Approximate Algorithms for Chamfer Distance Under Translation

2026-05-24

AI Summary

This work presents the first systematic study of minimizing the Chamfer distance between point sets under translation. The authors propose an exact quadratic-time algorithm for the one-dimensional case and efficient approximation algorithms for higher dimensions, achieving approximation ratios of (2+ε) and (1+ε) with nearly quadratic time complexity in certain regimes. Their approach integrates geometric approximation, grid discretization, and refined complexity analysis, and includes a fast decision procedure under a separation assumption. This study not only fills a theoretical gap in understanding Chamfer distance optimization under translation but also provides a practical algorithmic framework for high-dimensional point set registration.

Abstract

Given two sets of points A and B, $|A| = m$, $|B| = n$, the Chamfer distance from $A$ to $B$ is defined as $\operatorname{CD}(A,B) = \sum_{a\in A} \min_{b\in B} d(a,b)$, where $d$ is a distance metric. Chamfer distance is a popular measure of dissimilarity between two sets of points that has seen increasing usage in computer vision and information retrieval as a substitute for the more computationally demanding Earth Mover's distance. We propose a new problem, Chamfer distance under translation, defined as $\operatorname{CDuT}(A,B) :=\min_{t\in \mathbb{R}^d} \operatorname{CD}(A+t,B)$, where $A+t$ denotes the translation of every point in $A$ by $t$. Chamfer distance under translation is valuable in cases where translations capture aspects of the data unlikely to be relevant for dissimilarity, such as temporal, spatial, or other semantic information. For Chamfer distance under translation, we provide four algorithms: (1) an exact quadratic time algorithm in one dimension, (2) a near quadratic time ($2+\varepsilon$)-approximation algorithm in higher dimensions, (3) a $(1+\varepsilon)$-approximation algorithm with running time $\mathcal{O}(mn^2\varepsilon^{-(d+1)})$, and (4) a near-quadratic time $(1+\varepsilon)$-approximation algorithm for answering the decision version of $\operatorname{CDuT}$ given a separation assumption on $B$. We additionally explore the fine-grained complexity of $\operatorname{CDuT}$.

Problem

• Chamfer distance
• translation
• point sets
• approximation algorithms
• computational geometry

Innovation

• Chamfer distance under translation
• approximation algorithms
• fine-grained complexity
• point set registration
• computational geometry