Building Rome with Convex Optimization (2502.04640v3)

Abstract: Global bundle adjustment is made easy by depth prediction and convex optimization. We (i) propose a scaled bundle adjustment (SBA) formulation that lifts 2D keypoint measurements to 3D with learned depth, (ii) design an empirically tight convex semidfinite program (SDP) relaxation that solves SBA to certfiable global optimality, (iii) solve the SDP relaxations at extreme scale with Burer-Monteiro factorization and a CUDA-based trust-region Riemannian optimizer (dubbed XM), (iv) build a structure from motion (SfM) pipeline with XM as the optimization engine and show that XM-SfM compares favorably with existing pipelines in terms of reconstruction quality while being significantly faster, more scalable, and initialization-free.

• The paper proposes using convex semidefinite program relaxation with Burer-Monteiro factorization and a GPU optimizer (XM) to solve large-scale, nonconvex bundle adjustment problems efficiently.
• Empirical validation shows the method scales to thousands of camera frames and matches or surpasses existing techniques in accuracy and speed across various datasets.
• The research shows large-scale nonconvex bundle adjustment can be solved via convex relaxation, significantly impacting optimization theory and practical 3D reconstruction applications.

Building Rome with Convex Optimization: An Essay on Efficient Global Bundle Adjustment

The paper "Building Rome with Convex Optimization" presents an innovative approach to global bundle adjustment in structure from motion (SfM) applications. This paper explores a novel technique that capitalizes on convex optimization, yielding significant advancements in scalability, speed, and reliability, crucial for computer vision tasks where precise 3D reconstruction is necessary. Below is a detailed discussion of the methodologies, results, and implications of the paper for the field.

Methodological Advances

In the domain of SfM and simultaneous localization and mapping (SLAM), bundle adjustment (BA) traditionally poses significant challenges due to its nonconvex nature and potential large-scale requirements. This paper addresses these challenges through several pivotal contributions:

1. Scaled Bundle Adjustment (SBA) Formulation: The authors propose a technique that leverages depth prediction, transforming the 2D keypoint measurements into approximate 3D representations to simplify the bundle adjustment problem. By integrating learned depth into the process, SBA alleviates some of the constraints and complexities involved in traditional BA methods.
2. Convex Semidefinite Program (SDP) Relaxation: A key innovation is the introduction of an SDP relaxation technique. This approach converts the inherently nonconvex SBA problem into a convex one, facilitating a solution that is certifiable to global optimality. The authors achieve this by utilizing the Burer-Monteiro factorization for solving the SDP at scale through a GPU-based trust-region Riemannian optimizer named XM.
3. Implementation in SfM Pipeline: XM is incorporated into a complete SfM pipeline, demonstrating how convex optimization can uplift the conventional SfM frameworks to deliver higher quality reconstructions efficiently. This system competes favorably with existing pipelines by eliminating the need for initialization and accelerating the processing times significantly.

Empirical Validation

The researchers validate their approach using several datasets from different domains, demonstrating exceptional performance. XM's capacity to solve bundle adjustments involving up to tens of thousands of camera frames underscores its scalability. The results consistently show that the proposed method either matches or surpasses existing techniques concerning accuracy and speed.

In controlled settings like the BAL dataset, XM showcases its potential to outperform traditional solvers like Ceres when initialization is poor. For practically oriented datasets like Replica and IMC, XM provides competitive accuracy and demonstrates substantial reductions in processing time.

Theoretical and Practical Implications

The implications of this research are far-reaching and significant:

• Optimization Theory: This paper enriches the field of optimization by proving that large-scale nonconvex bundle adjustment problems can be addressed through convex relaxation techniques. The demonstrated empirical tightness of SDP relaxation—where feasible solutions to the relaxed problem can adequately solve the original nonconvex problem—sets a precedent for future optimization research.
• Practical Applications: The real-world applicability of this research is tangible. Efficient and scalable global BA allows for robust 3D reconstructions in diverse environments, catering to applications in robotics, autonomous navigation, virtual reality, and large-scale mapping.
• Future Directions in AI: The advances inspire further exploration into AI-driven applications where real-time and large-scale optimizations are crucial. The integration of learned depth into geometric optimization processes is a promising area that merits further investigation.

Conclusion

In summary, the paper presents a compelling case for using convex optimization in addressing long-standing challenges in SfM. By bringing computational tractability to large-scale nonconvex problems, the authors set a new benchmark in the field, delivering an approach that promises to enhance both theoretical research and practical implementations in computer vision and robotics. Future work could extend this foundation to incorporate dynamic scenes and further robustify against noisy input data, thereby expanding the versatility and application scope of the method.