# Scaling laws for Haralick texture features of linear gradients

## TL;DR

This paper introduces a new analytical framework to understand how texture features depend on image gradients and quantization, enabling consistent normalization for better machine learning and medical imaging applications.

## Contribution

The paper derives exact scaling laws for Haralick texture features based on image quantization, gradient magnitude, and displacement, enabling normalization independent of quantization.

## Key findings

- SA and DV scale linearly with gray-level quantization (Ng), while SV scales quadratically.
- Entropy follows a logarithmic trend with Ng, and numerical simulations validate these scaling laws.
- Normalization factors derived from these laws make Haralick features independent of quantization.

## Abstract

This study presents a novel analytical framework for understanding the relationship between the image gradients and the symmetries of the Gray Level Co-occurrence Matrix (GLCM). Analytical expression for four key features–sum average (SA), sum variance (SV), difference variance (DV), and entropy–were derived to capture their dependence on image’s gray-level quantization (Ng), the gradient magnitude (∇), and the displacement vector (d) through the corresponding GLCM. Scaling laws obtained from the exact analytical dependencies of Haralick features on Ng, ∇ and |d| show that SA and DV scale linearly with Ng, SV scales quadratically, and entropy follows a logarithmic trend. The scaling laws allow a consistent derivation of normalization factors that make Haralick features independent of the quantization scheme Ng. Numerical simulations using synthetic one-dimensional gradients validated our theoretical predictions. This theoretical framework establishes a foundation for consistent derivation of analytic expressions and scaling laws for Haralick features. Such an approach would streamline texture analysis across datasets and imaging modalities, enhancing the portability and interpretability of Haralick features in machine learning and medical imaging applications.