 2 theorems, 22 equations, 11 figures, 8 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. The STE Algorithm
  3. Notation
  4. Tyler's Estimator and its Application to RSR
  5. Motivation and Formulation of STE
  6. Computational Complexity
  7. Implementation Details
  8. STE fuses TME and a Variant of FMS
  9. Theory
  10. Applications to Structure from Motion
  11. Robust Fundamental Matrix Estimation
  12. Initial Camera Removal for SfM
  13. Conclusions
  14. Supplementary Details for the Numerical Experiments
  15. Supplementary Details for Fundamental Matrix Estimation
  16. ...and 6 more sections

Key Result

Theorem 1

Under assumptions 1-3, the sequence ${\mathbf{\Sigma}}^{(k)}$ generated by STE converges to ${\mathbf{U}}_{L_*}{\mathbf{\Sigma}}_{in,*}{\mathbf{U}}_{L_*}^\top$, the TME solution for the set of inliers ${\mathcal{X}}_{in}$. In addition, let $L^{(k)}$ be the subspace spanned by the top $d$ eigenvector

Figures (11)

  • Figure 1: Median (relative) rotation errors obtained by seven algorithms for the 14 datasets of Photo Tourism.
  • Figure 2: Mean (absolute) rotation errors (in degrees, left) and mean translation errors (in degrees, right) of LUD and four RSR methods used to initially screen bad cameras within LUD applied to the 14 datasets of Photo Tourism.
  • Figure 3: Median (top) and mean (bottom) relative rotation errors (in degrees) obtained by seven algorithms for the 14 datasets of Photo Tourism. The numerical results are reported in \ref{['tab:fund_rotation']}
  • Figure 4: Median (top) and mean (bottom) errors of direction vectors (in degrees) obtained by seven algorithms for the 14 datasets of Photo Tourism. The numerical results are reported in \ref{['tab:fund_translation']}
  • Figure 5: mAA($10^\circ$) obtained by seven algorithms for the 14 datasets of Photo Tourism.
  • ...and 6 more figures

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2