A hierarchical tensor framework for quantitative multiscale and multidirectional biomedical texture characterization

Histological images of skeletal muscle contain organization at several spatial scales and in several directions, from the edges of individual fibres to the arrangement of fibre bundles, yet texture analysis usually reports single-scale, orientation-averaged descriptors. This study formulates a hierarchical descriptor tensor indexed by descriptor channel, Gaussian scale, orientation and region of interest (ROI), derives its main analytical properties, and applies it to two published haematoxylin and eosin (H&E) cross-sections of normal skeletal muscle, one human and one from the fish Pangasianodon hypophthalmus. Ten channels (directional-derivative statistics, grey-level co-occurrence features and autocorrelation) were evaluated at four scales (1, 2, 4, 8 px) and four orientations, for one ROI (160 entries per image) and four ROIs (640 entries). The theory shows that the directional-energy block is a quadratic form of the regional structure matrix, so that four orientations determine the anisotropy index A and the texture axis exactly, and that pooling regions can only lower A. Against isotropic surrogates with the same radial spectrum, both images were anisotropic (A = 0.208 and 0.387; surrogate 95th percentile 0.008 and 0.011; P ≤ 0.005). The texture axis agreed with the structure-tensor field, with the ordering of directional correlation lengths and, for the fish image, with the long-axis orientation of 207 segmented fibres (difference 1.4°). Four ROIs revealed regional heterogeneity that a single ROI attenuated (13% in the human image). Gradient channels were nearly collinear (Spearman ρ ≥ 0.98), and co-occurrence contrast and correlation, linked by an exact variogram identity, decoupled on tissue. The results establish interpretable, internally consistent readings on real human and veterinary muscle histology; diagnostic value is not claimed.