UE毛发系统底层剖析

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作者：Heskey0
B站：https://space.bilibili.com/455965619
邮箱：3495759699@qq.com

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Unreal Engine - Hair Shading Model

Heskey0的个人空间

Papers

unreal-engine-hair-and-fur-whitepaper

SIGGRAPH 2016 Course: Physically Based Shading in Theory and Practice

unreal-engine-hair

dualscattering

NVIDIAsiggraph2008

EGSR2011

Marschner-Siggraph2003

一. Fur Structure

outer --> inner

• cuticle

• cortex : absorbs light

• medulla : scatters light

paths:

• R
• TT
• TRT

notation p represent R, TT or TRT

二. Radiance

curve irradiance : power per unit length

curve intensity : intensity per unit length

bidirectional scattering function :

\[S(\omega_i,\omega_r)=\frac{d\bar{L}_r(\omega_r)}{dE_i(\omega_i)} \]

the most prominent feature in the scattering function is the specular highlight that occurs approximately when $ \theta_r =-\theta_i $

BCSDF : bidirectional curve scattering distribution function

三. Basic concept

normal plane : 垂直于虚拟法线的平面

incidence plane : 由入射光线与入射点处的法线所构成的平面被称为入射面

\(\theta_i,\theta_r\) : inclinations with respect to the normal plane

\(\theta_d=\frac{\theta_i+\theta_r}{2}\) : the difference angle

\(\theta_h=\theta_i+\theta_r\)

\(\phi_i,\phi_r\) : azimuth

\(\phi=\phi_r-\phi_i\)

四. Theory

because of the symmetry of a cylinder, the 4D scattering function can be factored into a product of two 2D terms.

One term, capture the \(\theta\) dependence, called \(M\) : longitudinal scattering function , 纵向的散射方向

One term, capture the \(\phi\) dependence, called \(N\) : azimuthal scattering function , 头发横截面的散射方向

\[\begin{equation*} \begin{aligned} S(\phi_i,\theta_i;\phi_r,\theta_r) &= M_R(\theta_i,\theta_r)N_R(\eta^{\prime}(\theta_d);\phi_i,\phi_r)/cos^2\theta_d \\ &+ M_{TT}(\theta_i,\theta_r)N_{TT}(\eta^{\prime}(\theta_d);\phi_i,\phi_r)/cos^2\theta_d \\ &+ M_{TRT}(\theta_i,\theta_r)N_{TRT}(\eta^{\prime}(\theta_d);\phi_i,\phi_r)/cos^2\theta_d \end{aligned} \end{equation*} \]

for M, 符合高斯函数 :

[EGSR2011 An Energy-Conserving Hair Reflectance Model] 4.1

\[M_R=g(\beta_R,\theta_h-\alpha_R)\\ M_{TT}=g(\beta_{TT},\theta_h-\alpha_{TT})\\ M_{TRT}=g(\beta_{TRT},\theta_h-\alpha_{TRT}) \]

where g is a normalized Gaussian of longitudinal inclination \(\theta\), \(\beta\) is a roughness term

\[g(\beta,\theta)=\frac{e^{\theta^2/(2\beta^2)}}{\sqrt{2\pi}\beta} \]

for N, 可做一些简化 :

1. R

[EGSR2011 An Energy-Conserving Hair Reflectance Model] 4.1

\[N_R=\frac{|cos(\phi/2)|}{4}A(0,h) \]

where A : Attenuation term

\[A(0,h)=F(\eta,\sqrt{\frac{1}{2}+\frac{1}{2}(\omega_i\omega_r)}) \]

where F : Schlick Fresnel

\[F(\eta,x)=F_0+(1-F_0)(1-x)^5\\ F_0=\frac{(1-\eta)^2}{(1+\eta)^2} \]

other paths too complex and minor impact

2. TT

[A Data-Driven Light Scattering Model for Hair] 3.2

\[N_p(\theta_i,\theta_r,\phi)=\frac{1}{2}\int^1_{-1}A(p,h)D_p(\phi-\Phi(p,h))dh \]

where A : Attenuation term

\[A(p,h)=(1-f)^2f^{p-1}T(\mu_a,h)^p \]

where T : Absorption term

\[T(\mu_a,h)=exp(-2\mu_a\frac{1+cos(2\gamma_t)}{cos\theta_t}) \]

and

\[f=F(\eta,cos(\theta_d)\sqrt{1-h^2}) \]

float f = Hair_F( CosThetaD * sqrt( saturate( 1 - h*h ) ) );
float Fp = Pow2(1 - f);
float Tp = exp(-AbsorptionColor * 2 * abs(1 - Pow2(h * a) / CosThetaD));

we only need to evaluate h, D, T

由于复杂性，需要做一些简化：

2.1 D :

\[D_{TT}(\phi)=D(\phi,0.35,\pi)\\ D(\phi,s,\mu)=\frac{e^{\frac{\phi-\mu}{s}}}{s(1+e^{\frac{\phi-\mu}s})^2} \]

approximation :

\[D_{TT}(\phi)\approx e^{-3.65cos\phi-3.98} \]

 float Np = exp( -3.65 * CosPhi - 3.98 );

