Scaling laws for Haralick texture features of linear gradients

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 (N_g), the gradient magnitude (∇), and the displacement vector (d) through the corresponding GLCM. Scaling laws obtained from the exact analytical dependencies of Haralick features on N_g, ∇ and |d| show that SA and DV scale linearly with N_g, 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 N_g. 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.