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ikd-Tree: An Incremental K-D Tree for Robotic Applications

Yixi Cai, Wei Xu, Fu Zhang

TL;DR

The paper tackles the inefficiency of rebuilding static kd-trees for sequential robotic data by introducing ikd-Tree, an incremental kd-tree that supports point- and box-wise updates, downsampling, and lazy deletion with parallel partial re-building to preserve real-time performance. It provides a complete data-structure design, a two-thread re-balancing mechanism, and rigorous time/space complexity analyses. Experimental validation on randomized datasets and real-world LiDAR-inertial mapping demonstrates substantial speedups (order-of-magnitude) over static kd-trees and stable 100 Hz mapping in outdoor settings. The work offers practical, open-source tooling for real-time robotic perception and navigation tasks that require dynamic spatial partitioning.

Abstract

This paper proposes an efficient data structure, ikd-Tree, for dynamic space partition. The ikd-Tree incrementally updates a k-d tree with new coming points only, leading to much lower computation time than existing static k-d trees. Besides point-wise operations, the ikd-Tree supports several features such as box-wise operations and down-sampling that are practically useful in robotic applications. In parallel to the incremental operations (i.e., insert, re-insert, and delete), ikd-Tree actively monitors the tree structure and partially re-balances the tree, which enables efficient nearest point search in later stages. The ikd-Tree is carefully engineered and supports multi-thread parallel computing to maximize the overall efficiency. We validate the ikd-Tree in both theory and practical experiments. On theory level, a complete time complexity analysis is presented to prove the high efficiency. On experiment level, the ikd-Tree is tested on both randomized datasets and real-world LiDAR point data in LiDAR-inertial odometry and mapping application. In all tests, ikd-Tree consumes only 4% of the running time in a static k-d tree.

ikd-Tree: An Incremental K-D Tree for Robotic Applications

TL;DR

The paper tackles the inefficiency of rebuilding static kd-trees for sequential robotic data by introducing ikd-Tree, an incremental kd-tree that supports point- and box-wise updates, downsampling, and lazy deletion with parallel partial re-building to preserve real-time performance. It provides a complete data-structure design, a two-thread re-balancing mechanism, and rigorous time/space complexity analyses. Experimental validation on randomized datasets and real-world LiDAR-inertial mapping demonstrates substantial speedups (order-of-magnitude) over static kd-trees and stable 100 Hz mapping in outdoor settings. The work offers practical, open-source tooling for real-time robotic perception and navigation tasks that require dynamic spatial partitioning.

Abstract

This paper proposes an efficient data structure, ikd-Tree, for dynamic space partition. The ikd-Tree incrementally updates a k-d tree with new coming points only, leading to much lower computation time than existing static k-d trees. Besides point-wise operations, the ikd-Tree supports several features such as box-wise operations and down-sampling that are practically useful in robotic applications. In parallel to the incremental operations (i.e., insert, re-insert, and delete), ikd-Tree actively monitors the tree structure and partially re-balances the tree, which enables efficient nearest point search in later stages. The ikd-Tree is carefully engineered and supports multi-thread parallel computing to maximize the overall efficiency. We validate the ikd-Tree in both theory and practical experiments. On theory level, a complete time complexity analysis is presented to prove the high efficiency. On experiment level, the ikd-Tree is tested on both randomized datasets and real-world LiDAR point data in LiDAR-inertial odometry and mapping application. In all tests, ikd-Tree consumes only 4% of the running time in a static k-d tree.

Paper Structure

This paper contains 25 sections, 2 theorems, 4 equations, 7 figures, 2 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Related work
  3. ikd-Tree Design and Implementation
  4. Data Structure
  5. Building An Incremental K-D Tree
  6. Incremental Updates
  7. Pushdown and Pullup
  8. Point-wise Updates
  9. Box-wise Updates
  10. Downsample
  11. Re-balancing
  12. Balancing Criterion
  13. Re-build
  14. Parallel Re-build
  15. K-Nearest Neighbor Search
  16. ...and 10 more sections

Key Result

Lemma 1

An incremental k-d tree can handle a point-wise incremental operation with time complexity of $O(\log n)$ where $n$ is the tree size.

Figures (7)

  • Figure 1: Illustration of incremental k-d tree update and re-balancing. (a): an existing k-d tree (black dots) and new points (red triangles) to insert, blue cubes denote the space (i.e., branches) need to be re-balanced. (b): the k-d tree after points insertion and tree re-balancing, blue cubes denote the space after re-balancing while rest majority tree does not change.
  • Figure 2: Point Cloud Downsample. (a): the point cloud before down-sampling. (b): the point cloud after down-sampling
  • Figure 3: Re-build an unbalanced sub-tree
  • Figure 4: The time performance comparison between an ikd-Tree and a static k-d tree.
  • Figure 5: Fig. (a) and (b) illustrate new points (orange triangles) and points already on the k-d tree (blue dots). Fig. (c) shows the time for incremental updates of sparse and compact data on k-d trees of different size.
  • ...and 2 more figures

Theorems & Definitions (4)

  • Lemma 1: Point-wise Operations
  • proof
  • Lemma 2: Box-wise Operations
  • proof