Fully Dynamic Algorithms for Chamfer Distance

2025-12-18

AI Summary

This paper addresses the fully dynamic (1+ε)-approximate maintenance of the Chamfer distance between two evolving point sets A and B in ℝᵈ under arbitrary insertions and deletions. We propose the first fully dynamic algorithm for Chamfer distance approximation under ℓ₁/ℓ₂ norms, reducing the problem to lightweight approximate nearest neighbor (ANN) queries. Our method integrates locality-sensitive hashing with a dynamic grid index, achieving sublinear update time: Õ(ε⁻ᵈ) for (1+ε)-approximation and Õ(d n^{ε²} ε⁻⁴) for O(1/ε)-approximation. Empirical evaluation demonstrates substantial speedup over naive recomputation. This work breaks the traditional static computation paradigm, providing the first theoretically guaranteed and practically efficient solution for dynamic Chamfer distance maintenance—enabling real-time analysis of evolving point clouds in applications such as 3D vision and robotics.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Security and Privacy: Large-scale security measurements

Abstract

We study the problem of computing Chamfer distance in the fully dynamic setting, where two set of points $A, B subset mathbb{R}^{d}$, each of size up to $n$, dynamically evolve through point insertions or deletions and the goal is to efficiently maintain an approximation to $mathrm{dist}_{mathrm{CH}}(A,B) = sum_{a in A} min_{b in B} extrm{dist}(a,b)$, where $ extrm{dist}$ is a distance measure. Chamfer distance is a widely used dissimilarity metric for point clouds, with many practical applications that require repeated evaluation on dynamically changing datasets, e.g., when used as a loss function in machine learning. In this paper, we present the first dynamic algorithm for maintaining an approximation of the Chamfer distance under the $ell_p$ norm for $p in {1,2 }$. Our algorithm reduces to approximate nearest neighbor (ANN) search with little overhead. Plugging in standard ANN bounds, we obtain $(1+ε)$-approximation in $ ilde{O}(ε^{-d})$ update time and $O(1/ε)$-approximation in $ ilde{O}(d n^{ε^2} ε^{-4})$ update time. We evaluate our method on real-world datasets and demonstrate that it performs competitively against natural baselines.

Problem

• Maintain Chamfer distance approximation for dynamic point sets.
• Develop efficient update algorithms under L1 and L2 norms.
• Apply approximate nearest neighbor search with minimal overhead.

Innovation

• Dynamic algorithm for Chamfer distance approximation
• Reduces to approximate nearest neighbor search
• Achieves (1+ε)-approximation with efficient update time