[Paper Review] ikd-Tree: An Incremental K-D Tree for Robotic Applications

Yixi Cai, Wei Xu+1 more|arXiv (Cornell University)|Feb 22, 2021

Robotics and Sensor-Based LocalizationEngineering18 references68 citations

TL;DR

ikd-Tree incremental KD-tree updates points (and boxes), supports down-sampling, and balances via partial re-building with parallel re-builds, enabling real-time robotic mapping and navigation. Open-sourced at GitHub.

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.

Motivation & Objective

• Motivate the need for dynamic space partitioning in robotics with sequential data acquisition.
• Propose ikd-Tree to incrementally update a KD-tree with new points while down-sampling.
• Enable box-wise operations and lazy/deferred updates to maintain real-time performance.
• Develop a parallel partial-re-building strategy to rebalance the tree efficiently.
• Provide theoretical complexity analyses and empirical validation on randomized and real LiDAR data.

Proposed method

• Introduce a data structure for IKD-tree nodes with attributes to support incremental updates and down-sampling.
• Describe building a balanced KD-tree with maximal covariance axis selection and median splitting.
• Define point-wise and box-wise incremental updates with lazy labeling (deleted, treedeleted, pushdown) and pullup/pushdown propagation.
• Implement partial re-building to balance sub-trees, including a parallel two-thread rebuilding scheme to preserve real-time performance.
• Incorporate a down-sampling routine that performs box-wise search, center-point selection, and replacement of points in the down-sampling cube.
• Present a complexity framework showing O(log n) for point-wise ops, O(m log n) for box-wise insertions, and O(n) for parallel two-thread re-builds.

Experimental results

Research questions

• RQ1 How can a KD-tree be incrementally updated to reflect newly acquired robotic data without full reconstruction?
• RQ2 Can box-wise updates and down-sampling be integrated into an incremental KD-tree while preserving query efficiency?
• RQ3 What are the time/space complexities of incremental updates and partial re-balancing in ikd-Tree?
• RQ4 Does parallelized re-building maintain real-time performance in robotic mapping scenarios?

Key findings

• Incremental operations (point-wise) achieve O(log n) time complexity.
• Box-wise insertion of m points costs O(m log n); box-wise delete/re-insert have complexity dependent on data distribution and can leverage lazy updates.
• Down-sampling is achievable with O(log n) time due to small down-sampling cubes.
• Two-thread parallel re-building yields O(n) re-build time, improving real-time performance.
• In LiDAR-inertial odometry/mapping, ikd-Tree achieves nearly constant ~1.6 ms per update, vs faster growth for static trees.
• The framework consumes only 4% of running time for incremental updates compared to a static KD-tree and enables ~100 Hz mapping in practice.