Manifold k-NN: Accelerated k-NN Queries for Manifold Point Clouds

2026-05-04

AI Summary

This work addresses the limitations of traditional k-nearest neighbor (k-NN) methods, such as kd-trees, which neglect the intrinsic low-dimensional structure of manifold point clouds, and existing dynamic programming nearest neighbor search (DP-NNS) approaches that support only single-neighbor queries, thus failing to meet local neighborhood statistics requirements. The paper generalizes DP-NNS to arbitrary k by leveraging a key geometric insight: if a point is the nearest neighbor of a query, its second-nearest neighbor must reside either in its prefix set or successor list. Based on this observation, the authors devise a recursive k-NN algorithm that, for the first time, enables manifold-aware k-NN queries with arbitrary k, dynamic prefix subset querying, and point deletion—all without additional overhead. Integrated with incremental Voronoi diagrams, successor lists, recursive search, and local Delaunay updates, the method achieves 1–10× speedups over kd-trees in volume-to-surface queries, demonstrating its efficiency and versatility in dynamic geometry processing and modern graphics pipelines.

Abstract

k-nearest neighbor (k-NN) search is a fundamental primitive in geometry processing and computer graphics. While spatial partitioning structures such as kd-trees are standard, they are often manifold-blind, failing to exploit the intrinsic low-dimensional structure of points sampled from 2-manifolds. Recent advances in dynamic programming-based nearest neighbor search (DP-NNS) leverage incrementally constructed Voronoi diagrams to accelerate queries, where each site p maintains a list of successors that progressively refine its Voronoi cell. However, DP-NNS is restricted to single nearest neighbor (k=1) searches, precluding their adoption in applications that require local neighborhood statistics. In this paper, we generalize the DP-NNS framework to support arbitrary k-NN queries for manifold-aligned data. Our approach is founded on the geometric observation that if p_i is the nearest neighbor of a query q in P, then the second nearest neighbor of q must reside either within the prefix set P_{1:i-1} = {p_1, \dots, p_{i-1}} or within p_i's successor list. By recursively extending this principle, we introduce Manifold k-NN, a recursive algorithmic scheme that significantly outperforms conventional kd-trees for manifold-aligned data. Our method achieves a 1\times--10\times speedup in volume-to-surface query scenarios and inherently supports dynamic prefix queries -- enabling k-NN searches within any subset P_{1:m} (m \leq n) with zero overhead. Furthermore, we extend the framework to support point deletion via local Delaunay updates, providing a complete suite of dynamic operations for point set modification. Comprehensive experiments on diverse geometric datasets demonstrate the efficiency and broad applicability of our approach for modern graphics pipelines. Source code is available at https://github.com/sssomeone/manifold-knn.

Problem

• k-nearest neighbor
• manifold point clouds
• dynamic nearest neighbor search
• local neighborhood statistics
• Voronoi diagrams

Innovation

• Manifold k-NN
• dynamic nearest neighbor search
• Voronoi diagram
• k-nearest neighbors
• Delaunay update