Energy conservation improvements for generated GLSL

- Add mx_microfacet_ggx_directional_albedo for computing a directional albedo for the GGX/Smith/Schlick BRDF. - Replace mirror Fresnel with directional albedo for BSDF layering. - Move Fresnel computation into the lighting integral for filtered importance sampling. - Add mx_average_roughness for converting from anisotropic to isotropic roughness.
AcademySoftwareFoundation · Apr 9, 2020 · 9ab05b8 · 9ab05b8
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diff --git a/libraries/pbrlib/genglsl/lib/mx_bsdfs.glsl b/libraries/pbrlib/genglsl/lib/mx_bsdfs.glsl
@@ -1,3 +1,10 @@
+#include "pbrlib/genglsl/lib/mx_refraction_index.glsl"
+
+float mx_average_roughness(vec2 roughness)
+{
+    return sqrt(roughness.x * roughness.y);
+}
+
 float mx_orennayar(vec3 L, vec3 V, vec3 N, float NdotL, float roughness)
 {
     float LdotV = dot(L, V);
@@ -52,6 +59,22 @@ float mx_microfacet_ggx_smith_G(float NdotL, float NdotV, float alpha)
     return 2.0 / (lambdaL / NdotL + lambdaV / NdotV);
 }
 
+// https://www.unrealengine.com/blog/physically-based-shading-on-mobile
+vec3 mx_microfacet_ggx_directional_albedo(float NdotV, float roughness, vec3 F0, vec3 F90)
+{
+    const vec4 c0 = vec4(-1, -0.0275, -0.572, 0.022);
+    const vec4 c1 = vec4( 1,  0.0425,  1.04, -0.04 );
+    vec4 r = roughness * c0 + c1;
+    float a004 = min(r.x * r.x, exp2(-9.28 * NdotV)) * r.x + r.y;
+    vec2 AB = vec2(-1.04, 1.04) * a004 + r.zw;
+    return F0 * AB.x + F90 * AB.y;
+}
+
+float mx_microfacet_ggx_directional_albedo(float NdotV, float roughness, float ior)
+{
+    return mx_microfacet_ggx_directional_albedo(NdotV, roughness, vec3(mx_ior_to_f0(ior)), vec3(1.0)).x;
+}
+
 // http://blog.selfshadow.com/publications/s2017-shading-course/imageworks/s2017_pbs_imageworks_sheen.pdf (Equation 2)
 float mx_microfacet_sheen_NDF(float cosTheta, float roughness)
 {
@@ -117,37 +140,19 @@ vec3 mx_fresnel_schlick(float cosTheta, vec3 F0, vec3 F90, float exponent)
 
 vec3 mx_fresnel_schlick(float cosTheta, vec3 F0)
 {
-    if (cosTheta < 0.0)
-        return vec3(1.0);
-    float x = 1.0 - cosTheta;
-    float x2 = x*x;
-    float x5 = x2*x2*x;
+    float x = clamp(1.0 - cosTheta, 0.0, 1.0);
+    float x5 = mx_pow5(x);
     return F0 + (1.0 - F0) * x5;
 }
 
 float mx_fresnel_schlick(float cosTheta, float ior)
 {
-    if (cosTheta < 0.0)
-        return 1.0;
-    float F0 = (ior - 1.0) / (ior + 1.0);
-    F0 *= F0;
-    float x = 1.0 - cosTheta;
-    float x2 = x*x;
-    float x5 = x2*x2*x;
+    float x = clamp(1.0 - cosTheta, 0.0, 1.0);
+    float x5 = mx_pow5(x);
+    float F0 = mx_ior_to_f0(ior);
     return F0 + (1.0 - F0) * x5;
 }
 
