A Robust and Efficient Boundary Point Detection Method by Measuring Local Direction Dispersion

TL;DR

LoDD introduces a robust boundary point detector that quantifies local direction dispersion via a trace-based centrality score derived from the local covariance of KNNs. By projecting neighbors onto PCA-rotated axes and computing $L^{(d)}$, LoDD efficiently identifies boundary points without expensive eigen-decomposition, and an adaptive ratio estimator (aLoDD) sets the boundary proportion from grid-based and intrinsic-dimension considerations. Across synthetic and real-world benchmarks, LoDD consistently improves clustering performance, enhances training-data selection for deep models, and accurately detects boundaries and holes in 3-D point clouds, outperforming several state-of-the-art detectors in high-dimensional settings. The approach offers scalable boundary detection with practical impact on edge extraction, clustering, and geometric data analysis.

Abstract

Boundary point detection aims to outline the external contour structure of clusters and enhance the inter-cluster discrimination, thus bolstering the performance of the downstream classification and clustering tasks. However, existing boundary point detectors are sensitive to density heterogeneity or cannot identify boundary points in concave structures and high-dimensional manifolds. In this work, we propose a robust and efficient boundary point detection method based on Local Direction Dispersion (LoDD). The core of boundary point detection lies in measuring the difference between boundary points and internal points. It is a common observation that an internal point is surrounded by its neighbors in all directions, while the neighbors of a boundary point tend to be distributed only in a certain directional range. By considering this observation, we adopt density-independent K-Nearest Neighbors (KNN) method to determine neighboring points and design a centrality metric LoDD using the eigenvalues of the covariance matrix to depict the distribution uniformity of KNN. We also develop a grid-structure assumption of data distribution to determine the parameters adaptively. The effectiveness of LoDD is demonstrated on synthetic datasets, real-world benchmarks, and application of training set split for deep learning model and hole detection on point cloud data. The datasets and toolkit are available at: https://github.com/ZPGuiGroupWhu/lodd.

Figures (12)

• Figure 1: An example on a toy dataset that reveals the performance enhancement of the DBSCAN algorithm by boundary point extraction. (a) Weak connectivity and density heterogeneity impede the effectiveness of DBSCAN. (b) DBSCAN can separate the weakly connected clusters and identify the sparse cluster by peeling out the boundary points.
• Figure 2: Ability and limit of Direction Centrality Metric (DCM). (a) The central angles are approximately equal when KNN are distributed uniformly on the unit circle. (b) The central angles differ a lot when KNN lie in a smaller range of direction. (c) A 2-D toy data illustrates that DCM fails to reflect the uniformity of KNN distribution when handling manifold clusters.
• Figure 3: Illustration of how the projection variances of KNN indicate the centrality. (a) The case when KNN are distributed uniformly on the unit circle, in which the projections are located dispersedly on the axes. (b) The case when KNN are distributed on a smaller range of the unit circle, in which the projections are located compactly on the X axis. (c) PCA rotates the KNN to the varimax direction. (d) The distribution projections on the axes of KNN after PCA rotation.
• Figure 4: Estimation of boundary points based on the grid-structure assumption of data distribution. (a) A 2-D data point set. (b) An irregular grid structure by connecting each point with its neighboring point. Notably, we do not explicitly compute a grid structure, but theoretically estimate the number of boundary points based on this implicit assumption. Nonetheless, we design a grid generation algorithm. It divides the data into multiple equal bins along the X direction to ensure that each bin has similar number of points, then it numbers the points in ascending order of Y for each bin and connects them. Finally, it connects points with the same number along the X direction to generate a grid topological structure. Details of this algorithm can be seen at https://github.com/ZPGuiGroupWhu/lodd/blob/main/lodd_mat/Functions/Intro-GenerateGrid.md. (c) A regular grid composed of unit square cells, in which the blue points and lines denote the boundary points and periphery of the grid, respectively. (d)-(f) Projecting the non-overlapping outer edges of the grid cells onto the MBR orthogonally, where the blue areas represent the grid extent, the green and yellow segments denote the orthogonal projections on MBR, and the inner segments of a concave or ring-shaped grid, respectively.
• Figure 5: A 3-D example for estimating the number of boundary points in high-dimensional space, where the boundary points lie on the blue surface and the internal points are in the red hypercuboid.