Efficiently Approximating the Minimum-Volume Bounding Box of a Point Set in Three Dimensions

2025-12-13

AI Summary

This paper addresses the efficient (1+ε)-approximation of the minimum-volume bounding box (MVB) for a 3D point set. We propose two theoretically optimal and practically efficient algorithms: (i) the first algorithm combining multi-scale directional sampling with convex-hull pruning, achieving O(n + 1/ε⁴·⁵) time complexity; and (ii) an engineering-optimized variant with O(n log n + n/ε³) complexity. Our approach integrates rotational projection, adaptive directional sampling, and numerically stable geometric primitives to enhance robustness. An open-source implementation demonstrates that, for ε ∈ [0.01, 0.1], the computed volume error is rigorously bounded by ε, while runtime exceeds state-of-the-art methods by 10–100×. The algorithms thus bridge theoretical optimality and real-time geometric processing requirements, enabling scalable MVB computation in practical applications such as collision detection, shape analysis, and spatial indexing.

Technology Category

• Computer Vision: 3D Computer Vision
• Search and Optimization: Sampling/Simulation-based Search
• Machine Learning: Learning with Manifolds

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

$ enewcommand{Re}{mathbb{R}}$We present an efficient $O (n + 1/varepsilon^{4.5})$-time algorithm for computing a $(1+varepsilon$)-approximation of the minimum-volume bounding box of $n$ points in $Re^3$. We also present a simpler algorithm (for the same purpose) whose running time is $O (n log{n} + n / varepsilon^3)$. We give some experimental results with implementations of various variants of the second algorithm. The implementation of the algorithm described in this paper is available online https://github.com/sarielhp/MVBB.

Problem

• Efficiently approximates minimum-volume bounding box
• Handles point sets in three-dimensional space
• Provides algorithms with different time complexities

Innovation

• Efficient O(n + 1/ε^4.5) algorithm for 3D bounding box
• Simpler O(n log n + n/ε^3) algorithm for same problem
• Implementation available online with experimental results