# Bright results when importance sampling GGX NDF

I've been trying to importance sample the GGX NDF of my Cook-Torrance BRDF for some time now but I can't get it right.

Here's my code for the importance sampling part. It samples a direction and returns it. It also computes the corresponding probability for that direction and stores it in the pdf argument of cook_torrance_sample_direction. The PerRayData argument is only used to generate uniform random numbers.

inline vec3f __device__ cook_torrance_sample_direction(PerRayData& prd, vec3f& view_direction, vec3f& surface_normal, float roughness, float& pdf)
{
float roughness2 = roughness * roughness;

//rand_theta in [0.0, 1.0]
float rand_theta = prd.random();

float theta = atan(roughness * sqrtf(rand_theta / (1.0f - rand_theta)));
float phi = prd.random() * 2.0f * M_PI;//Isotropic case

float cos_theta = cos(theta);
float sin_theta = sin(theta);
float cos_phi = cos(phi);
float sin_phi = sin(phi);

//+0.00001f to avoid dividing by 0
float denom = (roughness2 - 1.0f) * cos_theta * cos_theta + 1.0f + 0.00001;
float pdf_half_vector = (2 * roughness2 * cos_theta * sin_theta) / (denom * denom);

vec3f sampled_normal_local = vec3f(cos_phi * sin_theta,
sin_phi * sin_theta,
cos_theta);

vec3f view_direction_local = world_to_local(surface_normal, view_direction);

//Section "One Extra Step" of the article
pdf = pdf_half_vector / (4 * dot(view_direction_local, sampled_normal_local) + 0.00001f);

vec3f surface_normal_local = vec3f(0.0f, 0.0f, 1.0f);
vec3f reflected_direction_local = normalize(-view_direction_local - 2 * dot(-view_direction_local, sampled_normal_local) * sampled_normal_local);
vec3f reflected_dir_world = local_to_world(surface_normal, reflected_direction_local);

return reflected_dir_world;
}


The sampled direction is then passed to the BRDF evaluation function (as the light_direction argument) which returns a color. I then divide this color by the PDF:

inline vec3f __device__ cook_torrance_brdf(const CookTorranceMaterial& material, const vec3f& view_dir, const vec3f& light_direction, const vec3f& normal)
{
vec3f halfway_vector = normalize(view_dir + light_direction);

float NoV = max(dot(normal, view_dir),            0.0f);
float NoL = max(dot(normal, light_direction),  0.0f);
float NoH = max(dot(normal, halfway_vector),      0.0f);
float VoH = max(dot(view_dir, halfway_vector),    0.0f);

vec3f F0 = (0.16f * (material.reflectance * material.reflectance));
//F0 for metals is equal to the albedo.
//We're going to lerp between the albedo and 0.04
//(previous value of F0) depending on
//the metalness of the material. A fully metalic material
//will thus have a F0 equal to its albedo
F0 = (1.0f - material.metallic) * F0 + material.metallic * material.albedo;

//Fresnel: reflected portion of the light (1 - transmitted)
vec3f F = schlick_approximation(VoH, F0);
float NDF = GGX_NDF(NoH, material.roughness);
float G = geometry_Smith(NoV, NoL, material.roughness);

vec3f kS = F;
vec3f kD = vec3f(1.0f) - kS;
kD *= 1.0 - material.metallic;

vec3f numerator = NDF * G * F;
//+0.0001f to avoid dividng by zero if the dot products are 0
float denominator = 4.0f * NoV * NoL + 0.0001f;
vec3f specular = numerator / denominator;

return (kD * material.albedo / (float)M_PI + specular);
}

inline vec3f __device__ schlick_approximation(float cos_theta, const vec3f& F0)
{
return F0 + (1.0f - F0) * powf((1.0f - cos_theta), 5.0f);
}

inline float __device__ GGX_NDF(float NoH, float roughness)
{
float alpha = roughness * roughness;
float alpha2 = alpha * alpha;
float NoH2 = NoH * NoH;

float num = alpha2;
float denom = (NoH2 * (alpha2 - 1.0f) + 1.0f);
denom = (float)M_PI * denom * denom;

return num / denom;
}

inline float __device__ geometry_Schlick_GGX(float NoX, float roughness)
{
float r = (roughness + 1.0);
float k = (r*r) / 8.0;

float num = NoX;
float denom = NoX * (1.0 - k) + k;

return num / denom;
}

inline float __device__ geometry_Smith(float NoV, float NoL, float roughness)
{
float ggx2 = geometry_Schlick_GGX(NoV, roughness);
float ggx1 = geometry_Schlick_GGX(NoL, roughness);

return ggx1 * ggx2;
}


However, this is giving me results brighter than expected for low roughness values. Here is a roughness of 0.05:

And for roughness lower than 0.013, black pixels start appearing (and eventually cover the entire image):

I noticed that for the too-bright case of the roughness = 0.05, my evaluation function returns color vectors whose RGB values are over 200 (sometimes way more than that, reaching thousands). I suppose this is okay and expected as long as the PDF counter balances it (which it doesn't because in those cases, the value of the PDF is only between ~5.0 and ~15.0)?