I'm reading through this article, and more specifically I'm trying to derive the equation that would explain the implementation the following shader (still in the same article):
#version 330 core
out vec4 FragColor;
in vec3 localPos;
uniform samplerCube environmentMap;
uniform float roughness;
const float PI = 3.14159265359;
float RadicalInverse_VdC(uint bits);
vec2 Hammersley(uint i, uint N);
vec3 ImportanceSampleGGX(vec2 Xi, vec3 N, float roughness);
void main()
{
vec3 N = normalize(localPos);
vec3 R = N;
vec3 V = R;
const uint SAMPLE_COUNT = 1024u;
float totalWeight = 0.0;
vec3 prefilteredColor = vec3(0.0);
for(uint i = 0u; i < SAMPLE_COUNT; ++i)
{
vec2 Xi = Hammersley(i, SAMPLE_COUNT);
vec3 H = ImportanceSampleGGX(Xi, N, roughness);
vec3 L = normalize(2.0 * dot(V, H) * H - V);
float NdotL = max(dot(N, L), 0.0);
if(NdotL > 0.0)
{
prefilteredColor += texture(environmentMap, L).rgb * NdotL;
totalWeight += NdotL;
}
}
prefilteredColor = prefilteredColor / totalWeight;
FragColor = vec4(prefilteredColor, 1.0);
}
Which is my understanding suppose to implement the computation of the integral
$$ I = \int_{\Omega} L(\omega_i)d\omega_i $$
However I'm not entirely sure since NdotL is also used in the computation so I might be wrong.
It seems to me the idea is given $N$ to sample a random halfway vector $H$, and based on this we can construct, by reflection, the light direction vector $L$, which would be random as well by construction. The article also mentions that $H$ is sampled from a normal distribution.
But anyway suppose I have samples $H^{1} ,\ldots, H^{N}$ and therefore $L^{1},\ldots, L^{N}$ samples, the latter I suppose would define $\omega_{i}^{1} \ldots, \omega_{i}^{N}$ (solid angles).
By importance sampling
$$ I \approx \frac{1}{N} \sum_{j=1}^{N} \frac{L(\omega_i^{j})}{pdf(\omega_i^{j})} \;\;\;\; (1) $$
If what I wrote is correct than I don't know where the dot product and the normalization factor
$$ W = \sum_{j=1}^{N} N\cdot L^{j} $$
comes from, namely how do I go from $(1)$ to
$$ I \approx \frac{\sum_{j=1}^{N}L(\omega_i^{j}) N \cdot L^{j}} {\sum_{j=1}^{N} N \cdot L^{j}} $$
Update: I'm pretty sure many of you know this already, but the shader above is for a pre-filtered environment map. Which means in this context the actual integral should be given by
$$ I(N) = \int_{\Omega} L(\omega_i) N \cdot L d\omega $$
This explains the factor $N \cdot L$ in the shader, the distribution function used is the GGX distribution, I assume plugging this stuff together might give me the formula.
Just updating because it might give a better clue.
