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Accelerating Globally Optimal Consensus Maximization in Geometric Vision

Xinyue Zhang, Liangzu Peng, Wanting Xu, Laurent Kneip

TL;DR

This work tackles the computational intractability of globally optimal consensus maximization in geometric vision by introducing Accelerated Consensus Maximization (ACM), which reduces the branching dimension and solves the remaining degree of freedom exactly via interval stabbing. ACM yields significantly tighter bounds and drastically fewer iterations, translating into substantial speedups (from twofold to over three orders of magnitude) across diverse problems including 3D-2D, 2D-2D, 3D-3D registrations, and rotation plus focal length estimation. The approach demonstrates robust performance under extreme outlier ratios and achieves near real-time, deterministic global optimization in several settings, with strong results on synthetic and real data such as KITTI and Stanford Bunny. Overall, ACM broadens the practical viability of deterministic CM for online applications by delivering general, problem-agnostic acceleration without relying on strong priors or problem-specific invariants.

Abstract

Branch-and-bound-based consensus maximization stands out due to its important ability of retrieving the globally optimal solution to outlier-affected geometric problems. However, while the discovery of such solutions caries high scientific value, its application in practical scenarios is often prohibited by its computational complexity growing exponentially as a function of the dimensionality of the problem at hand. In this work, we convey a novel, general technique that allows us to branch over an n-1 dimensional space for an n-dimensional problem. The remaining degree of freedom can be solved globally optimally within each bound calculation by applying the efficient interval stabbing technique. While each individual bound derivation is harder to compute owing to the additional need for solving a sorting problem, the reduced number of intervals and tighter bounds in practice lead to a significant reduction in the overall number of required iterations. Besides an abstract introduction of the approach, we present applications to four fundamental geometric computer vision problems: camera resectioning, relative camera pose estimation, point set registration, and rotation and focal length estimation. Through our exhaustive tests, we demonstrate significant speed-up factors at times exceeding two orders of magnitude, thereby increasing the viability of globally optimal consensus maximizers in online application scenarios.

Accelerating Globally Optimal Consensus Maximization in Geometric Vision

TL;DR

This work tackles the computational intractability of globally optimal consensus maximization in geometric vision by introducing Accelerated Consensus Maximization (ACM), which reduces the branching dimension and solves the remaining degree of freedom exactly via interval stabbing. ACM yields significantly tighter bounds and drastically fewer iterations, translating into substantial speedups (from twofold to over three orders of magnitude) across diverse problems including 3D-2D, 2D-2D, 3D-3D registrations, and rotation plus focal length estimation. The approach demonstrates robust performance under extreme outlier ratios and achieves near real-time, deterministic global optimization in several settings, with strong results on synthetic and real data such as KITTI and Stanford Bunny. Overall, ACM broadens the practical viability of deterministic CM for online applications by delivering general, problem-agnostic acceleration without relying on strong priors or problem-specific invariants.

Abstract

Branch-and-bound-based consensus maximization stands out due to its important ability of retrieving the globally optimal solution to outlier-affected geometric problems. However, while the discovery of such solutions caries high scientific value, its application in practical scenarios is often prohibited by its computational complexity growing exponentially as a function of the dimensionality of the problem at hand. In this work, we convey a novel, general technique that allows us to branch over an n-1 dimensional space for an n-dimensional problem. The remaining degree of freedom can be solved globally optimally within each bound calculation by applying the efficient interval stabbing technique. While each individual bound derivation is harder to compute owing to the additional need for solving a sorting problem, the reduced number of intervals and tighter bounds in practice lead to a significant reduction in the overall number of required iterations. Besides an abstract introduction of the approach, we present applications to four fundamental geometric computer vision problems: camera resectioning, relative camera pose estimation, point set registration, and rotation and focal length estimation. Through our exhaustive tests, we demonstrate significant speed-up factors at times exceeding two orders of magnitude, thereby increasing the viability of globally optimal consensus maximizers in online application scenarios.
Paper Structure (23 sections, 1 theorem, 69 equations, 13 figures, 2 tables, 2 algorithms)

This paper contains 23 sections, 1 theorem, 69 equations, 13 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Contributions
  3. Paper Structure
  4. Related Work
  5. Preliminaries
  6. Branch and Bound
  7. Interval Stabbing
  8. Accelerated Consensus Maximization
  9. Core of The Method
  10. Time Complexity Analysis
  11. ACM-0: 3D-2D Registration
  12. Synthetic Experiments
  13. ACM-1: 2D-2D Registration
  14. Synthetic Experiments
  15. Real Experiments
  16. ...and 8 more sections

Key Result

Proposition 1

Run the plain BnB (resp. ACM) algorithm with the splitting rule that divides cubes into $2^n$ (resp. $2^{n-1}$) congruent sub-cubes and with the stopping criterion that terminates at the maximal splitting depth $d$. Then the following holds:

Figures (13)

  • Figure 1: A visual example of (\ref{['eq:inverse']}): Given a threshold $\epsilon$ and sample $\mathbf{s}_i$, in general one can find a disjoint union of intervals (red, $x$-axis) for which every point $b$ satisfies $f(b,\mathbf{s}_i) \leq \epsilon$.
  • Figure 2: Search spaces of plain BnB (blue, top row) and ACM (green, bottom row). The left column shows example volumes and the right column the corresponding subcubes after splitting. Plain BnB splits an $n$-dimensional cube into $2^n$ sub-cubes. ACM splits an $(n-1)$-dimensional cube into $2^{n-1}$ sub-cubes.
  • Figure 3: Visualization of the lower bounds of plain BnB and ACM on a $2$-dimensional example. Plain BnB branches to search two variables $b_1$ and $b_2$. ACM searches over the space of $b_1$, and uses interval stabbing to determine $b_2$. The red curves are higher, i.e., the lower bounds of ACM are tighter.
  • Figure 4: Visual comparative analysis of ACM: The largest upper and lower bounds of ACM are tighter (\ref{['fig:bounds11']}); the upper bounds of ACM and plain BnB are comparable (\ref{['fig:bounds12']}); ACM prunes a higher percentage of cubes in earlier stages (\ref{['fig:bounds21']}); ACM maintains much fewer cubes during execution (\ref{['fig:bounds22']}).
  • Figure 5: 3D-2D registration experiments (Section \ref{['Sec:1DProblem']}) where we compare ACM-0 and the plain BnB method of jiao2020globally. ACM-0 is more than $200$ times faster in the high-outlier regime ( \ref{['fig:1Dtime']}) with comparable angular errors ( \ref{['fig:1Derror']}). ($200$ randomly generated noisy correspondences, $100$ trials)
  • ...and 8 more figures

Theorems & Definitions (9)

  • Definition 1: Interval Mappingscholz2011deterministic
  • Example 1
  • Example 2
  • Proposition 1
  • Example 3: Invalid Bounds
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4