Scaling of Haralick features with image bit depth and gray level co-occurrence matrix displacement vector for linear gradients.

BACKGROUND AND

objectivePerceptual studies have shown that textures that are indistinguishable based on their second-order statistics are perceived as equivalent by the human visual system. These statistics, which capture spatial correlations in pixel intensities, are more closely related to image gradients than to absolute pixel values. This study aims to derive analytic scaling laws for Haralick texture features to enable quantization-invariant, reproducible texture classification across heterogeneous imaging conditions.

methodsWe analyzed the symmetries of the Gray-Level Co-occurrence Matrix (GLCM) induced by linear image gradients of ∇ gray levels per pixel. Exploiting these structural regularities, we derived closed-form scaling laws for four widely used Haralick features-Energy, Contrast, Correlation, and Inverse Difference Moment. These laws yield theoretically justified normalization factors that reduce or eliminate dependence on image bit depth and gray-level quantization (N

resultsLinear gradients produce GLCM entries aligned along diagonals offset by ∇⋅|d|, where d is the displacement vector. This structure enabled the derivation of analytic expressions describing each feature's scaling behavior, validated through numerical simulations on synthetic images. The derived normalization factors outperform prior empirical approaches and explain observed discrepancies in real-world datasets.

conclusionsOur results provide a principled framework for the normalization of Haralick features, improving reproducibility, comparability, and interpretability across heterogeneous datasets. The scaling laws also inform feature selection and classifier design by identifying which features are most robust to quantization and displacement. These insights enable efficient and standardized texture analysis for applications in radiomics, remote sensing, and computer vision.