该文章以及代码主要来自
图形学论文解析与复现：【论文复现】An Efficient Representation for Irradiance Environment Maps
作者：Monica的小甜甜

与原文的不同：

• 对一些有问题的地方进行了修改
• 添加了注释
• 对有疑问的地方添加了疑问点
• 引入了其他一些Blog填充了原文中忽略的信息

1、预计算球面谐波函数系数

首先根据上一篇【球谐函数在环境光照中的使用原理】得到的最终公式：

我们需要预计算$L_l^m$的值。计算公式为：

$\Omega$为球面积分，这里对应对天空盒逐像素积分。
积分代码为：

void Harmonics::Evaluate()//求值
{m_Coefs = vector<glm::vec3>(m_Degree, glm::vec3());//6张图for (int k = 0; k < 6; k++){cv::Mat img = m_Images[k];int w = m_Images[k].cols;int h = m_Images[k].rows;//逐像素for (int j = 0; j < w; j++){for (int i = 0; i < h; i++){// 像素点位置float px = (float)i + 0.5;float py = (float)j + 0.5;// 像素点UV 【-1,1】：以摄像机正对位置的（0，0）float u = 2.0 * (px / (float)w) - 1.0;float v = 2.0 * (py / (float)h) - 1.0;// 像素间UV的一半的偏移量float d_x = 1.0 / (float)w;// (x0,y0)像素左下角 (x1,y1)像素右上角float x0 = u - d_x;float y0 = v - d_x;float x1 = u + d_x;float y1 = v + d_x;// 计算Cubemap的一个像素对应的立体角的大小float d_a = surfaceArea(x0, y0) - surfaceArea(x0, y1) - surfaceArea(x1, y0) + surfaceArea(x1, y1);// 纹理像素点 转化为 世界坐标点u = (float)j / (img.cols - 1);v = 1.0f - (float)i / (img.rows - 1);glm::vec3 p = CubeUV2XYZ({ k, u, v });// 获取当前像素颜色auto c = img.at<cv::Vec3f>(i, j);glm::vec3 color = {c[2], c[1], c[0]};// 得到基函数计算结果列表vector<float> Y = Basis(p);// 计算系数for (int i = 0; i < m_Degree; i++){m_Coefs[i] = m_Coefs[i] + Y[i] * color * d_a;}}}}
}

其中 计算Cubemap的一个像素对应的立体角的大小原理可参照
Solid Angle of A Cubemap Texel - 计算Cubemap的一个像素对应的立体角的大小

我们将得到的积分结果保存在一个文件中【SHCoefficients.txt】,用于之后读取。

2、预计算BRDF的LUT图

LUT（Look up Table）图，预计算了任意一个天空盒下，已知法线和视口的夹角以及材质粗糙度，查找得到Frenel项。

然而这个LUT图和IBL中的LUT有一些不同。
因为IBL中的LUT加入了 $n\cdot w$ 光照衰减项。
而在球谐函数中， $n\cdot w$ 作为$t_l参与运算$，因此在球谐函数的IBL中删除了$n\cdot w$。

main函数计算

	for(int i = 0; i < N; i++){for (int j = 0; j < N; j++){float NoV = (i + 0.5f) * (1.0f / N);float roughness = (j + 0.5f) * (1.0f / N);glm::vec2 eval = IntegrateBRDF(NoV, roughness);tex.store<glm::vec2>({ i, N - j - 1 }, 0, eval);}}

