quadrupolsplit

Quadrupolsplit is a data analysis concept describing a method to decompose a two-dimensional field into four directional components that reflect quadrupolar, or fourfold, symmetry around a reference point. The approach relies on projecting local field content onto a set of quadrupolar basis patterns, effectively separating information carried in four orthogonal angular directions.

Mathematically, quadrupolsplit considers a scalar or vector field defined on a plane and centers a neighborhood at a chosen point. A quartet of angular basis functions, typically derived from quadrupole-like patterns such as cos(2θ) and sin(2θ) and their combinations, is used to generate four filters. The local response is obtained by convolving the field with each filter or by projecting onto the corresponding basis, yielding four components labeled Q0, Q1, Q2 and Q3. In discrete form, these components are obtained through kernels designed to approximate a four-lobed angular pattern, often with a radial weighting to control scale.

Implementation involves selecting a center, choosing a scale parameter, applying the quadrupolar filters, and normalizing the results. The four components collectively capture local anisotropic structure and can be further analyzed to extract features such as energy in each quadrant, dominant orientation, and phase information. Quadrupolsplit is compatible with multiscale frameworks and can be integrated with other feature descriptors.

Applications are found in texture analysis, pattern recognition, edge orientation estimation, and geophysical or climate data interpretation where fourfold or quadrupolar patterns are informative. Limitations include sensitivity to the choice of center and scale, potential redundancy with other quadrature-based descriptors, and susceptibility to noise in small neighborhoods. See also quadrupole moments, spherical harmonics, and quadrature filters.