void weak_white_furnace_test(
size_t num_runs,
const float alpha_x,
const float alpha_y,
const float gamma,
const float angle_step,
WeakWhiteFurnaceTestResult& result)
{
result.m_min_G1 = numeric_limits<float>::max();
result.m_max_G1 = -numeric_limits<float>::max();
result.m_min_result = numeric_limits<float>::max();
result.m_max_result = -numeric_limits<float>::max();
MDF mdf;
for (size_t i = 0; i < num_runs; ++i)
{
static const size_t Bases[] = { 2 };
const Vector2f s = hammersley_sequence<float, 2>(Bases, num_runs, i);
const Vector3f v = sample_hemisphere_uniform(s);
const float G1 = mdf.G1(v, Vector3f(0.0f, 1.0, 0.0f), alpha_x, alpha_y, gamma);
result.m_min_G1 = std::min(result.m_min_G1, G1);
result.m_max_G1 = std::max(result.m_max_G1, G1);
const float cos_thetha_o_4 = std::abs(4.0f * v.y);
float integral = 0.0f;
for (float theta = 0.0f; theta < Pi<float>(); theta += angle_step)
{
const float cos_theta = std::cos(theta);
const float sin_theta = std::sin(theta);
for (float phi = 0.0f; phi < TwoPi<float>(); phi += angle_step)
{
const float cos_phi = std::cos(phi);
const float sin_phi = std::sin(phi);
const Vector3f l =
Vector3f::make_unit_vector(
cos_theta,
sin_theta,
cos_phi,
sin_phi);
const Vector3f h = normalize(v + l);
if (h.y > 0.0f)
integral += sin_theta * mdf.D(h, alpha_x, alpha_y, gamma) * G1 / cos_thetha_o_4;
}
}
// Result should be 1.
integral *= square(angle_step);
result.m_min_result = std::min(result.m_min_result, integral);
result.m_max_result = std::max(result.m_max_result, integral);
}
}
void weak_white_furnace_test(
size_t num_runs,
const double alpha_x,
const double alpha_y,
const double angle_step,
WeakWhiteFurnaceTestResult& result)
{
result.m_min_G1 = numeric_limits<double>::max();
result.m_max_G1 = -numeric_limits<double>::max();
result.m_min_result = numeric_limits<double>::max();
result.m_max_result = -numeric_limits<double>::max();
MDF mdf;
for (size_t i = 0; i < num_runs; ++i)
{
static const size_t Bases[] = { 2 };
const Vector2d s = hammersley_sequence<double, 2>(Bases, num_runs, i);
const Vector3d v = sample_hemisphere_uniform(s);
const double G1 = mdf.G1(v, Vector3d(0.0, 1.0, 0.0), alpha_x, alpha_y);
result.m_min_G1 = std::min(result.m_min_G1, G1);
result.m_max_G1 = std::max(result.m_max_G1, G1);
const double cos_thetha_o_4 = std::abs(4.0 * v.y);
double integral = 0.0;
for (double theta = 0.0; theta < Pi<double>(); theta += angle_step)
{
const double cos_theta = std::cos(theta);
const double sin_theta = std::sin(theta);
for (double phi = 0.0; phi < TwoPi<double>(); phi += angle_step)
{
const double cos_phi = std::cos(phi);
const double sin_phi = std::sin(phi);
const Vector3d l =
Vector3d::make_unit_vector(
cos_theta,
sin_theta,
cos_phi,
sin_phi);
const Vector3d h = normalize(v + l);
if (h.y > 0.0)
integral += sin_theta * mdf.D(h, alpha_x, alpha_y) * G1 / cos_thetha_o_4;
}
}
// Result should be 1.
integral *= square(angle_step);
result.m_min_result = std::min(result.m_min_result, integral);
result.m_max_result = std::max(result.m_max_result, integral);
}
}
static void compute_specular_albedo(
const MDF& mdf,
const Spectrum& rs,
const Vector3d& V,
Spectrum& albedo)
{
// V must lie above or in the surface.
assert(V.y >= 0.0);
albedo.set(0.0f);
for (size_t i = 0; i < AlbedoSampleCount; ++i)
{
// Generate a uniform sample in [0,1)^2.
static const size_t Bases[] = { 2 };
const Vector2d s = hammersley_sequence<double, 2>(Bases, i, AlbedoSampleCount);
// Sample the microfacet distribution to get an halfway vector H.
const Vector3d H = mdf.sample(s);
const double dot_HV = dot(H, V);
if (dot_HV <= 0.0)
continue;
// L is the reflection of V around H.
const Vector3d L = (dot_HV + dot_HV) * H - V;
// Reject L if it lies in or below the surface.
if (L.y <= 0.0)
continue;
// Evaluate the PDF of L.
const double dot_HN = H.y;
const double pdf_H = mdf.evaluate_pdf(dot_HN);
const double pdf_L = pdf_H / (4.0 * dot_HV);
assert(pdf_L >= 0.0);
if (pdf_L == 0.0)
continue;
// Sanity checks.
assert(is_normalized(V));
assert(is_normalized(H));
assert(is_normalized(L));
// Evaluate the specular component for this (L, V) pair.