其他被调用函数

const float PI = 3.14159265358979323846264338327950288;float RadicalInverse_VdC(unsigned int bits)
{bits = (bits << 16u) | (bits >> 16u);bits = ((bits & 0x55555555u) << 1u) | ((bits & 0xAAAAAAAAu) >> 1u);bits = ((bits & 0x33333333u) << 2u) | ((bits & 0xCCCCCCCCu) >> 2u);bits = ((bits & 0x0F0F0F0Fu) << 4u) | ((bits & 0xF0F0F0F0u) >> 4u);bits = ((bits & 0x00FF00FFu) << 8u) | ((bits & 0xFF00FF00u) >> 8u);return float(bits) * 2.3283064365386963e-10;
}glm::vec2 Hammersley(unsigned int i, unsigned int N)
{return glm::vec2(float(i) / float(N), RadicalInverse_VdC(i));
}glm::vec3 ImportanceSampleGGX(glm::vec2 Xi, float roughness, glm::vec3 N)
{float a = roughness * roughness;float phi = 2.0 * PI * Xi.x;float cosTheta = sqrt((1.0 - Xi.y) / (1.0 + (a*a - 1.0) * Xi.y));float sinTheta = sqrt(1.0 - cosTheta * cosTheta);// from spherical coordinates to cartesian coordinatesglm::vec3 H;H.x = cos(phi) * sinTheta;H.y = sin(phi) * sinTheta;H.z = cosTheta;// from tangent-space vector to world-space sample vectorglm::vec3 up = abs(N.z) < 0.999 ? glm::vec3(0.0, 0.0, 1.0) : glm::vec3(1.0, 0.0, 0.0);glm::vec3 tangent = normalize(cross(up, N));glm::vec3 bitangent = cross(N, tangent);glm::vec3 sampleVec = tangent * H.x + bitangent * H.y + N * H.z;return normalize(sampleVec);
}float GeometrySchlickGGX(float NdotV, float roughness)
{float a = roughness;float k = (a * a) / 2.0;float nom = NdotV;float denom = NdotV * (1.0 - k) + k;return nom / denom;
}float GeometrySmith(float roughness, float NoV, float NoL)
{float ggx2 = GeometrySchlickGGX(NoV, roughness);float ggx1 = GeometrySchlickGGX(NoL, roughness);return ggx1 * ggx2;
}glm::vec2 IntegrateBRDF(float NdotV, float roughness, unsigned int samples = 1024)
{glm::vec3 V;V.x = sqrt(1.0 - NdotV * NdotV);V.y = 0.0;V.z = NdotV;float A = 0.0;float B = 0.0;glm::vec3 N = glm::vec3(0.0, 0.0, 1.0);for (unsigned int i = 0u; i < samples; ++i){glm::vec2 Xi = Hammersley(i, samples);glm::vec3 H = ImportanceSampleGGX(Xi, roughness, N);glm::vec3 L = normalize(2.0f * dot(V, H) * H - V);float NoL = glm::max(L.z, 0.0f);float NoH = glm::max(H.z, 0.0f);float VoH = glm::max(dot(V, H), 0.0f);float NoV = glm::max(dot(N, V), 0.0f);if (NoL > 0.0){float G = GeometrySmith(roughness, NoV, NoL);float G_Vis = (G * VoH) / (NoH * NoV) / NoL;float Fc = pow(1.0 - VoH, 5.0);A += (1.0 - Fc) * G_Vis;B += Fc * G_Vis;}}return glm::vec2(A / float(samples), B / float(samples));
}

3、将计算数据传入Shader

• 传入BRDFLUT纹理
• 传入球谐函数系数列表

void CShadingPass::initV()
{auto m_LUTTexture = std::make_shared<ElayGraphics::STexture>();loadTextureFromFile("../Textures/BRDFLUT/BRDFLut.dds", m_LUTTexture);getCoefs();ElayGraphics::Camera::setMainCameraFarPlane(100);ElayGraphics::Camera::setMainCameraPos({ -1.57278, 0.244948, 0.367194 });ElayGraphics::Camera::setMainCameraFront({ 0.967832, -0.112856, -0.224865 });ElayGraphics::Camera::setMainCameraMoveSpeed(0.5);m_pShader = std::make_shared<CShader>("Sponza_VS.glsl", "Sponza_FS.glsl");m_pSponza = std::dynamic_pointer_cast<CSponza>(ElayGraphics::ResourceManager::getGameObjectByName("Sponza"));m_pShader->activeShader();m_pShader->setTextureUniformValue("u_BRDFLut", m_LUTTexture);m_pShader->setMat4UniformValue("u_ModelMatrix", glm::value_ptr(m_pSponza->getModelMatrix()));for (int i = 0; i < m_Coefs.size(); i++){m_pShader->setFloatUniformValue("u_Coef[" + std::to_string(i) + "]", m_Coefs[i].x, m_Coefs[i].y, m_Coefs[i].z);}m_pSponza->initModel(*m_pShader);
}

