spotsdiscrete

Spotsdiscrete is a computational framework used to represent and analyze spatial phenomena by decomposing a continuous field into a finite set of discrete spots. Each spot represents a localized influence region, typically modeled as a disk with an associated position, radius, and weight. The approach aims to provide a compact, flexible approximation of the target field while supporting efficient computation and storage.

Formally, a spotsdiscrete representation consists of a set S = { (xi, yi, ri, wi) | i = 1..n }. The estimated field value at a query point p is given by f(p) = sum over i of wi · K(||p − (xi, yi)||, ri), where K is a kernel with compact support, such as a truncated Gaussian or a disk-based kernel. The radius ri controls the spatial extent of each spot, and the kernel ensures contributions vanish beyond this extent. This enables local, scalable computations and easy compositionality of multiple fields.

Placement and sparsity are central concerns. Spots can be arranged on regular grids, placed via Poisson-disk sampling to enforce minimum distances, or adapted to data density. Trade-offs exist between fidelity and resource usage: higher spot counts improve accuracy but increase memory and processing costs; sometimes overlaps are allowed to improve smoothness, other times they are minimized to preserve sparsity.

Applications span digital halftoning, texture synthesis, image resampling, density estimation, and geographic information systems. Related concepts include splatting in computer graphics, radial basis function networks, and kernel density estimation.