Power Variable Projection for Initialization-Free Large-Scale Bundle Adjustment (2405.05079v5)

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.

• The paper introduces Power Variable Projection (PoVar), a novel algorithm extending inverse expansion and Riemannian manifold optimization for initialization-free large-scale bundle adjustment.
• Evaluation on BAL datasets shows PoVar achieves state-of-the-art speed and accuracy, effectively handling problems with thousands of camera views.
• The authors provide an open-source implementation, facilitating advancements in fields like autonomous systems and large-scale photogrammetry by offering a robust, scalable solution.

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

The paper presents a novel methodological advancement in the field of bundle adjustment (BA), specifically addressing the problem of initialization-free large-scale bundle adjustment. The authors introduce the Power Variable Projection (PoVar) algorithm, which is inspired by recent developments in inverse expansion methods based on power series. This research builds on the foundation of the Variable Projection (VarPro) algorithm, recognized for its capability to circumvent the necessity for initial estimates, which traditional Levenberg-Marquardt-based solvers require.

Contributions and Methodology

• Power Variable Projection (PoVar): The authors develop PoVar by extending the inverse expansion method through a power series. This extension includes theoretical integration with Riemannian manifold optimization principles. The core innovation allows PoVar to effectively address the scalability challenges of initialization-free BA problems.
• Theoretical and Practical Framework: The research introduces a projective framework that translates the power series expansion into the Riemannian manifold setting, allowing for efficient handling of large-scale BA without initialization. This is a non-trivial extension given the inherent differences in convergence behaviors and application scopes between traditional and initialization-free BA.
• Evaluation and Results: The authors validate their approach using real-world datasets from the BAL project. The results demonstrate that PoVar achieves state-of-the-art performance both in terms of computational speed and accuracy. The scalability is a significant highlight, with effective handling demonstrated even in scenarios involving thousands of camera views.

Numerical Results and Implications

The paper provides detailed numerical evaluations, emphasizing the efficiency of PoVar over existing methods. For varied problem tolerances, PoVar consistently outperforms traditional methods like iterative preconditioned conjugate gradient variants and direct factorization-based algorithms in scenarios demanding both low and high accuracy. The scalable nature of PoVar is highlighted through its smooth convergence properties and robustness across numerous large-scale problems.

Theoretical and Practical Implications

1. Theoretical Insight: The work contributes theoretical insights into the applicability and limitations of power series expansions within the VarPro framework. It addresses longstanding inefficiencies in managing large-scale BA, proposing a structured approach that is of particular interest to researchers exploring computational optimization in vision tasks.
2. Practical Impact: Practically, the introduction of PoVar has significant implications for fields reliant on structure-from-motion and 3D reconstruction techniques, which are increasingly demanding in terms of volume and complexity. Applications in autonomous systems, mixed reality, and large-scale photogrammetry stand to benefit from these advancements.
3. Open Source Contribution: The authors contribute an open-source implementation of their solver, available at the indicated GitHub repository. This availability facilitates further research and potential enhancements by other researchers and practitioners, providing a foundation for broader adoption and continued evolution in the community.

Speculations for Future Developments

Looking ahead, the integration of PoVar and its Riemannian extensions could inspire further investigations into initialization-free methods and stratified formulations, potentially expanding into other applications beyond BA. The fusion of expansion methods with manifold optimization strategies could also spur new hybrid approaches in related fields, such as nonlinear optimization and machine learning models that leverage manifold structures.

In conclusion, the introduction of Power Variable Projection marks a substantive step forward in the field of large-scale bundle adjustment. By addressing the critical challenge of initialization-free optimization, this research enhances both theoretical understanding and practical capability, setting the stage for extensive future exploration.