2.2 eta :

\[\eta^\prime=\frac{\sqrt{\eta^2-sin^2\theta_d}}{cos\theta_d} \]

approximation :

\[\eta=1.55\\ \eta^\prime\approx \frac{1.19}{cos\theta_d}+0.36cos\theta_d \]

error < 0.68%

2.3 h :

\[a=\frac{1}{\eta^\prime}\\ h_{TT}=\frac{sign(\phi)cos\frac\phi2}{\sqrt{1_a^2-2a*sign(\phi)sin\frac\phi2}} \]

approximation :

\[h_{TT}\approx(1+a(0.6-0.8cos\phi)cos\frac\phi2) \]

float h = CosHalfPhi * ( 1 + a * ( 0.6 - 0.8 * CosPhi ) );

2.4 T :

approximation :

\[T(\theta,\phi)=C^{\frac{\sqrt{1-h^2a^2}}{2cos\theta_d}} \]

3. TRT

h, the offset for Fresnel :

\[h_{TRT}=\frac{\sqrt3}{2} \]

absorption term :

\[T_{TRT}(\theta,\phi)=C^{\frac{0.8}{cos\theta_d}} \]

Distribution :

\[D_{TRT}(\phi)=\frac34D(\phi,0.15,0) D_{TRT}(\phi)\approx e^{17cos\phi-16.78} \]

But it’s still missing something ! We need to render a volume of hair not just a single strand

4. Multiple scattering

五. Implementation

file : HairBsdf.ush

function : HairShading()

【notice : HairShadingRef() is a Reference, 由宏REFERENCE控制是否开启，默认是关闭。

HairShading()是HairShadingRef()的近似估计, Lots of curve fitting and function aproximation】

Coding

//R
S += Mp * Np * Fp * (GBuffer.Specular * 2) * lerp(1, Backlit, saturate(-VoL));
//TT
S += Mp * Np * Fp * Tp * Backlit;
//TRT
S += Mp * Np * Fp * Tp;
//Multiple Scattering
float3 EvaluateHairMultipleScattering(
	const FHairTransmittanceData TransmittanceData,
	const float Roughness,
	const float3 Fs)
{
	return TransmittanceData.GlobalScattering * (Fs + TransmittanceData.LocalScattering) * TransmittanceData.OpaqueVisibility;
}
S  = EvaluateHairMultipleScattering(HairTransmittance, ClampedRoughness, S);
S += KajiyaKayDiffuseAttenuation(GBuffer, L, V, N, Shadow);

(Np * Fp) represent N : azimuthal scattering function

for M

//R
float Mp = Hair_g(B[0] * BScale, SinThetaL + SinThetaV - Shift);
//TT
float Mp = Hair_g( B[1], SinThetaL + SinThetaV - Alpha[1] );
//TRT
float Mp = Hair_g( B[2], SinThetaL + SinThetaV - Alpha[2] );

for N = (Np * Fp)

//R
float Np = 0.25 * CosHalfPhi;
float Fp = Hair_F(sqrt(saturate(0.5 + 0.5 * VoL)));
//TT
float Np = exp( -3.65 * CosPhi - 3.98 );
float h = CosHalfPhi * ( 1 + a * ( 0.6 - 0.8 * CosPhi ) );
float f = Hair_F( CosThetaD * sqrt( saturate( 1 - h*h ) ) );
float Fp = Pow2(1 - f);
//TRT
float Np = exp( 17 * CosPhi - 16.78 );
float f = Hair_F( CosThetaD * 0.5 );
float Fp = Pow2(1 - f) * f;

where

float Hair_g(float B, float Theta)
{
	return exp(-0.5 * Pow2(Theta) / (B * B)) / (sqrt(2 * PI) * B);
}

Parameters

• BaseColor
  • C in the equation
• Specular
  • Scales the R term
• Roughness
  • \(\beta_R=Roughness^2\)
  • \(\beta_{TT}=0.5Roughness^2\)
  • \(\beta_{TRT}=2Roughness^2\)
• Scatter
  • Scales multiple scattering term
• Shift
  • \(\alpha_R=-2Shift\)
  • \(\alpha_{TT}=Shift\)
  • \(\alpha_{TRT}=4Shift\)