Convex Relaxation for Robust Vanishing Point Estimation in Manhattan World

2025-05-07

AI Summary

This paper addresses the joint optimization problem of vanishing point (VP) estimation in Manhattan-world scenes—simultaneously localizing VPs and associating scene lines with them. We formulate the problem as a quadratically constrained quadratic program (QCQP) amenable to convex relaxation and propose GlobustVP, a novel framework featuring: (i) soft line-to-VP association to model matching uncertainty; (ii) truncated multi-hypothesis error minimization for outlier suppression; (iii) semidefinite programming (SDP)-based global initialization guaranteeing optimality; and (iv) iterative refinement under strict orthogonality constraints to ensure geometric consistency. Our method achieves both computational efficiency and robustness while rigorously enforcing orthogonal VP constraints. Extensive experiments on synthetic and real-world datasets demonstrate superior accuracy, robustness, and runtime performance over state-of-the-art methods. The source code is publicly available.

Technology Category

• Intelligent Robots: State Estimation
• Computer Vision: Learning & Optimization for CV
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

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 extbf{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 extbf{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.

Problem

• Robust vanishing point estimation in Manhattan world
• Jointly infer line-VP association and VP locations
• Balance efficiency, robustness, and global optimality

Innovation

• Convex relaxation for robust VP estimation
• Soft association via truncated multi-selection error
• Globally optimal outlier-robust iterative solver