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#ifndef UNITY_BSDF_INCLUDED
#define UNITY_BSDF_INCLUDED
// Note: All NDF and diffuse term have a version with and without divide by PI.
// Version with divide by PI are use for direct lighting.
// Version without divide by PI are use for image based lighting where often the PI cancel during importance sampling
//-----------------------------------------------------------------------------
// Fresnel term
//-----------------------------------------------------------------------------
float F_Schlick(float f0, float f90, float u)
{
float x = 1.0 - u;
float x2 = x * x;
float x5 = x * x2 * x2;
return (f90 - f0) * x5 + f0; // sub mul mul mul sub mad
}
float F_Schlick(float f0, float u)
{
return F_Schlick(f0, 1.0, u); // sub mul mul mul sub mad
}
float3 F_Schlick(float3 f0, float f90, float u)
{
float x = 1.0 - u;
float x2 = x * x;
float x5 = x * x2 * x2;
return f0 * (1.0 - x5) + (f90 * x5); // sub mul mul mul sub mul mad*3
}
float3 F_Schlick(float3 f0, float u)
{
return F_Schlick(f0, 1.0, u); // sub mul mul mul sub mad*3
}
// Does not handle TIR.
float F_Transm_Schlick(float f0, float f90, float u)
{
float x = 1.0 - u;
float x2 = x * x;
float x5 = x * x2 * x2;
return (1.0 - f90 * x5) - f0 * (1.0 - x5); // sub mul mul mul mad sub mad
}
// Does not handle TIR.
float F_Transm_Schlick(float f0, float u)
{
return F_Transm_Schlick(f0, 1.0, u); // sub mul mul mad mad
}
// Does not handle TIR.
float3 F_Transm_Schlick(float3 f0, float f90, float u)
{
float x = 1.0 - u;
float x2 = x * x;
float x5 = x * x2 * x2;
return (1.0 - f90 * x5) - f0 * (1.0 - x5); // sub mul mul mul mad sub mad*3
}
// Does not handle TIR.
float3 F_Transm_Schlick(float3 f0, float u)
{
return F_Transm_Schlick(f0, 1.0, u); // sub mul mul mad mad*3
}
//-----------------------------------------------------------------------------
// Specular BRDF
//-----------------------------------------------------------------------------
// With analytical light (not image based light) we clamp the minimun roughness in the NDF to avoid numerical instability.
#define UNITY_MIN_ROUGHNESS 0.002
float ClampRoughnessForAnalyticalLights(float roughness)
{
return max(roughness, UNITY_MIN_ROUGHNESS);
}
float D_GGXNoPI(float NdotH, float roughness)
{
float a2 = roughness * roughness;
float f = (NdotH * a2 - NdotH) * NdotH + 1.0;
return a2 / (f * f);
}
float D_GGX(float NdotH, float roughness)
{
return INV_PI * D_GGXNoPI(NdotH, roughness);
}
// Ref: Understanding the Masking-Shadowing Function in Microfacet-Based BRDFs, p. 19, 29.
float G_MaskingSmithGGX(float NdotV, float roughness)
{
// G1(V, H) = HeavisideStep(VdotH) / (1 + Λ(V)).
// Λ(V) = -0.5 + 0.5 * sqrt(1 + 1 / a²).
// a = 1 / (roughness * tan(theta)).
// 1 + Λ(V) = 0.5 + 0.5 * sqrt(1 + roughness² * tan²(theta)).
// tan²(theta) = (1 - cos²(theta)) / cos²(theta) = 1 / cos²(theta) - 1.
// Assume that (VdotH > 0), e.i. (acos(LdotV) < Pi).
float a2 = roughness * roughness;
float z2 = NdotV * NdotV;
return 1 / (0.5 + 0.5 * sqrt(1.0 + a2 * (1.0 / z2 - 1.0)));
}
// Ref: Understanding the Masking-Shadowing Function in Microfacet-Based BRDFs, p. 12.
