1.Microfacet微面元
在所有的微面元中，仅仅只有红色的面元即h的方向才能被v看到,h被称为半矢量\(\mathrm{h}=\frac{\mathrm{l}+\mathrm{v}}{\left | \mathrm{l}+\mathrm{v} \right |}\)，D和G都是影响h这个参数
\(f (\mathrm{l},\mathrm{v})=f_d(\mathrm{l},\mathrm{v})+\frac{F(\mathrm{v},\mathrm{h})D(\mathrm{h})G(\mathrm{l},\mathrm{v},\mathrm{h})}{4(\mathrm{n}\cdot \mathrm{l})(\mathrm{n}\cdot \mathrm{v})} \)
2.菲涅尔F
菲涅尔表示入射和折射的比例，这个数值和材质本身有关系也和光线入射角有关系
\( L_R=R_F(\theta_i)L_l \)
菲涅尔的计算公式[Schick,1994]
\( R_F(\theta_i)\approx R_F(0^{\circ})+(1-R_F(0^{\circ}))(1-\cos\theta_i)^5 \)
\(R_F(0^{\circ})\)表示入射光垂直于表面时菲涅尔的反射率的值,下图展示了菲涅尔在金属和非金属下不同角度的数值
3.法线分布项D
D就是法线的分布，简单的说就是h偏离法线n的程度，主要是由粗糙度决定，直接只用GGX的分布
NDF(Normal Distribution Function) GGX\( \alpha = Roughness^2 \)
\( D(h)=\frac{\alpha ^2}{\pi ((\mathrm{n} \cdot \mathrm{h})^2(\alpha ^2-1)+1)^2} \)
4.遮挡项G
G称为双向阴影遮挡函数，也就是v会被其他的微面元遮挡，G主要受粗糙度影响，是用Smith GGX分布
Geometry Factor Smith GGX\( \alpha = Roughness^2 \)
\( G_{GGX}(\mathrm{k})=\frac{2(\mathrm{n} \cdot \mathrm{k})}{(\mathrm{n} \cdot \mathrm{k}) + \sqrt{\alpha^2+(1-\alpha^2)(\mathrm{n} \cdot \mathrm{k})^2 }} \)
\( G(\mathrm{l},\mathrm{v},\mathrm{h})=G_1(\mathrm{l})G_1(\mathrm{v}) \)
5.漫反射Diffuse
Lambert模型
\( f_d(\mathrm{l},\mathrm{v}) = \frac{c_\mathrm{diff}}{\pi} \)
6.UE4材质模型
6.1 Basecolor
Basecolor就是材质的基础颜色，也就是材质的漫反射颜色，即漫反射公式中的\(c_{diff}\)，漫反射颜色的本质是，光线在照射材质后，光线进入材质内部经过复杂散射后出来的光，这个散射后的光丧失了方向。与之对应的是高光，高光直接在材质上反射，映射的直接是光的颜色
6.2 Roughness
粗糙度是影响D和G的重要参数
6.3 Metallic
金属是个很特殊的参数，影响的是漫反射diff和菲涅尔项F，当材质为纯金属的时候原来漫反射为0（可理解为纯金属的时候进入材质的的光和金属发生光电反应，全部以热能的形式消耗掉），而此时\(F_0\)就是diff。
非金属的情况，当完全是非金属的时候，漫反射就是diff，此时\(F_0\)就是下面的Specular6.4 Specular
由金属的参数性质可知，Specular只有在非金属的情况才生效，根据非金属的菲涅尔的图标可以看出\(F_0\)的范围大致在0.03到0.08之间
根据6.3和6.4可得出\( diffuseColor = baseColor*(1-metallic) \)
\( specularColor = mix(0.08*specular,baseColor,metallic) \)
7.精确光源
\( f=f_d+f_r \)
\( =\frac{c_{diff}}{\pi } + \frac{FDG}{4 (n\cdot l) (n \cdot v)} \)
\( =\frac{c_{diff}}{\pi } + (F_0+(1-F_0)(1-v\cdot h)^5)(\frac{\alpha ^2}{\pi ((\mathrm{n} \cdot \mathrm{h})^2(\alpha ^2-1)+1)^2})((\frac{2(\mathrm{n} \cdot \mathrm{l})}{(\mathrm{n} \cdot \mathrm{l}) + \sqrt{\alpha^2+(1-\alpha^2)(\mathrm{n} \cdot \mathrm{l})^2 }})(\frac{2(\mathrm{n} \cdot \mathrm{v})}{(\mathrm{n} \cdot \mathrm{v}) + \sqrt{\alpha^2+(1-\alpha^2)(\mathrm{n} \cdot \mathrm{v})^2 }}))\frac{1}{4 (n\cdot l) (n \cdot v)} \)
\( =\frac{c_{diff}}{\pi } + (F_0+(1-F_0)(1-v\cdot h)^5)(\frac{\alpha ^2}{\pi ((\mathrm{n} \cdot \mathrm{h})^2(\alpha ^2-1)+1)^2})((\frac{1}{(\mathrm{n} \cdot \mathrm{l}) + \sqrt{\alpha^2+(1-\alpha^2)(\mathrm{n} \cdot \mathrm{l})^2 }})(\frac{1}{(\mathrm{n} \cdot \mathrm{v}) + \sqrt{\alpha^2+(1-\alpha^2)(\mathrm{n} \cdot \mathrm{v})^2 }})) \)
\( L_r=f*L_i*\cos\theta =f*\frac{E}{r^2}*(n \cdot l) \)
precision highp float;
varying vec3 vNormal;//法线
varying vec3 vPosition;//位置
