1. What does the function hammersley do ?

2. He's sampling the environment map here...is that a TextureCube ? Or is this function being run for each cube face as a Texture2D ?

3. The input to this is the reflection vector R. How would it be calculated in this context ? I imagine similar to the direction vector in the AMD cubemapgen ?

1. hammersley generates psuedo random, fairly well spaced 2-D coordinates, that the GGX importance sample function then gathers into a region thats going to contribute the most for the given roughness.

2. Its a cubemap, for a rough surface an entire hemisphere is required. The nice thing about using a cubemap as an input is that its easy to render one in realtime.

3. The function is run for every pixel of a cubemap rendertarget, its convolving the environment for all directions.

Heres my attempt at implementing this entire process as presented by Epic at Siggraph last year (if anyone can point out errors, that would be awesome).

It uses SV_VERTEXID to generate fullscreen quads, and a GS with SV_RENDERTARGETARRAYINDEX to output to all 6 faces of a cubemap rendertarget simultaneously.

struct vs_out
{
float4 pos : SV_POSITION;
float2 uv : TEXCOORD0;
};
void vs_main(out vs_out o, uint id : SV_VERTEXID)
{
o.uv = float2((id << 1) & 2, id & 2);
o.pos = float4(o.uv * float2(2,-2) + float2(-1,1), 0, 1);
//o.uv = (o.pos.xy * float2(0.5,-0.5) + 0.5) * 4;
//o.uv.y = 1 - o.uv.y;
}
struct ps_in
{
float4 pos : SV_POSITION;
float3 nrm : TEXCOORD0;
uint face : SV_RENDERTARGETARRAYINDEX;
};
float3 UvAndIndexToBoxCoord(float2 uv, uint face)
{
float3 n = float3(0,0,0);
float3 t = float3(0,0,0);
if (face == 0) // posx (red)
{
n = float3(1,0,0);
t = float3(0,1,0);
}
else if (face == 1) // negx (cyan)
{
n = float3(-1,0,0);
t = float3(0,1,0);
}
else if (face == 2) // posy (green)
{
n = float3(0,-1,0);
t = float3(0,0,-1);
}
else if (face == 3) // negy (magenta)
{
n = float3(0,1,0);
t = float3(0,0,1);
}
else if (face == 4) // posz (blue)
{
n = float3(0,0,-1);
t = float3(0,1,0);
}
else // if (i.face == 5) // negz (yellow)
{
n = float3(0,0,1);
t = float3(0,1,0);
}
float3 x = cross(n, t);
uv = uv * 2 - 1;
n = n + t*uv.y + x*uv.x;
n.y *= -1;
n.z *= -1;
return n;
}
[maxvertexcount(18)]
void gs_main(triangle vs_out input[3], inout TriangleStream<ps_in> output)
{
for( int f = 0; f < 6; ++f )
{
for( int v = 0; v < 3; ++v )
{
ps_in o;
o.pos = input[v].pos;
o.nrm = UvAndIndexToBoxCoord(input[v].uv, f);
o.face = f;
output.Append(o);
}
output.RestartStrip();
}
}
SamplerState g_samCube
{
Filter = MIN_MAG_MIP_LINEAR;
AddressU = Clamp;
AddressV = Clamp;
};
TextureCube g_txEnvMap : register(t0);
cbuffer mip : register(b0)
{
float g_CubeSize;
float g_CubeLod;
float g_CubeLodCount;
};
// http://holger.dammertz.org/stuff/notes_HammersleyOnHemisphere.html
float radicalInverse_VdC(uint 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; // / 0x100000000
}
// http://holger.dammertz.org/stuff/notes_HammersleyOnHemisphere.html
float2 Hammersley(uint i, uint N) {
return float2(float(i)/float(N), radicalInverse_VdC(i));
}
static const float PI = 3.1415926535897932384626433832795;
// Image-Based Lighting
// http://www.unrealengine.com/files/downloads/2013SiggraphPresentationsNotes.pdf
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;
}
// M matrix, for encoding
const static float3x3 M = float3x3(
0.2209, 0.3390, 0.4184,
0.1138, 0.6780, 0.7319,
0.0102, 0.1130, 0.2969);
// Inverse M matrix, for decoding
const static float3x3 InverseM = float3x3(
6.0013, -2.700, -1.7995,
-1.332, 3.1029, -5.7720,
.3007, -1.088, 5.6268);
float4 LogLuvEncode(in float3 vRGB)
{
float4 vResult;
float3 Xp_Y_XYZp = mul(vRGB, M);
Xp_Y_XYZp = max(Xp_Y_XYZp, float3(1e-6, 1e-6, 1e-6));
vResult.xy = Xp_Y_XYZp.xy / Xp_Y_XYZp.z;
float Le = 2 * log2(Xp_Y_XYZp.y) + 127;
vResult.w = frac(Le);
vResult.z = (Le - (floor(vResult.w*255.0f))/255.0f)/255.0f;
return vResult;
}
float3 LogLuvDecode(in float4 vLogLuv)
{
float Le = vLogLuv.z * 255 + vLogLuv.w;
float3 Xp_Y_XYZp;
Xp_Y_XYZp.y = exp2((Le - 127) / 2);
Xp_Y_XYZp.z = Xp_Y_XYZp.y / vLogLuv.y;
Xp_Y_XYZp.x = vLogLuv.x * Xp_Y_XYZp.z;
float3 vRGB = mul(Xp_Y_XYZp, InverseM);
return max(vRGB, 0);
}
// Ignacio Castano via http://the-witness.net/news/2012/02/seamless-cube-map-filtering/
float3 fix_cube_lookup_for_lod(float3 v, float cube_size, float lod)
{
float M = max(max(abs(v.x), abs(v.y)), abs(v.z));
float scale = 1 - exp2(lod) / cube_size;
if (abs(v.x) != M) v.x *= scale;
if (abs(v.y) != M) v.y *= scale;
if (abs(v.z) != M) v.z *= scale;
return v;
}
// Pre-Filtered Environment Map
// http://www.unrealengine.com/files/downloads/2013SiggraphPresentationsNotes.pdf
float4 ps_main(in ps_in i) : SV_TARGET0
{
float3 N = fix_cube_lookup_for_lod(normalize(i.nrm), g_CubeSize, g_CubeLod);
float Roughness = (float)g_CubeLod / (float)(g_CubeLodCount-1);
float4 totalRadiance = float4(0,0,0,0);
const uint C = 1024;
[loop]
for (uint j = 0; j < C; ++j)
{
float2 Xi = Hammersley(j,C);
float3 H = ImportanceSampleGGX( Xi, Roughness, N );
float3 L = 2 * dot( N, H ) * H - N;
float nDotL = saturate(dot(L, N));
[branch]
if (nDotL > 0)
{
float4 pointRadiance = (g_txEnvMap.SampleLevel( g_samCube, L, 0 ));
totalRadiance.rgb += pointRadiance.rgb * nDotL;
totalRadiance.w += nDotL;
}
}
return LogLuvEncode(totalRadiance.rgb / totalRadiance.w);
}

**Edited by Digitalfragment, 17 April 2014 - 12:58 AM.**