float D_GGX_Visible(float NdotH, float NdotV, float VdotH, float roughness)
{
return D_GGX(NdotH, roughness) * G_MaskingSmithGGX(NdotV, roughness) * VdotH / NdotV;
}
// Precompute part of lambdaV
float GetSmithJointGGXPartLambdaV(float NdotV, float roughness)
{
float a2 = roughness * roughness;
return sqrt((-NdotV * a2 + NdotV) * NdotV + a2);
}
// Note: V = G / (4 * NdotL * NdotV)
// Ref: http://jcgt.org/published/0003/02/03/paper.pdf
float V_SmithJointGGX(float NdotL, float NdotV, float roughness, float partLambdaV)
{
float a2 = roughness * roughness;
// Original formulation:
// lambda_v = (-1 + sqrt(a2 * (1 - NdotL2) / NdotL2 + 1)) * 0.5
// lambda_l = (-1 + sqrt(a2 * (1 - NdotV2) / NdotV2 + 1)) * 0.5
// G = 1 / (1 + lambda_v + lambda_l);
// Reorder code to be more optimal:
float lambdaV = NdotL * partLambdaV;
float lambdaL = NdotV * sqrt((-NdotL * a2 + NdotL) * NdotL + a2);
// Simplify visibility term: (2.0 * NdotL * NdotV) / ((4.0 * NdotL * NdotV) * (lambda_v + lambda_l));
return 0.5 / (lambdaV + lambdaL);
}
float V_SmithJointGGX(float NdotL, float NdotV, float roughness)
{
float partLambdaV = GetSmithJointGGXPartLambdaV(NdotV, roughness);
return V_SmithJointGGX(NdotL, NdotV, roughness, partLambdaV);
}
// Inline D_GGX() * V_SmithJointGGX() together for better code generation.
float DV_SmithJointGGX(float NdotH, float NdotL, float NdotV, float roughness, float partLambdaV)
{
float a2 = roughness * roughness;
float f = (NdotH * a2 - NdotH) * NdotH + 1.0;
float2 D = float2(a2, f * f); // Fraction without the constant (1/Pi)
float lambdaV = NdotL * partLambdaV;
float lambdaL = NdotV * sqrt((-NdotL * a2 + NdotL) * NdotL + a2);
float2 G = float2(1, lambdaV + lambdaL); // Fraction without the constant (0.5)
return (INV_PI * 0.5) * (D.x * G.x) / (D.y * G.y);
}
float DV_SmithJointGGX(float NdotH, float NdotL, float NdotV, float roughness)
{
float partLambdaV = GetSmithJointGGXPartLambdaV(NdotV, roughness);
return DV_SmithJointGGX(NdotH, NdotL, NdotV, roughness, partLambdaV);
}
// Precompute a part of LambdaV.
// Note on this linear approximation.
// Exact for roughness values of 0 and 1. Also, exact when the cosine is 0 or 1.
// Otherwise, the worst case relative error is around 10%.
// https://www.desmos.com/calculator/wtp8lnjutx
float GetSmithJointGGXPartLambdaVApprox(float NdotV, float roughness)
{
float a = roughness;
return NdotV * (1 - a) + a;
}
float V_SmithJointGGXApprox(float NdotL, float NdotV, float roughness, float partLambdaV)
{
float a = roughness;
float lambdaV = NdotL * partLambdaV;
float lambdaL = NdotV * (NdotL * (1 - a) + a);
return 0.5 / (lambdaV + lambdaL);
}
float V_SmithJointGGXApprox(float NdotL, float NdotV, float roughness)
{
float partLambdaV = GetSmithJointGGXPartLambdaVApprox(NdotV, roughness);
return V_SmithJointGGXApprox(NdotL, NdotV, roughness, partLambdaV);
}
// roughnessT -> roughness in tangent direction
// roughnessB -> roughness in bitangent direction
float D_GGXAnisoNoPI(float TdotH, float BdotH, float NdotH, float roughnessT, float roughnessB)
{
float aT2 = roughnessT * roughnessT;
float aB2 = roughnessB * roughnessB;
float f = TdotH * TdotH / aT2 + BdotH * BdotH / aB2 + NdotH * NdotH;
return 1.0 / (roughnessT * roughnessB * f * f);
}
float D_GGXAniso(float TdotH, float BdotH, float NdotH, float roughnessT, float roughnessB)
{
return INV_PI * D_GGXAnisoNoPI(TdotH, BdotH, NdotH, roughnessT, roughnessB);
}
float GetSmithJointGGXAnisoPartLambdaV(float TdotV, float BdotV, float NdotV, float roughnessT, float roughnessB)
{
float aT2 = roughnessT * roughnessT;
float aB2 = roughnessB * roughnessB;
return sqrt(aT2 * TdotV * TdotV + aB2 * BdotV * BdotV + NdotV * NdotV);
}
// Note: V = G / (4 * NdotL * NdotV)
// Ref: https://cedec.cesa.or.jp/2015/session/ENG/14698.html The Rendering Materials of Far Cry 4
float V_SmithJointGGXAniso(float TdotV, float BdotV, float NdotV, float TdotL, float BdotL, float NdotL, float roughnessT, float roughnessB, float partLambdaV)
{
float aT2 = roughnessT * roughnessT;
float aB2 = roughnessB * roughnessB;
float lambdaV = NdotL * partLambdaV;
float lambdaL = NdotV * sqrt(aT2 * TdotL * TdotL + aB2 * BdotL * BdotL + NdotL * NdotL);
return 0.5 / (lambdaV + lambdaL);
}
float V_SmithJointGGXAniso(float TdotV, float BdotV, float NdotV, float TdotL, float BdotL, float NdotL, float roughnessT, float roughnessB)
{
float partLambdaV = GetSmithJointGGXAnisoPartLambdaV(TdotV, BdotV, NdotV, roughnessT, roughnessB);
return V_SmithJointGGXAniso(TdotV, BdotV, NdotV, TdotL, BdotL, NdotL, roughnessT, roughnessB, partLambdaV);
}
// Inline D_GGXAniso() * V_SmithJointGGXAniso() together for better code generation.