uniform vec3 uLightPosition;//点灯光位置
uniform vec3 uLightColor;//点灯光颜色
uniform float uLightRadius;//点灯光半径
uniform vec3 uCamPosition;//摄像机位置
uniform vec3 uBaseColor;//基础色
uniform float uRoughness;//粗糙度
uniform float uMetallic;//金属
uniform float uSpecular;//高光
#define PI 3.14159265359
#define invPI 0.3183098861837697
#define invTWO_PI 0.15915494309
#define saturate(x) clamp(x, 0.0, 1.0)
vec3 Diffuse_Lambert( vec3 DiffuseColor )
{
return DiffuseColor * (1.0 / PI);
}
float D_GGX( float Roughness, float NoH )
{
float m = Roughness * Roughness;
float m2 = m * m;
float d = ( NoH * m2 - NoH ) * NoH + 1.0; // 2 mad
return m2 / ( PI*d*d ); // 4 mul, 1 rcp
}
float Vis_Smith( float Roughness, float NoV, float NoL )
{
float a = Roughness * Roughness ;
float a2 = a*a;
float Vis_SmithV = NoV + sqrt( NoV * (NoV - NoV * a2) + a2 );
float Vis_SmithL = NoL + sqrt( NoL * (NoL - NoL * a2) + a2 );
return 1.0 / ( Vis_SmithV * Vis_SmithL );
}
vec3 F_Schlick( vec3 SpecularColor, float VoH )
{
float Fc = pow( 1.0 - VoH, 5.0 );
return Fc + (1.0 - Fc) * SpecularColor;
}
float getAttenuation( vec3 lightPosition, vec3 vertexPosition, float lightRadius )
{
float r = lightRadius;
vec3 L = lightPosition - vertexPosition;
float dist = length(L);
float d = max( dist - r, 0.0 );
L /= dist;
float denom = d / r + 1.0;
float attenuation = 1.0 / (denom*denom);
float cutoff = 0.0052;
attenuation = (attenuation - cutoff) / (1.0 - cutoff);
attenuation = max(attenuation, 0.0);
return attenuation;
}
void main(void){
vec3 N = normalize( vNormal );
vec3 V = normalize( uCamPosition - vPosition );
vec3 L = normalize( uLightPosition - vPosition );
vec3 H = normalize(V + L);
float NoL = saturate( dot( N, L ) );
float NoV = saturate( dot( N, V ) );
float VoH = saturate( dot( V, H ) );
float NoH = saturate( dot( N, H ) );
vec3 diffuseColor = uBaseColor - uBaseColor * uMetallic;
vec3 specularColor = mix( vec3( 0.08 * uSpecular ), uBaseColor, uMetallic );
float attenuation = getAttenuation( uLightPosition, vPosition, uLightRadius );
float D = D_GGX(uRoughness,NoH);
float Vis = Vis_Smith(uRoughness,NoV,NoL);
vec3 F = F_Schlick(specularColor,VoH);
vec3 diffusePoint = Diffuse_Lambert(diffuseColor);
vec3 specularPoint = D * Vis * F;
vec3 colorPoint = uLightColor * ( diffusePoint + specularPoint ) * NoL * attenuation;
vec4 infoUv = vec4(colorPoint * 1.0 ,1.0);
gl_FragColor = infoUv;
}
8.环境光源
8.0 蒙特卡洛方法
对于一个连续函数f，它的积分公式
\( F=\int_{a}^{b}f(x)dx \)
对应f的蒙特卡洛积分公式
\( F^N=\frac{1}{N}\sum_{i=1}^{N}\frac{f(X_i)}{pdf(X_i)} \)
8.1 基本公式
\( \int _\Omega L_i(l)f(l,v)\cos\theta _i \mathrm{d}l \approx \frac{1}{N}\sum_{k=1}^{N}\frac{L_i(l_k)f(l_k,v)\cos\theta_l{_k}}{pdf(l_k,v)} \)
\( pdf=\frac{D(n \cdot h)}{4(v \cdot h)} \)
float3 ImportanceSampleGGX( float2 Xi, float Roughness , float3 N )
{
float a = Roughness * Roughness;
