Hyperbolic geometry of structure matrices in a hierarchical texture-descriptor tensor: multiscale texture of grapevine leaves and wood affected by esca

Esca of grapevine produces foliar symptoms and internal wood necrosis that are scored visually, by severity class. This study extends a hierarchical texture-descriptor tensor, indexed by descriptor channel, Gaussian scale, orientation and region of interest, with four results needed for plant tissue, and applies it to published colour photographs of Vitis vinifera ‘Malvasia Fina’ leaves and trunk cross-sections at four severity levels (control, asymptomatic, chlorotic, scorched), and to an external esca-affected trunk section with zones labelled by its authors. The theory shows (i) that the affine-invariant distance between two structure matrices splits exactly into an energy term and a hyperbolic term in which the pair (anisotropy A, doubled texture axis 2ψ) are Beltrami–Klein disk coordinates, so that changes of contrast, of anisotropy and of orientation are separated, and only the first changes under a global photographic gain; (ii) that the orientation profile of the mean absolute directional derivative contains the second and fourth circular moments of gradient orientation with weights 1/3 and 1/15, the fourth being recoverable only with eight orientations; (iii) that pooling over colour channels obeys the same inequality as pooling over regions; and (iv) that a moving radial–tangential frame removes the cancellation of radial anatomy in pooled estimates. On leaves, the global anisotropy index depended on where the analysed region lay relative to the main veins, whereas the mean local coherence fell from 0.30–0.34 in symptomless leaves to 0.19 in symptomatic leaves (Kendall τ = −0.38 across 141 blocks of the four leaves, a within-image descriptive test, P < 10−8), and the decrease persisted after a common blur; the asymptomatic leaf of an infected vine was not distinguished from the control (P = 0.60). In trunk sections, ray-dominated texture (radiality index ρ > 0) in the outer wood gave way to concentric texture with increasing severity (outer mean ρ from +0.13 to −0.21). In the external section, white rot, the interaction border and sectorial necrosis were separated from healthy wood mainly by rotation, anisotropy and energy, respectively, and the texture axis turned by about 90° about 110 px inside the colour edge. The results provide interpretable, contrast-aware texture readings on the analysed agronomic images; they are exploratory, and diagnostic value is not claimed.