qubins

QuBins are a theoretical construct used in computational geometry and data analysis to describe a partition of a space into small, axis-aligned, non-overlapping cells called qubins. In an n-dimensional space, select, for each dimension i, a partition of its coordinate range into mi equal-length intervals. The qubin with index (i1, ..., in) is the Cartesian product of the chosen intervals along each axis. The family of all such products forms a regular grid that covers the space; every point belongs to exactly one qubin.

Properties: QuBins create a grid with total count M = m1 × m2 × ... × mn. They are simple to implement and support constant-time location queries via index arithmetic. Adaptive qubins generalize this idea by varying interval lengths based on data density; hierarchical qubins (quadtree-, octree-like structures) allow multiscale representations by recursively subdividing qubins.

Applications: QuBins are used for spatial indexing, discretization in numerical integration, voxel-based visualization, and feature extraction in machine learning where continuous data are binned to create histograms or grid-based representations.

Examples: In 2D, with m1 = m2 = 4, a unit square is partitioned into 16 equal sub-squares. The qubin at (2,3) corresponds to the second interval along x and third along y.

History and etymology: The term qubin combines “quantum” or “quanta” with “bin” and was proposed in theoretical contexts to describe discretization schemes in multi-dimensional spaces.

See also: voxel, histogram bin, grid cell, spatial index, kd-tree.