Fully Dynamic Algorithms for Chamfer Distance

TL;DR

The paper addresses maintaining a (1+ε)-accurate estimate of the Chamfer distance dist_CH(A,B) for fully dynamic point sets in ℝ^d under insertions and deletions for p ∈ {1,2}. It reduces the problem to approximate nearest neighbor queries via a dynamic NN oracle and builds a dynamic quad-tree-based importance-sampling framework that implicitly tracks a sampling distribution without maintaining a full assignment A→B. The main contributions include a provably correct, near-update-time dynamic algorithm with (1+α+ε) approximation guarantees and high-probability boosting, plus empirical validation across multiple real-world datasets showing robustness to outliers and favorable performance relative to baselines. The work advances dynamic proximity problems in point clouds and provides a practical primitive for machine learning and vision tasks where Chamfer distance is used as a loss or similarity measure.

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 $\mathrm{dist}_{\mathrm{CH}}(A,B) = \sum_{a \in A} \min_{b \in B} \textrm{dist}(a,b)$, where $\textrm{dist}$ 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 $\tilde{O}(ε^{-d})$ update time and $O(1/ε)$-approximation in $\tilde{O}(d n^{ε^2} ε^{-4})$ update time. We evaluate our method on real-world datasets and demonstrate that it performs competitively against natural baselines.

Figures (7)

• Figure 1: Relative error curves for datasets without outliers. These experiments are independently run for $5$ times, and we report the average (the dot), max and min value (the shaded area) after every $\frac{w}{5}$ to $\frac{w}{3}$ updates (depending on the dataset) where $w$ is the window size.
• Figure 2: Relative error curves for datasets with outliers, with the same setup as in \ref{['fig:exp:err']}.
• Figure 3: Average running time per window update for all algorithms on datasets without outliers.
• Figure 4: The value $\mathop{\mathrm{dist}}\nolimits(a, B) / \sum_{a' \in A} \mathop{\mathrm{dist}}\nolimits(a', B)$ over all points $a \in A$ for SIFT dataset, which are the "ideal" probabilities for importance sampling.
• Figure 5: Average running time per window update for all algorithms on datasets with outliers.

Theorems & Definitions (21)

• Theorem 1
• Definition 1: Chamfer distance
• Definition 2: Dynamic nearest-neighbor data structure
• Lemma 1: Dynamic Weighted Sampler
• proof
• Theorem 2
• Lemma 2
• Claim 1
• proof
• Claim 2