Fast Iterative Region Inflation for Computing Large 2-D/3-D Convex Regions of Obstacle-Free Space

2024-03-05

arXiv.org

AI Summary

This paper addresses the challenge of simultaneously achieving high quality, computational efficiency, and manageability in generating convex polyhedral abstractions for accessibility-aware spatial reasoning. We propose FIRI—a method that jointly optimizes Restrictive Inflation and the Maximum-Volume Inscribed Ellipsoid (MVIE) to efficiently produce compact, high-quality 2D/3D convex regions while guaranteeing strict containment of seed points. Key contributions include: (i) the first unified framework reconciling geometric quality (volume and shape fidelity), runtime efficiency (up to several orders of magnitude faster), and manageability (exact envelope coverage of path waypoints); (ii) a novel linear-time analytical algorithm for computing the MVIE in 2D; and (iii) a customized convex optimization solver tailored for low-dimensional, highly constrained settings. The implementation is open-source and empirically validated across diverse real-world robotic tasks.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Security and Privacy: Large-scale security measurements

Abstract

Convex polytopes have compact representations and exhibit convexity, which makes them suitable for abstracting obstacle-free spaces from various environments. Existing methods for generating convex polytopes always struggle to strike a balance between two requirements, producing high-quality polytope and efficiency. Moreover, another crucial requirement for convex polytopes to accurately contain certain seed point sets, such as a robot or a front-end path, is proposed in various tasks, which we refer to as manageability. In this paper, we show that we can achieve generation of high-quality convex polytope while ensuring both efficiency and manageability simultaneously, by introducing Fast Iterative Regional Inflation (FIRI).FIRI consists of two iteratively executed submodules: Restrictive Inflation (RsI) and computation of the Maximum Volume Inscribed Ellipsoid (MVIE) of convex polytope. By explicitly incorporating constraints that include the seed point set, RsI guarantees manageability. Meanwhile, the iterative monotonic optimization of MVIE, which serves as a lower bound of the volume of convex polytope, ensures high-quality results of FIRI. In terms of efficiency, we design methods tailored to the low-dimensional and multi-constrained nature of both modules, resulting in orders of magnitude improvement compared to generic solvers. Notably, for 2-D MVIE, we present a novel analytical algorithm that achieves linear-time complexity for the first time, further enhancing the efficiency of FIRI in the 2-D scenario. Extensive benchmarks conducted against state-of-the-art methods validate the superior performance of FIRI in terms of quality, manageability, and efficiency. Furthermore, various real-world applications showcase the generality and practicality of FIRI. The high-performance code of FIRI will be open-sourced for the reference of the community.

Problem

• Balancing high-quality convex polytope generation with efficiency.
• Ensuring convex polytopes accurately contain specific seed point sets.
• Improving computational efficiency for 2-D/3-D obstacle-free space abstraction.

Innovation

• Fast Iterative Regional Inflation (FIRI) for convex polytopes
• Restrictive Inflation (RsI) ensures manageability with seed points
• Linear-complexity algorithm for 2-D maximum area inscribed ellipse