Pointsthe

Pointsthe is a term used in computational geometry and data analysis to denote a scalar value assigned to each point in a finite set, intended to summarize local geometric information. The concept combines aspects of local density and directional structure to produce a per-point feature that can be used in machine learning, clustering, and shape analysis.

Definition and construction

For a finite point set P in Euclidean space, and a chosen parameter k, let N_k(p) denote the k nearest neighbors of a point p in P. The pointsthe value s(p) is formed by blending a density component with an orientation component. The density component typically uses a distance-based kernel over the neighborhood, such as a Gaussian with bandwidth σ, while the orientation component derives information from the local covariance of N_k(p), often summarized by the eigenvalue spread or by an anisotropy measure. The final s(p) is normalized to lie within a standard range, enabling comparison across points and datasets.

Computation

A typical workflow to compute pointsthe values is: (1) build a k-nearest-neighbors structure for P; (2) for each p in P, identify N_k(p); (3) compute the local covariance of N_k(p) and its eigenvalues; (4) evaluate the density and orientation components and combine them into s(p); (5) normalize all s(p) values. The overall complexity depends on the kNN structure but is commonly near O(n log n) for n points.

Properties and use

Pointsthe values are designed to be shift-invariant and, with normalization, scale-aware. They tend to be high in densely populated or anisotropic regions and lower in sparse or isotropic areas. Applications include feature extraction for point-cloud classification, segmentation, anomaly detection, and as inputs to learning algorithms that operate on unstructured geometric data.

See also

Point cloud, density estimation, k-nearest neighbors, local intrinsic dimensionality.