Anisotropic Spherical Gaussian Distribution

• ASG Distribution is a parametrized function on the unit sphere that generalizes isotropic models by using distinct tangent bandwidths to capture anisotropic, elliptical highlight shapes.
• It enables precise control of shape and anisotropy through parameters like sharpness and ellipticity, supporting advanced neural rendering, pose estimation, and spatial statistics.
• Its analytic structure and efficient normalization allow closed-form integration and differentiation, facilitating real-time rendering and robust probabilistic inference.

The anisotropic spherical Gaussian (ASG) is a parametrized function on the unit sphere that generalizes the familiar isotropic spherical Gaussian by introducing distinct bandwidths along two orthogonal tangent axes. This directional flexibility allows the ASG to represent sharply peaked, elongated, or elliptical lobes and has enabled accurate modeling of anisotropic reflectance, advanced appearance fields in neural rendering, and various geostatistical and machine learning applications. The ASG's mathematical properties and implementation efficiency have led to its adoption in fields ranging from neural radiance field rendering to 3D Gaussian splatting, pose estimation, and spatial statistics.

1. Mathematical Definition and Parametric Structure

The ASG lobe is defined on the unit sphere $S^2$ using an orthonormal frame $[x, y, z]$, where $z$ is the main axis (lobe direction, typically a reflection or mean vector), and $x, y$ are local tangent axes. The canonical ASG takes the form

$$\mathrm{ASG}(v; [x, y, z], [\lambda, \mu], c) = c \cdot \max(v \cdot z, 0) \cdot \exp\left[ -\lambda (v \cdot x)^2 - \mu (v \cdot y)^2 \right]$$

with parameters:

• $v \in S^2$: unit direction under evaluation,
• $z$: mean/principal lobe axis,
• $x, y$: minor/major tangent axes,
• $\lambda, \mu > 0$: sharpness (inverse variance) along $x, y$,
• $c$: amplitude, optionally absorbs normalization or physical quantities,
• $\max(v \cdot z, 0)$: hemisphere gating (visibility term).

When $\lambda = \mu$, the ASG reduces to an isotropic spherical Gaussian; for $\lambda \neq \mu$, level sets become ellipses on the sphere, enabling explicit modeling of anisotropic highlight shapes (Barreiro et al., 25 Dec 2025, Du et al., 19 Feb 2025).

Alternate parametrizations replace $[\lambda, \mu]$ with overall concentration $\kappa$ and ellipticity $e$, for example

$$\lambda = \frac{1}{2}\kappa(1 + e), \quad \mu = \frac{1}{2}\kappa(1 - e)$$

where $e = 0$ yields the isotropic case, and $e \to 1$ maximally stretches the lobe along $y$ (Barreiro et al., 25 Dec 2025).

2. Normalization and Probability Density Formulations

For applications requiring a normalized probability density, the ASG's amplitude $c$ is set so that

$\int_{S^2} \mathrm{ASG}(v)\, dv = 1$

The normalization constant, denoted as $Z(\lambda, \mu)$, for the most common ASG kernel with hemispherical gating, admits the form

$$Z(\lambda, \mu) = \int_0^{2\pi} \int_0^{\pi/2} \exp\left[-\lambda \sin^2\theta \cos^2\phi - \mu \sin^2\theta \sin^2\phi \right] \sin\theta\, d\theta\, d\phi$$

Closed-form solutions exist for special cases using error functions (Du et al., 19 Feb 2025), but in general numerical quadrature or tabulation is employed. For the isotropic case, the normalization recovers the classic von Mises-Fisher constant:

$$\mathrm{SG}_{\mathrm{iso}}(d; \kappa, n, c) = c \exp[\kappa(n \cdot d)]$$

with normalization $4\pi \sinh \kappa / \kappa$ (Clausen et al., 2024).

3. Anisotropy and Shape Control

The principal distinction between the ASG and isotropic SG lies in explicit anisotropy:

• Sharpness control ($\kappa$, $\lambda$, $\mu$): adjusts overall concentration about the main axis; higher values produce narrower, more localized lobes.
• Ellipticity (or anisotropy parameter): controlled either directly by $\lambda \neq \mu$ or by derived parameters such as ellipticity $e$.
• Orientation: $x, y$ axes are typically aligned with surface features such as tangent and bitangent directions to model material microstructure or brushed appearance (Barreiro et al., 25 Dec 2025, Du et al., 19 Feb 2025).

This structure enables explicit encoding of elliptical highlights and lobe orientation, crucial for modeling materials with directional structure, such as brushed metal or anisotropic fibers (Yang et al., 2024, Du et al., 19 Feb 2025).

