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Convex Relaxation for Robust Vanishing Point Estimation in Manhattan World

CVPR 2025 · Learning theory and algorithms › Combinatorial and search optimization · approach: Heuristics for integer programming · 5 citations (OpenAlex)

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Applies convex (SDP) relaxation with a truncated multi-selection error to jointly estimate Manhattan-world vanishing points and line-VP associations, solved by an iterative per-VP solver (GlobustVP).

Problem

Vanishing point estimation in a Manhattan world requires jointly inferring line-VP associations and VP locations; existing methods are either suboptimal local/RANSAC solvers or globally optimal BnB methods that are slow, and learning-based methods lack optimality and generalize poorly.

Approach

Introduces a soft truncated multi-selection error that makes the primal problem a QCQP, relaxes it to an SDP, and solves it iteratively by finding one VP and its inlier lines per SDP block, followed by Manhattan orthogonality post-refinement.

Key results

• On YUD, GlobustVP reaches AA@10° of 96.1%, the best among compared methods (PARSAC 85.6%).
• On SU3, GlobustVP attains AA@3° of 80.2%, below learning-based NeurVPS (93.2%) and PARSAC (86.2%), i.e., competitive rather than best.
• Runtime is reported as on par with prior methods (about 50 ms on YUD example in Fig. 1), while being faster than BnB at larger line counts with 6-line sampling.

Measured effects reported in the paper

Surprising findings

• On SU3 the learning-based NeurVPS outperforms GlobustVP, which the authors attribute to training on similar data.

Limitations

• Global optimality holds only under mild conditions (distinguishability assumption) and is not guaranteed in noisy/outlier-heavy cases in general.
• SDP cost grows cubically with problem size, so line sampling is used for large inputs.
• Performance on SU3 lags learning-based methods trained on that data.

Experimental setup

Datasets: YUD (York Urban Database) SU3 (SceneCity Urban 3D wireframe) synthetic line data

Benchmarks: YUD SU3 synthetic dataset

Metrics: precision recall F1-score consistency error angular accuracy AA@3/5/10 runtime

Baselines: RANSAC J-Linkage T-Linkage BnB Quasi-VP Contrario-VP Zhai et al. NeurVPS CONSAC Lin et al. PARSAC

Builds on: Branch-and-Bound (Bazin et al.) RANSAC Quasi-VP Contrario-VP

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