How Spherical Harmonics (SH) used in 3D Gaussian Splatting

1. Introduction to Spherical Harmonics (SH)

Spherical Harmonics (SH) are a mathematical basis used to represent functions defined on the surface of a sphere. They are most commonly used in graphics for representing lighting environments, especially for low-frequency lighting such as diffuse or ambient lighting.

• SH are defined by two indices: degree $l$ and order $m$.
• The number of SH terms up to degree $l$ is $(l+1{)}^2$.
• SH are orthonormal, meaning each SH basis function is independent of the others.

Applications of SH in 3D Graphics:

• Ambient Lighting: Precomputed lighting environments stored in SH coefficients are used to approximate how light interacts with objects in a scene.
• Diffuse Lighting: SH functions can represent how light scatters from surfaces in all directions.
• Light Probes: SH coefficients are used to store lighting information at different points in a scene, allowing for real-time lighting evaluation.

Key Properties of SH:

• Low-Frequency Approximation: SH are effective at representing smooth, slowly-varying lighting environments, making them ideal for diffuse lighting.
• Compact Representation: By truncating the SH expansion to a low degree (e.g., degree 3), we reduce the computational complexity and memory footprint while maintaining reasonable accuracy.

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2. Mathematical Background of Spherical Harmonics

Spherical harmonics $Y_l^m(\theta ,\varphi )$ are defined in terms of two angular coordinates:

• $\theta$: The polar angle (angle from the z-axis).
• $\varphi$: The azimuthal angle (angle from the x-axis in the x-y plane).

For degree $l$ and order $m$, Spherical Harmonics $Y_l^m(\theta ,\varphi )$ are given by:

$$Y_l^m(\theta ,\varphi )=\sqrt{\frac{(2l+1)}{4\pi }\cdot \frac{(l-m)!}{(l+m)!}}$$