float DV_SmithJointGGXAniso(float TdotH, float BdotH, float NdotH,
float TdotV, float BdotV, float NdotV,
float TdotL, float BdotL, float NdotL,
float roughnessT, float roughnessB, float partLambdaV)
{
float aT2 = roughnessT * roughnessT;
float aB2 = roughnessB * roughnessB;
float f = TdotH * TdotH / aT2 + BdotH * BdotH / aB2 + NdotH * NdotH;
float2 D = float2(1, roughnessT * roughnessB * f * f); // Fraction without the constant (1/Pi)
float lambdaV = NdotL * partLambdaV;
float lambdaL = NdotV * sqrt(aT2 * TdotL * TdotL + aB2 * BdotL * BdotL + NdotL * NdotL);
float2 G = float2(1, lambdaV + lambdaL); // Fraction without the constant (0.5)
return (INV_PI * 0.5) * (D.x * G.x) / (D.y * G.y);
}
float DV_SmithJointGGXAniso(float TdotH, float BdotH, float NdotH,
float TdotV, float BdotV, float NdotV,
float TdotL, float BdotL, float NdotL,
float roughnessT, float roughnessB)
{
float partLambdaV = GetSmithJointGGXAnisoPartLambdaV(TdotV, BdotV, NdotV, roughnessT, roughnessB);
return DV_SmithJointGGXAniso(TdotH, BdotH, NdotH,
TdotV, BdotV, NdotV,
TdotL, BdotL, NdotL,
roughnessT, roughnessB, partLambdaV);
}
//-----------------------------------------------------------------------------
// Diffuse BRDF - diffuseColor is expected to be multiply by the caller
//-----------------------------------------------------------------------------
float LambertNoPI()
{
return 1.0;
}
float Lambert()
{
return INV_PI;
}
float DisneyDiffuseNoPI(float NdotV, float NdotL, float LdotV, float perceptualRoughness)
{
// (2 * LdotH * LdotH) = 1 + LdotV
// float fd90 = 0.5 + 2 * LdotH * LdotH * perceptualRoughness;
float fd90 = 0.5 + (perceptualRoughness + perceptualRoughness * LdotV);
// Two schlick fresnel term
float lightScatter = F_Schlick(1.0, fd90, NdotL);
float viewScatter = F_Schlick(1.0, fd90, NdotV);
// Normalize the BRDF for polar view angles of up to (Pi/4).
// We use the worst case of (roughness = albedo = 1), and, for each view angle,
// integrate (brdf * cos(theta_light)) over all light directions.
// The resulting value is for (theta_view = 0), which is actually a little bit larger
// than the value of the integral for (theta_view = Pi/4).
// Hopefully, the compiler folds the constant together with (1/Pi).
return rcp(1.03571) * (lightScatter * viewScatter);
}
float DisneyDiffuse(float NdotV, float NdotL, float LdotV, float perceptualRoughness)
{
return INV_PI * DisneyDiffuseNoPI(NdotV, NdotL, LdotV, perceptualRoughness);
}
// Ref: Diffuse Lighting for GGX + Smith Microsurfaces, p. 113.
float3 DiffuseGGXNoPI(float3 albedo, float NdotV, float NdotL, float NdotH, float LdotV, float roughness)
{
float facing = 0.5 + 0.5 * LdotV; // (LdotH)^2
float rough = facing * (0.9 - 0.4 * facing) * (0.5 / NdotH + 1);
float transmitL = F_Transm_Schlick(0, NdotL);
float transmitV = F_Transm_Schlick(0, NdotV);
float smooth = transmitL * transmitV * 1.05; // Normalize F_t over the hemisphere
float single = lerp(smooth, rough, roughness); // Rescaled by PI
float multiple = roughness * (0.1159 * PI); // Rescaled by PI
return single + albedo * multiple;
}
float3 DiffuseGGX(float3 albedo, float NdotV, float NdotL, float NdotH, float LdotV, float roughness)
{
// Note that we could save 2 cycles by inlining the multiplication by INV_PI.
return INV_PI * DiffuseGGXNoPI(albedo, NdotV, NdotL, NdotH, LdotV, roughness);
}
#endif // UNITY_BSDF_INCLUDED