Latticemapped

Latticemapped is a term used to describe a representation that associates data values with points on a lattice, a regular, discrete grid. Formally, a latticemapped object can be described as a mapping f from a lattice L to a set of values V, often written as f: L -> V. The lattice is typically a subset of Z^d with a fixed dimensionality d, providing coordinates for each grid point and a notion of neighborhood. This framing emphasizes the dual role of the lattice as both coordinate framework and storage medium for values.

In practice, latticemapped data are stored as multi-dimensional arrays or as sparse structures keyed by lattice coordinates. Computations on a latticemapped field often rely on local stencils that aggregate values from neighboring lattice points, as in finite difference methods, cellular automata, or image filters. Interpolation between lattice points, such as nearest-neighbor, bilinear, or trilinear interpolation, allows mapping between continuous space and lattice values and vice versa.

Applications span numerical simulation, image processing, geospatial grids, and grid-based machine learning. Variants include latticemapped embeddings, lattice-structured graphs, and discretizations used in physics and engineering. Key considerations include memory layout, boundary conditions, and numerical stability. The term emphasizes the lattice’s role as both the spatial scaffold and the data-organization scheme for values defined on discrete points.