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Rotation Averaging: A Primal-Dual Method and Closed-Forms in Cycle Graphs

Gabriel Moreira, Manuel Marques, João Paulo Costeira

TL;DR

This work sets forth a novel primal-dual method for synchronizing rotations, motivated by the widely accepted spectral initialization, and characterize stationary points of rotation averaging in cycle graphs topologies and contextualize this result within spectral graph theory.

Abstract

A cornerstone of geometric reconstruction, rotation averaging seeks the set of absolute rotations that optimally explains a set of measured relative orientations between them. In addition to being an integral part of bundle adjustment and structure-from-motion, the problem of synchronizing rotations also finds applications in visual simultaneous localization and mapping, where it is used as an initialization for iterative solvers, and camera network calibration. Nevertheless, this optimization problem is both non-convex and high-dimensional. In this paper, we address it from a maximum likelihood estimation standpoint and make a twofold contribution. Firstly, we set forth a novel primal-dual method, motivated by the widely accepted spectral initialization. Further, we characterize stationary points of rotation averaging in cycle graphs topologies and contextualize this result within spectral graph theory. We benchmark the proposed method in multiple settings and certify our solution via duality theory, achieving a significant gain in precision and performance.

Rotation Averaging: A Primal-Dual Method and Closed-Forms in Cycle Graphs

TL;DR

This work sets forth a novel primal-dual method for synchronizing rotations, motivated by the widely accepted spectral initialization, and characterize stationary points of rotation averaging in cycle graphs topologies and contextualize this result within spectral graph theory.

Abstract

A cornerstone of geometric reconstruction, rotation averaging seeks the set of absolute rotations that optimally explains a set of measured relative orientations between them. In addition to being an integral part of bundle adjustment and structure-from-motion, the problem of synchronizing rotations also finds applications in visual simultaneous localization and mapping, where it is used as an initialization for iterative solvers, and camera network calibration. Nevertheless, this optimization problem is both non-convex and high-dimensional. In this paper, we address it from a maximum likelihood estimation standpoint and make a twofold contribution. Firstly, we set forth a novel primal-dual method, motivated by the widely accepted spectral initialization. Further, we characterize stationary points of rotation averaging in cycle graphs topologies and contextualize this result within spectral graph theory. We benchmark the proposed method in multiple settings and certify our solution via duality theory, achieving a significant gain in precision and performance.

Paper Structure

This paper contains 17 sections, 6 theorems, 86 equations, 6 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notation
  3. Related Work
  4. Problem statement
  5. MLE formulation of PGO
  6. From non-convex MLE to strong duality
  7. Primal-Dual Method
  8. Primal update
  9. Dual update
  10. Cycle graphs
  11. One-parameter subgroups of SO(3)
  12. Optimization in SO(3)
  13. Experimental results
  14. Optimality
  15. Cycle graphs
  16. ...and 2 more sections

Key Result

Lemma 1

The dual iterates $\Lambda_{(k)}$ of GPM defined by the recursion (eq:dual_gpm) verify $\mathop{\mathrm{tr}}\nolimits(\Lambda_{(k+1)}) \geq \mathop{\mathrm{tr}}\nolimits(\Lambda_{(k)})$ (proof in the appendix).

Figures (6)

  • Figure 1: Left: ground-truth graph consisting of a mesh with 441 vertices. Edges of this colored grid correspond to noiseless pairwise measurements. One diagonal measurement (black) is added, which underestimates the true diagonal by 70%. Right: optimized vertex positions.
  • Figure 2: Left: ground-truth graph, with 100 vertices represented on the complex circle group. Colored edges correspond to noiseless measurements. The horizontal black measurement is added, with a phase error of $-\pi/4$ over the ground-truth $\pi$. Right: vertex positions of the phase synchronization global optimum.
  • Figure 3: The cycle graph problem on the left can be transformed into the problem on the right via a change-of-basis.
  • Figure 4: Primal-dual convergence to the optimum in SO(3) (color represents frequency). Vertical axis: Graph's Fiedler value (same as the 4th eigenvalue of the latent connection Laplacian $\underline{L}$). Horizontal axis: operator norm of the difference between the latent and measured connection adjacency matrices, $\underline{A}$ and $\tilde{A}$, respectively. Axes in multiples of $n$.
  • Figure 5: Convergence of absolute value of the smallest eigenvalue of $\Lambda_{(k)}-\tilde{A}$ for the PGO datasets Sphere (0.36s), Torus3D (0.35s), Cubicle (0.46s) and Grid3D (1.78s), made available in Carlone2015_dualCarlone2015.
  • ...and 1 more figures

Theorems & Definitions (8)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Theorem 4
  • proof
  • Theorem 5
  • proof
  • Theorem 6