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 $\operatorname{CDuT}$.

翻译：给定两个点集$A$和$B$，$|A| = m$，$|B| = n$，从$A$到$B$的Chamfer距离定义为$\operatorname{CD}(A,B) = \sum_{a\in A} \min_{b\in B} d(a,b)$，其中$d$为距离度量。Chamfer距离是衡量两个点集差异性的常用指标，近年来在计算机视觉和信息检索领域作为计算复杂度更高的推土机距离的替代方案得到广泛应用。我们提出一个新问题——平移下Chamfer距离，定义为$\operatorname{CDuT}(A,B) :=\min_{t\in \mathbb{R}^d} \operatorname{CD}(A+t,B)$，其中$A+t$表示将$A$中每个点平移$t$。当平移仅反映数据中与差异性无关的方面（如时间、空间或其他语义信息）时，平移下Chamfer距离具有重要价值。针对该问题，我们提出四种算法：(1) 一维空间中的精确二次时间算法；(2) 高维空间中的近二次时间$(2+\varepsilon)$-近似算法；(3) 运行时间为$\mathcal{O}(mn^2\varepsilon^{-(d+1)})$的$(1+\varepsilon)$-近似算法；(4) 在$B$满足分离假设条件下，用于回答$\operatorname{CDuT}$判定问题的近二次时间$(1+\varepsilon)$-近似算法。此外，我们探讨了$\operatorname{CDuT}$的细粒度计算复杂度。