4. Learning and Integration in Neural and Rendering Pipelines

ASG kernels have been incorporated into neural rendering, inverse rendering, and appearance field models, where the bandwidths, amplitudes, and axes are spatially varying and learned:

• Neural fields (ShinyNeRF, AniSDF): The spatial MLP predicts per-point normals, tangent orientation, sharpness, and anisotropy. The lobe is then approximated by a mixture of isotropic vMF kernels for differentiable integration and efficient volume rendering (Barreiro et al., 25 Dec 2025, Gao et al., 2024).
• 3D Gaussian Splatting (Spec-Gaussian, GlossGau): The low-frequency limitations of spherical harmonics are overcome by parametrizing each 3D Gaussian's view-dependent channel as a sum of learned ASG lobes. Parameters are decoded via compact MLPs per Gaussian, enabling efficient modeling of specular and anisotropic surfaces at real-time rates (Yang et al., 2024, Du et al., 19 Feb 2025).
• Path Guiding (Normalized ASG Mixtures): For density estimation or sampling, the normalized ASG or mixtures thereof offer analyticity (enabling closed-form evaluation, sampling, and gradients) and sufficient expressivity for capturing illumination or BSDF products in physically-based rendering (Huang et al., 2023).

A summary table of key learning/integration strategies across applications:

Application	Parameter Learning	ASG Role
Neural Radiance Fields	MLP regresses normal, tangent, κ, e	Reflectance encoding
3D Gaussian Splatting	MLP decodes {λ, μ, ξ} from features	Appearance field
Importance Sampling	Density net learns λ, a, axis, weight	Irradiance / proposal pdf
Inverse Rendering	Surface MLP + analytic closed-form	Microfacet NDF modeling

5. Applications Across Disciplines

Neural Rendering and Inverse Rendering:

ASG kernels have demonstrated empirical superiority in modeling specular and anisotropic scenes compared to low-order spherical harmonics. Spec-Gaussian achieves 0.5–3.6 dB PSNR improvements over vanilla 3D-GS on challenging datasets and captures highlights missed by prior art (Yang et al., 2024). GlossGau and ShinyNeRF use per-splat or per-point ASGs to model the microfacet normal distribution, delivering physically interpretable, editable material representations and robust inverse rendering (Du et al., 19 Feb 2025, Barreiro et al., 25 Dec 2025).

Directional Statistics and Pose Estimation:

In machine learning, ASG-distributed label smoothing yields unbiased mean directions for directional regression problems (e.g., facial pose estimation), overcoming biases in previous interval or elementwise Gaussian regularizers. The network learns λ, η per-sample, giving supervised flexibility adapted to pose-dependent uncertainty (Cao et al., 2022).

Geostatistics:

The ASG formalism, when construed as a covariance kernel (i.e., product of isotropic and latitude-dependent kernels), enables anisotropic Gaussian processes on the sphere. Applications include kriging and covariance modeling for global climate fields, where latitude-longitude anisotropy is prominent in atmospheric datasets (Venet et al., 2019).

6. Theoretical Extensions and Formulations

Geodesic-Normal Law:

A Riemannian geometric perspective formulates the ASG as the geodesic normal distribution, parameterized by a mean on the sphere and a positive-definite covariance on the tangent space. This supports direct sampling and analytic elliptical contours (principal directions and axis-ratios on the sphere), bypassing curvature bias and chart dependence (Chacón et al., 2024). This approach provides precise control of anisotropy on arbitrary-dimensional hyperspheres.

Practical Considerations:

• Efficient evaluation and differentiation are practical due to closed-form normalization or analytic gradients in standard ASG forms (Huang et al., 2023, Clausen et al., 2024).
• In rendering, the hemisphere gating term (e.g., max($v\cdot z,0$)) ensures compatibility with reflection models and maintains a single lobe (Barreiro et al., 25 Dec 2025, Du et al., 19 Feb 2025).
• Numerical stability is maintained by pretabulation, analytic formulas, or constrained parameters (e.g., enforcing positivity of λ, μ) (Du et al., 19 Feb 2025, Cao et al., 2022).

7. Limitations and Comparative Perspective

• Spherical Harmonics vs. ASG: SH provides global, smooth approximations but cannot model high-frequency, localized, or highly anisotropic reflectance without impractically high degree (Yang et al., 2024).
• Isotropic Spherical Gaussians vs. ASG: A single isotropic SG cannot represent strongly elongated or directional lobes; mixtures may mitigate this but at significant parameter/memory cost (Barreiro et al., 25 Dec 2025, Yang et al., 2024).
• Fit to Riemannian Geometry: Curvature-correct ASG as direct geodesic-normal PDFs provide improved numerical properties over Euclidean-fit anisotropies, especially away from the pole (Chacón et al., 2024).

The ASG's capacity for explicit, analytically tractable anisotropy, compact parameterization, and closed-form integration positions it as a foundational tool for modern appearance modeling, probabilistic inference, and geostatistical analysis on the sphere.