Half Vector

The half vector H is the microsurface normal $\displaystyle \omega_m$ in the formulation while the normal N is the geometric normal $\displaystyle \omega_g$ in the formulation.

According to the "9.8 BRDF Models for Surface Reflection" of Real-Time Rendering Fourth Edition, [Hammon 2017] proposed a method to calculate the NdotH and LdotH without the knowledge of the half vector H itself.

    // |L + V|^2 = L^2 + V^2 + 2L·V = 1 + 1 + 2L·V
    // N·H = N·((L + V)/(|L + V|)) = (N·L + N·V)/(|L + V|)
    // L·H = L·((L + V)/(|L + V|)) = (L^2 + L·V)/(|L + V|)
    // V·H = V·((L + V)/(|L + V|)) = (L·V + V^2)/(|L + V|)
    // 2L·H = 2V·H = L·H + V·H = (L^2 + L·V + L·V + V^2)/(|L + V|) = (|L + V|^2)/(|L + V|) = |L + V| = 1 + 1 + 2L·V
    // ⇒ L·H = 0.5 * |L + V| = (0.5 * |L + V|^2)/(|L + V|) =(0.5 * (1 + 1 + 2L·V))/(|L + V|) = 1/(|L + V|) + (L·V)/(|L + V|)
    // UE: [Init](https://github.com/EpicGames/UnrealEngine/blob/4.27/Engine/Shaders/Private/BRDF.ush#L31)
    // U3D: [GetBSDFAngle](https://github.com/Unity-Technologies/Graphics/blob/v10.8.0/com.unity.render-pipelines.core/ShaderLibrary/CommonLighting.hlsl#L361)
    highp float invLenH = inversesqrt(2.0 + 2.0 * NdotL);
    highp float NdotH = clamp((NdotL + NdotV) * invLenH, 0.0, 1.0);
    highp float LdotH = clamp(invLenH * VdotL + invLenH, 0.0, 1.0);

The NdotH and the LdotH are calculated by GetBSDFAngle in Unity3D and Init in UE4.

Specular AA

By "Equation (9.76)" of Real-Time Rendering Fourth Edition, the NDF by averaging the normal and the roughness separately is NOT the averaging NDF.

NormalCurvatureToRoughness in UE4

TextureNormalVariance in U3D

Trowbridge-Reitz

NDF

According to the "9.8.1 Normal Distribution Functions" of Real-Time Rendering Fourth Edition, the GGX should technically be called the Trowbridge-Reitz which is also adopted by the "8.4 Microfacet Models" of PBR Book.

G

[Heitz 2014]

References

[Heitz 2014] Eric Heitz. "Understanding the Masking-Shadowing Function in Microfacet-Based BRDFs." JCGT 2014.
[Hammon 2017] Earl Hammon. "PBR Diffuse Lighting for GGX+Smith Microsurfaces." GDC 2017.