float Phi = 2 * PI * Xi.x;
float CosTheta = sqrt( (1 - Xi.y) / ( 1 + (a*a - 1) * Xi.y ) );
float SinTheta = sqrt( 1 - CosTheta * CosTheta );
float3 H;
H.x = SinTheta * cos( Phi );
H.y = SinTheta * sin( Phi );
H.z = CosTheta;
float3 UpVector = abs(N.z) < 0.999 ? float3(0,0,1) : float3(1,0,0);
float3 TangentX = normalize( cross( UpVector , N ) );
float3 TangentY = cross( N, TangentX );
// Tangent to world space
return TangentX * H.x + TangentY * H.y + N * H.z;
}
float3 SpecularIBL( float3 SpecularColor , float Roughness , float3 N, float3 V )
{
float3 SpecularLighting = 0;
const uint NumSamples = 1024;
for( uint i = 0; i < NumSamples; i++ )
{
float2 Xi = Hammersley( i, NumSamples );
float3 H = ImportanceSampleGGX( Xi, Roughness , N );
float3 L = 2 * dot( V, H ) * H - V;
float NoV = saturate( dot( N, V ) );
float NoL = saturate( dot( N, L ) );
float NoH = saturate( dot( N, H ) );
float VoH = saturate( dot( V, H ) );
if( NoL > 0 )
{
float3 SampleColor = EnvMap.SampleLevel( EnvMapSampler , L, 0 ).rgb;
float G = G_Smith( Roughness , NoV, NoL );
float Fc = pow( 1 - VoH, 5 );
float3 F = (1 - Fc) * SpecularColor + Fc;
// Incident light = SampleColor * NoL
// Microfacet specular = D*G*F / (4*NoL*NoV)
// pdf = D * NoH / (4 * VoH)
SpecularLighting += SampleColor * F * G * VoH / (NoH * NoV);
}
}
return SpecularLighting / NumSamples;
}
8.2 展开优化
\( \frac{1}{N}\sum_{k=1}^{N}\frac{L_i(l_k)f(l_k,v)\cos\theta_l{_k}}{pdf(l_k,v)} \approx \left ( \frac{1}{N}\sum_{k=1}^{N}L_i(l_k) \right )\left ( \frac{1}{N}\sum_{k=1}^{N} \frac{f(l_k,v)\cos\theta_l{_k}}{pdf(l_k,v)}\right ) \)
8.3 前部优化 Pre-Filtered Environment Map
对于前半部分，做如下处理，假定 n=v=r
\( \frac{1}{N}\sum_{k=1}^{N}L_i(l_k) \approx \mathrm{Cubemap. Sample (r, mip) } \)
float3 PrefilterEnvMap( float Roughness , float3 R )
{
float3 N = R;
float3 V = R;
float3 PrefilteredColor = 0;
const uint NumSamples = 1024;
for( uint i = 0; i < NumSamples; i++ )
{
float2 Xi = Hammersley( i, NumSamples );
float3 H = ImportanceSampleGGX( Xi, Roughness , N );
float3 L = 2 * dot( V, H ) * H - V;
float NoL = saturate( dot( N, L ) );
if( NoL > 0 )
{
PrefilteredColor += EnvMap.SampleLevel( EnvMapSampler , L, 0 ).rgb * NoL;
TotalWeight += NoL;
}
}
return PrefilteredColor / TotalWeight;
}
8.3 后部优化
8.3.1 基本做法
\( \frac{1}{N}\sum_{k=1}^{N} \frac{f(l_k,v)\cos\theta_l{_k}}{pdf(l_k,v)} = \frac{1}{N}\sum_{k=1}^{N} S \)
分解S
\( S = \frac{DFV \cos\theta }{pdf}\)
对F进行分解处理
\( F = F_0 +(1-F_0)(1-\cos\theta)^5 = F_0(1-(1-\cos\theta)^5) + (1-\cos\theta)^5 \)
\( = F_0(1-(1-v\cdot h)^5) + (1-v\cdot h)^5 = F_0(1-F_c) + F_c \)
其中\( F_c = （1-v\cdot h)^5 \)
处理pdf\( pdf = \frac{D(n \cdot h)}{4(v \cdot h)}\)
对S最终分解处理
\( S= \frac{4DFV(n \cdot l)(v \cdot h)}{D(n \cdot h)} = F \frac{4V(n \cdot l)(v \cdot h)}{(n \cdot h)}\)