-float mx_fresnel_schlick_roughness(float cosTheta, float ior, float roughness)
-{
-    cosTheta = abs(cosTheta);
-    float F0 = (ior - 1.0) / (ior + 1.0);
-    F0 *= F0;
-    float x = 1.0 - cosTheta;
-    float x2 = x*x;
-    float x5 = x2*x2*x;
-    return F0 + (max(1.0 - roughness, F0) - F0) * x5;
-}
-
 // https://seblagarde.wordpress.com/2013/04/29/memo-on-fresnel-equations/
 float mx_fresnel_dielectric(float cosTheta, float ior)
 {

diff --git a/libraries/pbrlib/genglsl/lib/mx_environment_fis.glsl b/libraries/pbrlib/genglsl/lib/mx_environment_fis.glsl
@@ -30,7 +30,7 @@ vec3 mx_latlong_map_lookup(vec3 dir, mat4 transform, float lod, sampler2D sample
 }
 
 // Only GGX is supported for now and the distribution argument is ignored
-vec3 mx_environment_radiance(vec3 N, vec3 V, vec3 X, vec2 roughness, int distribution)
+vec3 mx_environment_radiance(vec3 N, vec3 V, vec3 X, vec2 roughness, vec3 F0, vec3 F90, int distribution)
 {
     vec3 Y = normalize(cross(N, X));
     X = cross(Y, N);
@@ -60,20 +60,19 @@ vec3 mx_environment_radiance(vec3 N, vec3 V, vec3 X, vec2 roughness, int distrib
         float lod = mx_latlong_compute_lod(L, pdf, $envRadianceMips - 1, $envRadianceSamples);
         vec3 sampleColor = mx_latlong_map_lookup(L, $envMatrix, lod, $envRadiance);
 
+        // Compute the Fresnel term.
+        vec3 F = mx_fresnel_schlick(VdotH, F0, F90, 5.0);
+
         // Compute the geometric term.
-        float G = mx_microfacet_ggx_smith_G(NdotL, NdotV, max(roughness.x, roughness.y));
+        float G = mx_microfacet_ggx_smith_G(NdotL, NdotV, mx_average_roughness(roughness));
 
-        // Fresnel is applied outside the lighting integral for now.
-        // TODO: Move Fresnel term into the lighting integral.
-        float F = 1.0;
-
         // Add the radiance contribution of this sample.
         // From https://cdn2.unrealengine.com/Resources/files/2013SiggraphPresentationsNotes-26915738.pdf
         //   incidentLight = sampleColor * NdotL
-        //   microfacetSpecular = D * G * F / (4 * NdotL * NdotV)
+        //   microfacetSpecular = D * F * G / (4 * NdotL * NdotV)
         //   pdf = D * NdotH / (4 * VdotH)
         //   radiance = incidentLight * microfacetSpecular / pdf
-        radiance += sampleColor * G * F * VdotH / (NdotV * NdotH);
+        radiance += sampleColor * F * G * VdotH / (NdotV * NdotH);
     }
 
     // Normalize and return the final radiance.

diff --git a/libraries/pbrlib/genglsl/lib/mx_environment_none.glsl b/libraries/pbrlib/genglsl/lib/mx_environment_none.glsl
@@ -1,4 +1,4 @@
-vec3 mx_environment_radiance(vec3 N, vec3 V, vec3 X, vec2 roughness, int distribution)
+vec3 mx_environment_radiance(vec3 N, vec3 V, vec3 X, vec2 roughness, vec3 F0, vec3 F90, int distribution)
 {
     return vec3(0.0);
 }

diff --git a/libraries/pbrlib/genglsl/lib/mx_environment_prefilter.glsl b/libraries/pbrlib/genglsl/lib/mx_environment_prefilter.glsl
@@ -27,10 +27,13 @@ vec3 mx_latlong_map_lookup(vec3 dir, mat4 transform, float lodBias, sampler2D sa
     return vec3(0.0);
 }
 
