Approximate Algorithms for Chamfer Distance Under Translation
This work addresses the need for efficient computation of Chamfer distance under translation, which is valuable for computer vision and information retrieval tasks where translation invariance is desired.
The paper introduces the Chamfer distance under translation (CDuT) problem and provides four algorithms, including an exact quadratic-time algorithm in 1D and a near-quadratic time (2+ε)-approximation in higher dimensions, as well as a (1+ε)-approximation with running time O(m n^2 ε^{-(d+1)}).
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 $\operatorname{CDuT}$.