\( = F_0(1-F_c)\frac{4V(n \cdot l)(v \cdot h)}{(n \cdot h)} + F_c\frac{4V(n \cdot l)(v \cdot h)}{(n \cdot h)} \)
此时给定\(NOV\)和\(Roughness\),\(Roughness\)在重要性采样中确定\(h\),那么\(n、v、h\)都有即可推算出\(l\)，那么上述式子的结果就只有和\(NOV、Roughness、F_0\)相关
通过数学技巧可以写成\(F_0*Scale+Offset,Scale和Offset\)这种线性的和Roughness、NoV相关，可以通过查表或者LUT的方式快速计算
float Vis_SmithJointApprox(float Roughness, float NoV, float NoL) {
float a = Roughness * Roughness;
float Vis_SmithV = NoL * (NoV * (1.0 - a) + a);
float Vis_SmithL = NoV * (NoL * (1.0 - a) + a);
return 0.5 / (Vis_SmithV + Vis_SmithL);
}
Vector2 IntegrateBRDF(Vector2 Random, float Roughness, float NoV){
Vector3 V;
V.x = sqrt(1.0 - NoV * NoV);// sin;
V.y = 0;
V.z = NoV;// # cos
float A = 0;
float B = 0;
int NumSamples = 128;
for (int i = 0;i < NumSamples;i++){
Vector2 E = Hammersley(i, NumSamples, Random);
Vector3 H = ImportanceSampleGGX(E, Roughness);
Vector3 L = H * (2 * V.dot(H)) - V;
float NoL = saturate(L.z);
float NoH = saturate(H.z);
float VoH = saturate(V.dot(H));
if (NoL > 0){
float Vis = Vis_SmithJointApprox(Roughness, NoV, NoL);
//#Incident light = NoL
//#pdf = D * NoH / (4 * VoH)
//#NoL * Vis / pdf
float NoL_Vis_PDF = NoL * Vis * (4 * VoH / NoH);
float Fc = pow(1 - VoH, 5);
A += (1 - Fc) * NoL_Vis_PDF;
B += Fc * NoL_Vis_PDF;
}
}
return Vector2(A / NumSamples, B / NumSamples);
}
最终生成的图片，运行时直接采样图片
8.3.2 再次优化
LUT的曲线本身是平滑的，所以可以用数学的方式（泰勒公式）来拟合曲线，下面是UE4的拟合代码
Vector2 EnvBRDFApprox(float Roughness,float NoV){
//# [ Lazarov 2013, "Getting More Physical in Call of Duty: Black Ops II" ]
//# Adaptation to fit our G term.
Vector4 c0 = Vector4(-1, -0.0275, -0.572, 0.022);
Vector4 c1 = Vector4(1, 0.0425, 1.04, -0.04);
Vector4 r = c0 * (Roughness) + c1;
float a004 = min(r.x * r.x, pow(2, -9.28 * NoV)) * r.x + r.y;
Vector2 AB = Vector2(-1.04, 1.04)*(a004) + Vector2(r.z, r.w);
return AB;
}
用上述的方式，拟合出来的图片可以和上图来对比
8.4 环境光的Diffuse
直接对原始的环境光图片按照consine分布采样，然后直接累加颜色值，然后重新生成Diffuse的环境光图片，使用的时候，只需要用点的normal去采样这个环境光图片，就是diffuse的光照结果，代码如下
Vector4 DiffuseIBL(Vector2 Random, Vector3 N, BitmapData &img, bool hdr){
Vector4 DiffuseLighting;
unsigned NumSamples = 4096 * 2;
//for i in range(NumSamples) :
for (unsigned int i = 0; i < NumSamples; i++){
Vector2 E = Hammersley(i, NumSamples, Random);
Vector3 C = CosineSampleHemisphere(E);
Vector3 L = TangentToWorld(C, N);
float NoL = saturate(N.dot(L));;
//#print("NoL", NoL);
if (NoL > 0){
Vector2 uv2 = pos2uv(L);
Vector4 result = img.getPixel32Int(uv2.x * img.width, uv2.y * img.height);
//#lambert = DiffuseColor * NoL / PI
//#pdf = NoL / PI
float e = 1.0;
if (hdr)
{
e = result.w - 128.0;
e = pow(2, e);
}
DiffuseLighting.x += result.x * e;
DiffuseLighting.y += result.y * e;
DiffuseLighting.z += result.z * e;
DiffuseLighting.w += 255;
}
}
float Weight = 1.0 / NumSamples;
DiffuseLighting = DiffuseLighting * (Weight);
return DiffuseLighting;
}
至此就是UE4关于物理渲染的整个推导流程