-vec3 mx_environment_radiance(vec3 N, vec3 V, vec3 X, vec2 roughness, int distribution)
+vec3 mx_environment_radiance(vec3 N, vec3 V, vec3 X, vec2 roughness, vec3 F0, vec3 F90, int distribution)
 {
-    vec3 dir = reflect(-V, N);
-    return mx_latlong_map_lookup(dir, $envMatrix, max(roughness.x, roughness.y), $envRadiance);
+    vec3 L = reflect(-V, N);
+    float NdotV = dot(N, V);
+
+    vec3 dirAlbedo = mx_microfacet_ggx_directional_albedo(NdotV, mx_average_roughness(roughness), F0, F90);
+    return mx_latlong_map_lookup(L, $envMatrix, mx_average_roughness(roughness), $envRadiance) * dirAlbedo;
 }
 
 vec3 mx_environment_irradiance(vec3 N)

diff --git a/libraries/pbrlib/genglsl/lib/mx_refraction_index.glsl b/libraries/pbrlib/genglsl/lib/mx_refraction_index.glsl
@@ -1,3 +1,9 @@
+// Convert a real-valued index of refraction to normal-incidence reflectivity.
+float mx_ior_to_f0(float ior)
+{
+    return mx_square((ior - 1.0) / (ior + 1.0));
+}
+
 // "Artist Friendly Metallic Fresnel", Ole Gulbrandsen, 2014
 // http://jcgt.org/published/0003/04/03/paper.pdf
 

diff --git a/libraries/pbrlib/genglsl/mx_conductor_brdf.glsl b/libraries/pbrlib/genglsl/mx_conductor_brdf.glsl
@@ -1,5 +1,4 @@
 #include "pbrlib/genglsl/lib/mx_bsdfs.glsl"
-#include "pbrlib/genglsl/lib/mx_refraction_index.glsl"
 
 void mx_conductor_brdf_reflection(vec3 L, vec3 V, float weight, vec3 reflectivity, vec3 edge_color, vec2 roughness, vec3 N, vec3 X, int distribution, out BSDF result)
 {
@@ -21,19 +20,17 @@ void mx_conductor_brdf_reflection(vec3 L, vec3 V, float weight, vec3 reflectivit
 
     vec3 H = normalize(L + V);
     float NdotH = dot(N, H);
-
-    float D = mx_microfacet_ggx_NDF(X, Y, H, NdotH, roughness.x, roughness.y);
-    float G = mx_microfacet_ggx_smith_G(NdotL, NdotV, max(roughness.x, roughness.y));
+    float VdotH = dot(V, H);
 
     vec3 ior_n, ior_k;
     mx_artistic_to_complex_ior(reflectivity, edge_color, ior_n, ior_k);
 
-    float VdotH = dot(V, H);
+    float D = mx_microfacet_ggx_NDF(X, Y, H, NdotH, roughness.x, roughness.y);
     vec3 F = mx_fresnel_conductor(VdotH, ior_n, ior_k);
-    F *= weight;
+    float G = mx_microfacet_ggx_smith_G(NdotL, NdotV, mx_average_roughness(roughness));
 
     // Note: NdotL is cancelled out
-    result = F * D * G / (4 * NdotV);
+    result = D * F * G * weight / (4 * NdotV);
 }
 
 void mx_conductor_brdf_indirect(vec3 V, float weight, vec3 reflectivity, vec3 edge_color, vec2 roughness, vec3 N, vec3 X, int distribution, out vec3 result)
@@ -47,8 +44,7 @@ void mx_conductor_brdf_indirect(vec3 V, float weight, vec3 reflectivity, vec3 ed
     vec3 ior_n, ior_k;
     mx_artistic_to_complex_ior(reflectivity, edge_color, ior_n, ior_k);
 
