Momentand

Momentand is a hypothetical construct used in mathematics and theoretical physics to summarize a field’s geometric structure together with its temporal evolution. It is intended as a compact descriptor that integrates static shape information with dynamic behavior, enabling comparison and reconstruction of complex spatiotemporal patterns without storing full field histories.

Formally, a momentand is an ordered pair M = (m, n). The static component m consists of a chosen set of spatial geometric moments of a scalar field φ(x) defined on a domain D, such as m_pq = ∫∫_D x^p y^q φ(x, y) dx dy for p+q ≤ k. The dynamic component n comprises temporal moments derived from the time history of φ, for example n_tk = ∫ t^k φ(x0, t) dt at representative locations x0 or integrals over the domain of time derivatives, capturing rates of change and persistence. Together, the pair aims to capture both the static geometry and dynamic evolution of the system in a single descriptor.

Momentands have several mathematical properties. They behave linearly with respect to changes in the field, and, under rigid motions, their spatial part m transforms in a predictable way while the temporal part n encodes the corresponding time-domain effects. Normalization can yield invariance to scale and orientation. In many models, the momentand acts as a sufficient statistic for certain classes of spatiotemporal fields, within the assumptions of those models.

Computation typically proceeds from discrete samples. Moments are evaluated by numerical quadrature or summation over pixels or voxels, and temporal moments from time series at selected locations or aggregated over the domain. Reconstruction from a momentand is approximate and may require regularization to mitigate noise and incomplete data.

Applications envisioned for momentands include computer vision for shape and gesture representation, fluid dynamics for characterizing evolving structures, and signal processing for compactly summarizing time-varying fields.

See also: moments, geometric moments, cumulants, time moments.