Graduf

Graduf, short for gradient uniformity factor, is a dimensionless metric used to quantify the variation of the gradient of a scalar field across a region. It is designed to capture how smoothly the slope of a field changes in space, with low values indicating a uniform gradient and high values signaling abrupt changes or noise.

In practice, graduf has no universal formula. Most implementations compute gradient vectors ∇f on a discretized domain, then summarize the dispersion of these vectors or the dispersion of their magnitudes. Common approaches include the angular dispersion of gradient directions or the coefficient of variation of gradient magnitudes, often with a normalization by the mean gradient to achieve scale invariance. Because different definitions exist, the numerical range of graduf can vary between datasets and software.

Interpretation of graduf is context-dependent. Lower values suggest smoother, more uniform gradients, while higher values indicate irregularities, edges, or noise. It is used as a diagnostic measure in several fields to assess smoothing quality, to compare data preprocessing pipelines, or to inform algorithmic choices in tasks that assume gradient smoothness.

Applications and limitations: Graduf appears in image denoising and texture analysis, terrain or topographic quality assessment, and numerical optimization as a proxy for gradient smoothness. However, it is not a standardized metric; results depend on data resolution, noise levels, and the chosen computation method. It can be sensitive to outliers and should be interpreted alongside other measures such as gradient magnitude, total variation, or Sobolev norms.

Origin and reception: The term graduf has appeared in scholarly discussions and software documentation in recent years, but there is no universal definition or broad consensus. See also Gradient, Gradient magnitude, Smoothness, Total variation, Sobolev norm, Regularization.