-    vec3 Li = mx_environment_radiance(N, V, X, roughness, distribution);
+    vec3 Li = mx_environment_radiance(N, V, X, roughness, vec3(1.0), vec3(1.0), distribution);
     vec3 F = mx_fresnel_conductor(dot(N, V), ior_n, ior_k);
-    F *= weight;
-    result = Li * F;
+    result = Li * F * weight;
 }
diff --git a/libraries/pbrlib/genglsl/mx_dielectric_brdf.glsl b/libraries/pbrlib/genglsl/mx_dielectric_brdf.glsl
@@ -20,17 +20,17 @@ void mx_dielectric_brdf_reflection(vec3 L, vec3 V, float weight, vec3 tint, floa
 
     vec3 H = normalize(L + V);
     float NdotH = dot(N, H);
+    float VdotH = dot(V, H);
 
     float D = mx_microfacet_ggx_NDF(X, Y, H, NdotH, roughness.x, roughness.y);
-    float G = mx_microfacet_ggx_smith_G(NdotL, NdotV, max(roughness.x, roughness.y));
-
-    float VdotH = dot(V, H);
     float F = mx_fresnel_schlick(VdotH, ior);
-    F *= weight;
+    float G = mx_microfacet_ggx_smith_G(NdotL, NdotV, mx_average_roughness(roughness));
+
+    float dirAlbedo = mx_microfacet_ggx_directional_albedo(NdotV, mx_average_roughness(roughness), ior);
 
     // Note: NdotL is cancelled out
-    result = tint * D * G * F / (4 * NdotV) // Top layer reflection
-           + base * (1.0 - F);              // Base layer reflection attenuated by top fresnel
+    result = D * F * G * tint * weight / (4 * NdotV)    // Top layer reflection
+           + base * (1.0 - dirAlbedo * weight);         // Base layer reflection attenuated by top directional albedo
 }
 
 void mx_dielectric_brdf_transmission(vec3 V, float weight, vec3 tint, float ior, vec2 roughness, vec3 N, vec3 X, int distribution, BSDF base, out BSDF result)
@@ -46,11 +46,10 @@ void mx_dielectric_brdf_transmission(vec3 V, float weight, vec3 tint, float ior,
     // inverse of top layer reflectance.
 
     // Abs here to allow transparency through backfaces
-    float NdotV = abs(dot(N,V));
-    float F = mx_fresnel_schlick(NdotV, ior);
-    F *= weight;
+    float NdotV = abs(dot(N, V));
+    float dirAlbedo = mx_microfacet_ggx_directional_albedo(NdotV, mx_average_roughness(roughness), ior);
 
-    result = base * (1.0 - F); // Base layer transmission attenuated by top fresnel
+    result = base * (1.0 - dirAlbedo * weight); // Base layer transmission attenuated by top directional albedo
 }
 
 void mx_dielectric_brdf_indirect(vec3 V, float weight, vec3 tint, float ior, vec2 roughness, vec3 N, vec3 X, int distribution, BSDF base, out BSDF result)
@@ -61,12 +60,11 @@ void mx_dielectric_brdf_indirect(vec3 V, float weight, vec3 tint, float ior, vec
         return;
     }
 
-    vec3 Li = mx_environment_radiance(N, V, X, roughness, distribution);
+    vec3 Li = mx_environment_radiance(N, V, X, roughness, vec3(mx_ior_to_f0(ior)), vec3(1.0), distribution);
 
-    float NdotV = dot(N,V);
-    float F = mx_fresnel_schlick_roughness(NdotV, ior, max(roughness.x, roughness.y));
-    F *= weight;
+    float NdotV = dot(N, V);
+    float dirAlbedo = mx_microfacet_ggx_directional_albedo(NdotV, mx_average_roughness(roughness), ior);
 
-    result = Li * tint * F     // Top layer reflection
-           + base * (1.0 - F); // Base layer reflection attenuated by top fresnel
+    result = Li * tint * weight                 // Top layer reflection
+           + base * (1.0 - dirAlbedo * weight); // Base layer reflection attenuated by top directional albedo
 }
diff --git a/libraries/pbrlib/genglsl/mx_generalized_schlick_brdf.glsl b/libraries/pbrlib/genglsl/mx_generalized_schlick_brdf.glsl
@@ -20,18 +20,18 @@ void mx_generalized_schlick_brdf_reflection(vec3 L, vec3 V, float weight, vec3 c
 
