Gradigradi

Gradigradi is a term used in mathematical modeling to describe a class of gradient-based fields defined on grids or lattices. It emphasizes the representation of spatial variation by combining gradient information across multiple scales and discrete steps, allowing modeling of how a quantity changes across space in both continuous and discrete settings.

Concepts and variants: A gradigradi field assigns to each grid node a gradient vector. In the multi-scale approach, gradients are represented at several levels of granularity, with smooth transitions between levels. There are continuous gradigradi fields, which use interpolated gradient values, and discrete gradigradi fields, which assign gradients only at grid points. Operators in this framework, such as gradigradi-operators, compute gradient magnitude and direction in a way that is robust to sampling and grid irregularities. The framework supports linear combinations and thresholding, enabling feature extraction and segmentation.

Applications: In image processing, gradigradi methods can support edge-preserving smoothing and texture analysis. In geographic information systems, they can characterize terrain slopes and watershed dynamics. In climate and environmental modeling, they help visualize and analyze spatial gradients in temperature, precipitation, or pollution fields. In machine learning, gradigradi features can serve as inputs for classifiers and regressors.

Relationship and status: Gradigradi relates to gradient, gradient magnitude, and gradient-domain processing, but focuses on a multi-scale, grid-aware formulation. The term has appeared in a limited set of discussions and is not widely standardized; usage varies across communities and software.