GlobalPointer: Large-Scale Plane Adjustment with Bi-Convex Relaxation (2407.13537v2)

Abstract: Plane adjustment (PA) is crucial for many 3D applications, involving simultaneous pose estimation and plane recovery. Despite recent advancements, it remains a challenging problem in the realm of multi-view point cloud registration. Current state-of-the-art methods can achieve globally optimal convergence only with good initialization. Furthermore, their high time complexity renders them impractical for large-scale problems. To address these challenges, we first exploit a novel optimization strategy termed \textit{Bi-Convex Relaxation}, which decouples the original problem into two simpler sub-problems, reformulates each sub-problem using a convex relaxation technique, and alternately solves each one until the original problem converges. Building on this strategy, we propose two algorithmic variants for solving the plane adjustment problem, namely \textit{GlobalPointer} and \textit{GlobalPointer++}, based on point-to-plane and plane-to-plane errors, respectively. Extensive experiments on both synthetic and real datasets demonstrate that our method can perform large-scale plane adjustment with linear time complexity, larger convergence region, and robustness to poor initialization, while achieving similar accuracy as prior methods. The code is available at https://github.com/wu-cvgl/GlobalPointer.

• The paper proposes a Bi-Convex Relaxation method that decomposes plane adjustment into convex sub-problems, enhancing scalability and robustness.
• It introduces GlobalPointer (point-to-plane) and GlobalPointer++ (plane-to-plane) variants that improve convergence and computational efficiency.
• Extensive experiments on synthetic and real datasets demonstrate linear time complexity and high robustness to poor initialization in 3D computer vision tasks.

GlobalPointer: Large-Scale Plane Adjustment with Bi-Convex Relaxation

The paper "GlobalPointer: Large-Scale Plane Adjustment with Bi-Convex Relaxation," authored by Bangyan Liao, Zhenjun Zhao, Lu Chen, Haoang Li, Daniel Cremers, and Peidong Liu, addresses the complex problem of plane adjustment (PA) in multi-view point cloud registration. This task is integral to numerous 3D computer vision applications, such as LiDAR-based simultaneous localization and mapping (SLAM) and 3D reconstruction. The authors introduce an innovative optimization strategy known as Bi-Convex Relaxation, which significantly enhances the practical scalability and robustness of PA.

Contributions and Methodology

The core contribution of the paper lies in the proposed Bi-Convex Relaxation technique, which decomposes the non-convex PA problem into two simpler sub-problems. Each sub-problem is reformulated using convex relaxation techniques and solved alternately. This approach effectively circumvents the computational intractability of large-scale PA problems and mitigates the strong dependence on initialization observed in current state-of-the-art methods.

Building on Bi-Convex Relaxation, the authors introduce two variants of PA algorithms:

1. GlobalPointer: This variant utilizes point-to-plane errors and demonstrates a larger convergence region and higher robustness to poor initialization.
2. GlobalPointer++: This variant leverages plane-to-plane errors, offering better computational efficiency.

Numerical Results and Experimental Evaluation

The paper details extensive experiments conducted on both synthetic and real datasets to validate the performance of GlobalPointer and GlobalPointer++. Crucially, these experiments reveal that the proposed methods achieve linear time complexity in large-scale scenarios, facilitating practical deployment in real-world applications.

Key Performance Metrics:

• Accuracy: Across various synthetic scenarios with different noise levels, GlobalPointer consistently reached global convergence, often outperforming contemporary methods like BALM2, EF, ESO, PA-Full, and PA-Decoupled in terms of achieving lower total point-to-plane error.
• Robustness: The approach's robustness to initialization ensures that even under high noise levels, the proposed methods maintain empirical global optimality.
• Time Complexity: The linear time complexity with respect to the number of planes and poses underscores the method's scalability. In larger settings, GlobalPointer++ specifically achieves significant computational speedups.

Theoretical Implications

The theoretical underpinnings of the proposed method involve decoupling the PA problem into SDP sub-problems. Each of these sub-problems is guaranteed to converge to the global minimum in noise-free scenarios. Despite the overall non-convexity of the original PA problem, the Bi-Convex Relaxation method substantially enlarges the convergence basin, enhancing the likelihood of attaining empirical global optimality.

Practical Implications and Future Directions

Practically, the robustness of the GlobalPointer and GlobalPointer++ methods to poor initialization simplifies the adoption of PA in real-time applications, where initial guesses are often unreliable. The method's ability to handle large-scale problems efficiently addresses a significant bottleneck in deploying multi-frame point cloud registration in dynamic, large environments.

Looking forward, future work could explore:

• Hybrid Approaches: Combining both point-to-plane and plane-to-plane error formulations within a single framework to leverage the strengths of both methods.
• Real-time Implementation: Optimizing the algorithms further for real-time embedded systems.
• Extended Validation: Broadening the experimental validation across diverse environments and sensor configurations to generalize the practical applicability of the algorithms.

In conclusion, the work presents substantial advancements in the field of plane adjustment for point cloud registration, marked by improved scalability, accuracy, and robustness. The GlobalPointer and GlobalPointer++ frameworks hold significant promise for enhancing performance in existing 3D computer vision systems and fostering innovation in areas relying on detailed 3D environmental understanding.