     vec3 H = normalize(L + V);
     float NdotH = dot(N, H);
+    float VdotH = dot(V, H);
 
     float D = mx_microfacet_ggx_NDF(X, Y, H, NdotH, roughness.x, roughness.y);
-    float G = mx_microfacet_ggx_smith_G(NdotL, NdotV, max(roughness.x, roughness.y));
-
-    float VdotH = dot(V, H);
     vec3 F = mx_fresnel_schlick(VdotH, color0, color90, exponent);
-    F *= weight;
-    float avgF = dot(F, vec3(1.0 / 3.0));
+    float G = mx_microfacet_ggx_smith_G(NdotL, NdotV, mx_average_roughness(roughness));
+
+    vec3 dirAlbedo = mx_microfacet_ggx_directional_albedo(NdotV, mx_average_roughness(roughness), color0, color90);
+    float avgDirAlbedo = dot(dirAlbedo, vec3(1.0 / 3.0));
 
     // Note: NdotL is cancelled out
-    result = D * G * F / (4 * NdotV)    // Top layer reflection
-           + base * (1.0 - avgF);       // Base layer reflection attenuated by top fresnel
+    result = D * F * G * weight / (4 * NdotV)		// Top layer reflection
+           + base * (1.0 - avgDirAlbedo * weight);	// Base layer reflection attenuated by top directional albedo
 }
 
 void mx_generalized_schlick_brdf_transmission(vec3 V, float weight, vec3 color0, vec3 color90, float exponent, vec2 roughness, vec3 N, vec3 X, int distribution, BSDF base, out BSDF result)
@@ -47,12 +47,11 @@ void mx_generalized_schlick_brdf_transmission(vec3 V, float weight, vec3 color0,
     // inverse of top layer reflectance.
 
     // Abs here to allow transparency through backfaces
-    float NdotV = abs(dot(N,V)); 
-    vec3 F = mx_fresnel_schlick(NdotV, color0, color90, exponent);
-    F *= weight;
-    float avgF = dot(F, vec3(1.0 / 3.0));
+    float NdotV = abs(dot(N, V)); 
+    vec3 dirAlbedo = mx_microfacet_ggx_directional_albedo(NdotV, mx_average_roughness(roughness), color0, color90);
+    float avgDirAlbedo = dot(dirAlbedo, vec3(1.0 / 3.0));
 
-    result = base * (1.0 - avgF); // Base layer transmission attenuated by top fresnel
+    result = base * (1.0 - avgDirAlbedo * weight); // Base layer transmission attenuated by top directional albedo
 }
 
 void mx_generalized_schlick_brdf_indirect(vec3 V, float weight, vec3 color0, vec3 color90, float exponent, vec2 roughness, vec3 N, vec3 X, int distribution, BSDF base, out BSDF result)
@@ -63,13 +62,12 @@ void mx_generalized_schlick_brdf_indirect(vec3 V, float weight, vec3 color0, vec
         return;
     }
 
-    vec3 Li = mx_environment_radiance(N, V, X, roughness, distribution);
+    vec3 Li = mx_environment_radiance(N, V, X, roughness, color0, color90, distribution);
 
-    float NdotV = dot(N,V);
-    vec3 F = mx_fresnel_schlick(NdotV, color0, color90, exponent);
-    F *= weight;
-    float avgF = dot(F, vec3(1.0 / 3.0));
+    float NdotV = dot(N, V);
+    vec3 dirAlbedo = mx_microfacet_ggx_directional_albedo(NdotV, mx_average_roughness(roughness), color0, color90);
+    float avgDirAlbedo = dot(dirAlbedo, vec3(1.0 / 3.0));
 
-    result = Li * F                 // Top layer reflection
-           + base * (1.0 - avgF);   // Base layer reflection attenuated by top fresnel
+    result = Li * weight                            // Top layer reflection
+           + base * (1.0 - avgDirAlbedo * weight);  // Base layer reflection attenuated by top directional albedo
 }