Pointsis

Pointsis is a theoretical construct used in geometry and data analysis to describe and quantify local relationships within a finite set of points in Euclidean space. The concept provides a compact descriptor of a point cloud by aggregating information about each point’s neighborhood, enabling comparisons and feature extraction without requiring a full mesh or surface representation.

A common instantiation defines a point cloud P = {p1, …, pn} in R^d and fixes an integer k. For each i, let N_k(i) be the set of indices of the k nearest neighbors of pi. The pointsis descriptor S(P) is then the n-dimensional vector with components S_i = (1/k) ∑_{j in N_k(i)} ||pi − pj||, i.e., the average distance from each point to its k nearest neighbors. Variants may replace distances with directional vectors to neighbors, or incorporate higher-order moments of the local neighborhood. The descriptor is typically used as a locally sensitive feature that can be fed into downstream tasks such as matching or clustering.

Computation relies on efficient nearest-neighbor searches, with practical costs dominated by the data structure used for kNN queries. The basic descriptor is translation-invariant when distances are used, and scale can be controlled by pre-normalizing the data. Pointsis is often valued for its simplicity and robustness to irregular sampling, though it can be sensitive to outliers or density variations, motivating weighted or percentile-based refinements.

Applications include shape comparison, point-cloud registration, segmentation, and integration into machine learning pipelines for 3D data. See also point cloud, local descriptor, and k-nearest neighbors.