Anisotropic Texture Richness (ATR)

• ATR is a composite characterization that quantifies how textures distribute structure, roughness, and detail across directions and scales.
• It utilizes operator scaling Gaussian random fields and anisotropic Besov spaces, incorporating metrics like the generalized Hurst index for directional analysis.
• The approach underpins both stochastic texture analysis and learned scene representations by highlighting directional scaling and localized smoothness variations.

Anisotropic Texture Richness (ATR) denotes, in the body of work considered here, a quantitative description of how a texture distributes structure, roughness, and detail across directions and scales. The term is not explicitly introduced in the foundational analysis of anisotropic self-similar Gaussian textures, but that analysis provides the central ingredients for such a notion: an anisotropy operator $E_0$, a generalized Hurst index $H_0$, and an anisotropic local critical exponent $\alpha_{f,loc}(D,p,q)$ whose dependence on the analysis anisotropy $D$ yields a directional smoothness profile (Clausel et al., 2013). In later rendering literature, closely related ideas appear in adaptive anisotropic textured Gaussians, where texture capacity is made directional and content-dependent at the level of individual primitives (Hsu et al., 14 Jan 2026, Wei et al., 16 Dec 2025).

1. Conceptual scope

ATR is best understood as a composite characterization rather than a single canonical scalar. In the mathematical setting of anisotropic self-similar textures, the relevant components are the field’s directional scaling law, its roughness or self-similarity exponent, and the contrast between analysis directions that are aligned or misaligned with the intrinsic geometry. In the algorithmic setting of textured Gaussian splatting, the analogous components are per-primitive aspect ratio, per-axis texture resolution, and gradient- or error-driven allocation of texels.

This suggests two closely related meanings. In stochastic texture analysis, ATR describes the geometry of sample paths through anisotropic scaling and anisotropic Besov regularity. In learned scene representations, ATR describes the ability of a representation to encode more high-frequency detail along certain directions than others while allocating capacity non-uniformly. The two viewpoints are compatible: both treat anisotropy as directional scaling structure and richness as multiscale complexity that is not uniformly distributed.

A useful separation, already implicit in the literature, is between anisotropy itself and anisotropic texture richness. Anisotropy identifies preferred directions and unequal scaling laws; richness adds the question of how much multiscale variation is present and how sharply it concentrates around those preferred directions.

2. Stochastic and functional-analytic foundations

The foundational model is the class of Operator Scaling Gaussian Random Fields (OSGRF), defined on $\mathbb{R}^2$ by a harmonizable representation

$$X_{\rho,E_0,H_0}(\underline{x}) = \int_{\mathbb{R}^2} \bigl(e^{i\langle \underline{x}, \underline{\xi}\rangle}-1\bigr) \,\rho(\underline{\xi})^{-(H_0+1)} \,d\widehat W(\underline{\xi}),$$

where $E_0\in\mathcal E^+$ has positive-real-part eigenvalues and $\mathrm{Tr}(E_0)=2$, $$H_0\in\bigl(0,\min_{\lambda\in\operatorname{Sp}(E_0)}\mathrm{Re}(\lambda)\bigr)$$, and $\rho$ is continuous, positive, and $H_0$0-homogeneous in the sense that

$H_0$1

The field satisfies the operator-scaling property

$H_0$2

so $H_0$3 encodes directional scaling and $H_0$4 is the generalized Hurst index (Clausel et al., 2013).

A concrete example uses

$H_0$5

with $H_0$6. In that case horizontal and vertical frequencies are weighted differently, and anisotropy is explicit. The rotated version studied in the hyperbolic-wavelet analysis introduces an angle $H_0$7, so the intrinsic anisotropy axes need not coincide with the image axes (Roux et al., 2013).

The functional framework used to measure this geometry is the family of anisotropic Besov spaces $H_0$8, where $H_0$9 is a diagonalizable anisotropy matrix. Their norms are defined through directional finite differences along eigenvectors of $\alpha_{f,loc}(D,p,q)$0, and the corresponding local critical exponent

$\alpha_{f,loc}(D,p,q)$1

measures the maximal local smoothness of $\alpha_{f,loc}(D,p,q)$2 when regularity is assessed with anisotropy $\alpha_{f,loc}(D,p,q)$3 (Clausel et al., 2013).

These objects furnish the canonical ingredients for ATR in the self-similar Gaussian setting. The matrix $\alpha_{f,loc}(D,p,q)$4 describes directional scaling, $\alpha_{f,loc}(D,p,q)$5 describes overall roughness, and the map $\alpha_{f,loc}(D,p,q)$6 describes how texture regularity changes as the analysis geometry is varied.

3. Strong optimality and the geometry of anisotropic smoothness

The central theorem establishes that the sharpest smoothness measurement is obtained when the analysis anisotropy matches the field’s true anisotropy. For (