Spectrum fr_spec;
fr_spec = fresnel_dielectric_schlick(rs, dot_HV);
fr_spec *= static_cast<float>((L.y * mdf.evaluate(dot_HN)) / (4.0 * pdf_L * dot_HV * dot_HV));
albedo += fr_spec;
}
albedo /= static_cast<float>(AlbedoSampleCount);
}
void plot(
MapleFile& file,
const string& name,
const MDF& mdf,
const size_t point_count,
const size_t sample_count)
{
vector<double> angles(point_count);
vector<double> densities(point_count);
for (size_t i = 0; i < point_count; ++i)
{
const double angle =
fit(
static_cast<double>(i), 0.0, static_cast<double>(point_count - 1),
-HalfPi, +HalfPi);
const double cos_angle = cos(angle);
angles[i] = rad_to_deg(angle);
densities[i] = mdf.evaluate(cos_angle) * cos_angle;
}
vector<double> angle_samples(sample_count);
vector<double> density_samples(sample_count);
for (size_t i = 0; i < sample_count; ++i)
{
static const size_t Bases[] = { 2 };
const Vector2d s = hammersley_sequence<double, 2>(Bases, i, sample_count);
const Vector3d w = mdf.sample(s);
const double cos_angle = w.y;
const double angle = acos(cos_angle) * (w.x < 0.0 ? -1.0 : 1.0);
angle_samples[i] = rad_to_deg(angle);
density_samples[i] = mdf.evaluate(cos_angle) * cos_angle;
}
file.define(name, angles, densities);
file.define(name + "_samples", angle_samples, density_samples);
file.plot(
make_vector(
MaplePlotDef(name)
.set_legend("Microfacet Distribution Function (" + name + ")"),
MaplePlotDef(name + "_samples")
.set_legend("Integration Samples")
.set_style("point")
.set_color("red")));
}
double integrate_sampling(
const MDF& mdf,
const Sampler& sampler,
const size_t sample_count)
{
double integral = 0.0;
for (size_t i = 0; i < sample_count; ++i)
{
static const size_t Bases[] = { 2 };
const Vector2d s = hammersley_sequence<double, 2>(Bases, i, sample_count);
const Vector3d w = sampler.sample(s);
const double pdf = sampler.pdf(w);
const double cos_theta = w.y;
const double value = mdf.evaluate(cos_theta);
const double sample = value / pdf;
integral += sample * cos_theta;
}
integral /= static_cast<double>(sample_count);
return integral;
}
static void evaluate_fr_spec(
const MDF& mdf,
const Spectrum& rs,
const double dot_HL, // cos_beta in the paper
const double dot_HN,
Spectrum& fr_spec)
{
assert(dot_HL >= 0.0);
assert(dot_HN >= 0.0);
fr_spec = fresnel_dielectric_schlick(rs, dot_HL);
fr_spec *= static_cast<float>(mdf.evaluate(dot_HN) / (4.0 * dot_HL * dot_HL));
}
bool is_positive(
const MDF& mdf,
const size_t sample_count)
{
for (size_t i = 0; i < sample_count; ++i)
{
const double theta = radical_inverse_base2<double>(i) * HalfPi;
const double cos_theta = cos(theta);
const double value = mdf.evaluate(cos_theta);
if (value < 0.0)
return false;
}
return true;
}
bool is_positive(
const MDF& mdf,
const typename MDF::ValueType alpha_x,
const typename MDF::ValueType alpha_y,
const size_t sample_count)
{
for (size_t i = 0; i < sample_count; ++i)
{
static const size_t Bases[] = { 2 };
const Vector2d s = hammersley_sequence<double, 2>(Bases, sample_count, i);
const Vector<typename MDF::ValueType, 3> h = sample_hemisphere_uniform(s);
const double value = mdf.D(h, alpha_x, alpha_y);
if (value < 0.0)
return false;
}
return true;
}
bool is_positive(
const MDF& mdf,
const float alpha_x,
const float alpha_y,
const size_t sample_count,
const float gamma = 0.0f)
{
for (size_t i = 0; i < sample_count; ++i)
{
static const size_t Bases[] = { 2 };
const Vector2f s = hammersley_sequence<float, 2>(Bases, sample_count, i);
const Vector3f h = sample_hemisphere_uniform(s);
const float value = mdf.D(h, alpha_x, alpha_y, gamma);
if (value < 0.0f)
return false;
}
return true;
}
float integrate(
const MDF& mdf,
const float alpha,
const size_t sample_count,
const float gamma = 0.0f)
{
float integral = 0.0f;
for (size_t i = 0; i < sample_count; ++i)
{
const float theta = radical_inverse_base2<float>(i) * HalfPi<float>();
const Vector3f h(0.0f, cos(theta), 0.0f);
const float value = mdf.D(h, alpha, alpha, gamma);
integral += value * h.y * sin(theta);
}
integral *= HalfPi<float>() / sample_count; // integration over theta
integral *= TwoPi<float>(); // integration over phi
return integral;
}
double integrate_quadrature(
const MDF& mdf,
const size_t sample_count)
{
double integral = 0.0;
for (size_t i = 0; i < sample_count; ++i)
{
const double theta = radical_inverse_base2<double>(i) * HalfPi;
const double cos_theta = cos(theta);
const double sin_theta = sin(theta);
const double value = mdf.evaluate(cos_theta);
integral += value * cos_theta * sin_theta;
}
integral *= HalfPi / sample_count; // integration over theta
integral *= TwoPi; // integration over phi
return integral;
}
typename MDF::ValueType integrate(
const MDF& mdf,
const typename MDF::ValueType alpha,
const size_t sample_count)
{
typedef typename MDF::ValueType ValueType;
ValueType integral = ValueType(0.0);
for (size_t i = 0; i < sample_count; ++i)
{
const ValueType theta = radical_inverse_base2<ValueType>(i) * HalfPi<ValueType>();
const Vector<ValueType, 3> h(ValueType(0.0), cos(theta), ValueType(0.0));
const ValueType value = mdf.D(h, alpha, alpha);
integral += value * h.y * sin(theta);
}
integral *= HalfPi<ValueType>() / sample_count; // integration over theta
integral *= TwoPi<ValueType>(); // integration over phi
return integral;
}
