Convex Relaxation for Robust Vanishing Point Estimation in Manhattan World

Abstract: Determining the vanishing points (VPs) in a Manhattan world, as a fundamental task in many 3D vision applications, consists of jointly inferring the line-VP association and locating each VP. Existing methods are, however, either sub-optimal solvers or pursuing global optimality at a significant cost of computing time. In contrast to prior works, we introduce convex relaxation techniques to solve this task for the first time. Specifically, we employ a "soft" association scheme, realized via a truncated multi-selection error, that allows for joint estimation of VPs' locations and line-VP associations. This approach leads to a primal problem that can be reformulated into a quadratically constrained quadratic programming (QCQP) problem, which is then relaxed into a convex semidefinite programming (SDP) problem. To solve this SDP problem efficiently, we present a globally optimal outlier-robust iterative solver (called GlobustVP), which independently searches for one VP and its associated lines in each iteration, treating other lines as outliers. After each independent update of all VPs, the mutual orthogonality between the three VPs in a Manhattan world is reinforced via local refinement. Extensive experiments on both synthetic and real-world data demonstrate that GlobustVP achieves a favorable balance between efficiency, robustness, and global optimality compared to previous works. The code is publicly available at https://github.com/WU-CVGL/GlobustVP.

Convex Relaxation for Robust Vanishing Point Estimation in Manhattan World

Vanishing point estimation is an essential component of many 3D vision applications, underpinning tasks like SLAM, camera calibration, and structural understanding. In the context of the Manhattan world assumption—where only three mutually orthogonal line directions exist—accurate vanishing point estimation becomes crucial for ensuring precision in computational tasks. The paper "Convex Relaxation for Robust Vanishing Point Estimation in Manhattan World" introduces a novel method leveraging convex relaxation techniques to optimize vanishing point estimation, avoiding the drawbacks of traditional methods that are either computationally expensive or sub-optimal.

Key Contributions

1. Convex Relaxation Approach: This paper uniquely integrates convex relaxation methods into vanishing point estimation, offering a first-time exploration of this strategy for solving such tasks. The authors reformulate the optimization problem into a quadratically constrained quadratic programming (QCQP) problem, which is subsequently relaxed into a convex semidefinite programming (SDP) problem.

2. Novel Error Formulation: The introduction of the truncated multi-selection error enables the joint estimation of vanishing points and their associated line directions. This formulation allows the system to identify both inliers and outliers, minimizing bias and enhancing robustness against noise.

3. GlobustVP Solver: An iterative solver named GlobustVP is proposed, addressing the computational efficiency and robustness issues inherent in previous methods. This solver independently searches for one vanishing point and its associated lines at each iteration, treating other lines as outliers, thus ensuring global optimality under mild conditions.

4. Manhattan Post-Refinement: After identifying vanishing points, a post-refinement step reinforces the mutual orthogonality of discovered vanishing points, ensuring consistency with the Manhattan world assumption.

Findings and Implications

The rigorous evaluation conducted on both synthetic and real-world data highlights that GlobustVP achieves a commendable balance between efficiency, robustness, and accuracy. The results demonstrate superior performance over traditional methods, particularly under conditions involving high noise levels and outliers. With the SDP providing a globally optimal solution in polynomial time, the efficiency and reliability of vanishing point estimation are substantially improved, offering practical advantages in real-world applications such as augmented reality, autonomous navigation, and architectural modeling.

Theoretical Advancements and Future Directions

This research opens promising pathways for future developments in AI and computational geometry. By overcoming the limitations of existing methods through a novel convex optimization technique, it sets a precedent for further exploration of convex relaxation in 3D vision tasks. Future research could investigate broader applications of this relaxation method, explore enhanced solvers for large-scale scenes, and integrate learning-based approaches to complement the traditional optimization methods, further increasing robustness and generalization.

The convergence of theoretical precision with practical efficiency marks a significant step forward in vanishing point estimation, underlining the potential for extended applications in robust 3D vision frameworks.