4、 Draw

#version 430 corein  vec3 v2f_FragPosInViewSpace;
in  vec2 v2f_TexCoords;
in  vec3 v2f_ViewSpaceNormal;
in  vec3 v2f_WorldSpaceNormal;layout (location = 0) out vec4 Albedo_;const float PI = 3.1415926535897932384626433832795;
uniform vec3 u_Coef[16];
uniform vec3 u_DiffuseColor;
uniform sampler2D u_BRDFLut;vec3 FresnelSchlickRoughness(float cosTheta, vec3 F0, float roughness)
{return F0 + (max(vec3(1.0 - roughness), F0) - F0) * pow(max(1.0 - cosTheta, 0.0), 5.0);
}  void main()
{	if((abs(v2f_ViewSpaceNormal.x) < 0.0001f) && (abs(v2f_ViewSpaceNormal.y) < 0.0001f) && (abs(v2f_ViewSpaceNormal.z) < 0.0001f)){Albedo_ = vec4(0, 0, 0, 1);return;}float Basis[9];float x = v2f_WorldSpaceNormal.x;float y = v2f_WorldSpaceNormal.y;float z = v2f_WorldSpaceNormal.z;float x2 = x * x;float y2 = y * y;float z2 = z * z;//这里所有系数应该为乘PI------------------个人认为Basis[0] = 1.f / 2.f * sqrt(1.f / PI);Basis[1] = 2.0 / 3.0 * sqrt(3.f / 4.f * PI) * z;Basis[2] = 2.0 / 3.0 * sqrt(3.f / 4.f * PI) * y;Basis[3] = 2.0 / 3.0 * sqrt(3.f / 4.f * PI) * x;Basis[4] = 1.0 / 4.0 * 1.f / 2.f * sqrt(15.f * PI) * x * z;Basis[5] = 1.0 / 4.0 * 1.f / 2.f * sqrt(15.f * PI) * z * y;Basis[6] = 1.0 / 4.0 * 1.f / 4.f * sqrt(5.f * PI) * (-x2 - z2 + 2 * y2);Basis[7] = 1.0 / 4.0 * 1.f / 2.f * sqrt(15.f * PI) * y * x;Basis[8] = 1.0 / 4.0 * 1.f / 4.f * sqrt(15.f * PI) * (x2 - z2);vec3 Diffuse = vec3(0,0,0);vec3 F0 = vec3(0.2,0.2,0.2);float Roughness = 0.5;vec3 N = normalize(vec4(v2f_ViewSpaceNormal,1.0f)).xyz;//viewMatrix * vec3 V = -normalize(v2f_FragPosInViewSpace);//vec3 R = reflect(-V, N); F0        = FresnelSchlickRoughness(max(dot(N, V), 0.0), F0, Roughness);vec2 EnvBRDF  = texture(u_BRDFLut, vec2(max(dot(N, V), 0.0), Roughness)).rg;vec3 LUT = (F0 * EnvBRDF.x + EnvBRDF.y);for (int i = 0; i < 9; i++)Diffuse += u_Coef[i] * Basis[i] * (1-LUT);Albedo_ = vec4(Diffuse);
}

结果展示

只有漫反射的效果：

只有镜面反射的效果：