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Power Variable Projection for Initialization-Free Large-Scale Bundle Adjustment

Simon Weber, Je Hyeong Hong, Daniel Cremers

TL;DR

This work tackles large-scale bundle adjustment without good initialization, a setting where traditional LM-based BA struggles. It introduces Power Variable Projection (PoVar), which merges a power-series-based inverse expansion with the VarPro framework, and extends it to a Riemannian optimization context (RiPoBA) for the second-stage refinement. The authors prove theoretical guarantees for the PoVar expansion and its Riemannian counterpart, and demonstrate superior speed and accuracy on real BAL data compared to state-of-the-art initialization-free methods and VarPro baselines. The approach achieves scalable initialization-free structure-from-motion by leveraging a two-stage stratified BA and memory-efficient storage, and they release an open-source implementation. Overall, PoVar and RiPoBA establish a new scalable paradigm for initialization-free BA with potential impact on large-scale 3D reconstruction tasks.

Abstract

Most Bundle Adjustment (BA) solvers like the Levenberg-Marquardt algorithm require a good initialization. Instead, initialization-free BA remains a largely uncharted territory. The under-explored Variable Projection algorithm (VarPro) exhibits a wide convergence basin even without initialization. Coupled with object space error formulation, recent works have shown its ability to solve small-scale initialization-free bundle adjustment problem. To make such initialization-free BA approaches scalable, we introduce Power Variable Projection (PoVar), extending a recent inverse expansion method based on power series. Importantly, we link the power series expansion to Riemannian manifold optimization. This projective framework is crucial to solve large-scale bundle adjustment problems without initialization. Using the real-world BAL dataset, we experimentally demonstrate that our solver achieves state-of-the-art results in terms of speed and accuracy. To our knowledge, this work is the first to address the scalability of BA without initialization opening new venues for initialization-free structure-from-motion.

Power Variable Projection for Initialization-Free Large-Scale Bundle Adjustment

TL;DR

This work tackles large-scale bundle adjustment without good initialization, a setting where traditional LM-based BA struggles. It introduces Power Variable Projection (PoVar), which merges a power-series-based inverse expansion with the VarPro framework, and extends it to a Riemannian optimization context (RiPoBA) for the second-stage refinement. The authors prove theoretical guarantees for the PoVar expansion and its Riemannian counterpart, and demonstrate superior speed and accuracy on real BAL data compared to state-of-the-art initialization-free methods and VarPro baselines. The approach achieves scalable initialization-free structure-from-motion by leveraging a two-stage stratified BA and memory-efficient storage, and they release an open-source implementation. Overall, PoVar and RiPoBA establish a new scalable paradigm for initialization-free BA with potential impact on large-scale 3D reconstruction tasks.

Abstract

Most Bundle Adjustment (BA) solvers like the Levenberg-Marquardt algorithm require a good initialization. Instead, initialization-free BA remains a largely uncharted territory. The under-explored Variable Projection algorithm (VarPro) exhibits a wide convergence basin even without initialization. Coupled with object space error formulation, recent works have shown its ability to solve small-scale initialization-free bundle adjustment problem. To make such initialization-free BA approaches scalable, we introduce Power Variable Projection (PoVar), extending a recent inverse expansion method based on power series. Importantly, we link the power series expansion to Riemannian manifold optimization. This projective framework is crucial to solve large-scale bundle adjustment problems without initialization. Using the real-world BAL dataset, we experimentally demonstrate that our solver achieves state-of-the-art results in terms of speed and accuracy. To our knowledge, this work is the first to address the scalability of BA without initialization opening new venues for initialization-free structure-from-motion.
Paper Structure (29 sections, 4 theorems, 35 equations, 11 figures)

This paper contains 29 sections, 4 theorems, 35 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Related Work
  3. Problem Statement and Motivation
  4. Inverse expansion method
  5. Initialization-free bundle adjustment
  6. First stage: separable nonlinear optimization with projective camera.
  7. Limitations and proposed method
  8. Power Variable Projection
  9. Revisited variable projection
  10. Power series for VarPro
  11. Power Riemannian manifold optimization
  12. Experiments
  13. Implementation
  14. Experimental settings
  15. Setup.
  16. ...and 14 more sections

Key Result

proposition thmcounterproposition

Let $M$ be an $n \times n$ matrix. If the spectral radius of $M$ satisfies $\lVert M \rVert <1$, then where the error matrix satisfies

Figures (11)

  • Figure 1: In contrast to traditional bundle adjustment problem, initialization-free BA is a largely under-explored problem. It does not assume any approximation of pose and landmark parameters, making the problem much harder to solve. From a random initialization (left figure), and given only image measurements, we aim to recover pose and landmark parameters. Our approach, that extends inverse expansion method, is motivated by the lack of scalability of existing solvers. On the real-world BAL problems (e.g. Venice-89, right figure), we demonstrate the efficiency of the proposed combination of our novel solver Power Variable Projection (PoVar) and Riemannian manifold optimization framework for expansion method to solve the stratified BA problem.
  • Figure 1: With $\eta = 0.2$, performance profiles for all real-world BAL problems for solving the first stage (6). Given a tolerance $\tau \in \{0.01, 0.003, 0.001 \}$, it represents the percentage of solved problems ($y$-axis) with relative runtime $\alpha$ ($x$-axis). Our solver PoVar is very competitive, and most notably for the highest accuracy $\tau = 0.001$ and $\tau=0.003$.
  • Figure 2: Average performance profiles across all BAL problems for solving the first stage Eq. \ref{['eq:ba_first_stage']}. Given a tolerance $\tau \in \{0.01, 0.003, 0.001 \}$, it represents the percentage of solved problems ($y$-axis) with relative runtime $\alpha$ ($x$-axis). Expansion methods PoVar and PoBA show outstanding speed-accuracy results. Our solver PoVar is competitive, and most notably for the highest accuracy $\tau = 0.001$.
  • Figure 2: With $\eta = 0.3$, performance profiles for all real-world BAL problems for solving the first stage (6).
  • Figure 3: Convergence plots of Dubrovnik-253 (left) from BAL datasets with 253 poses and Ladybug-412 with 412 poses, for solving the first stage (Eq. \ref{['eq:ba_first_stage']}). The dotted lines correspond to cost thresholds for tolerance $\tau \in \{0.01, 0.003, 0.001 \}$.
  • ...and 6 more figures

Theorems & Definitions (6)

  • proposition thmcounterproposition
  • lemma thmcounterlemma
  • proof
  • lemma thmcounterlemma
  • proof
  • theorem thmcountertheorem