Uniformly Sampling A Hemisphere
1. Vector UniformSampleHemisphere(float u1, float u2);
Vector UniformSampleHemisphere(float u1, float u2) {
float z = u1;
float r = sqrtf(max(0.f, 1.f - z*z));
float phi = 2 * M_PI * u2;
float x = r * cosf(phi);
float y = r * sinf(phi);
return Vector(x, y, z);
}作用：
（传入u1，u2，范围都是[0，1]，均匀地选择半球上的一个方向，具体的数学原来可以参考《PBRT》P663，这里就不抄出来）
choosing a direction on the hemisphere uniformly with respect to solid angle.
2. float UniformHemispherePdf();
float UniformHemispherePdf() {
return INV_TWOPI;
}作用：
（采样半球方向的pdf）
For hemispheres (and all other directional sampling), these functions return
values with respect to solid angle. For the hemisphere, the solid angle PDF is a constant
p(ω) = 1/(2π).
Uniformly Sampling Full Sphere
1. Vector UniformSampleSphere(float u1, float u2);
Vector UniformSampleSphere(float u1, float u2) {
float z = 1.f - 2.f * u1;
float r = sqrtf(max(0.f, 1.f - z*z));
float phi = 2.f * M_PI * u2;
float x = r * cosf(phi);
float y = r * sinf(phi);
return Vector(x, y, z);
}
作用：
（传入u1，u2，范围都是[0，1]，均匀地选择整一个球上的一个方向）
Sampling the full sphere uniformly over its area follows almost exactly the same derivation,
which we omit here.
2. float UniformSpherePdf();
float UniformSpherePdf() {
return 1.f / (4.f * M_PI);
}作用：
（采样整一个球方向的pdf）
Uniformly Sampling Disk
(a) When the obvious but incorrect mapping of uniform random variables to points on
the disk is used, the resulting distribution is not uniform and the samples are more likely to be near
the center of the disk. (b) The correct mapping gives a uniform distribution of points.
1. void UniformSampleDisk(float u1, float u2, float *x, float *y);
void UniformSampleDisk(float u1, float u2, float *x, float *y) {
float r = sqrtf(u1);
float theta = 2.0f * M_PI * u2;
*x = r * cosf(theta);
*y = r * sinf(theta);
}作用：
（上图的a，不能均匀采样一个圆盘）
2. void ConcentricSampleDisk(float u1, float u2, float *dx, float *dy);
void ConcentricSampleDisk(float u1, float u2, float *dx, float *dy) {
float r, theta;
// Map uniform random numbers to $[-1,1]^2$
float sx = 2 * u1 - 1;
float sy = 2 * u2 - 1;
// Map square to $(r,\theta)$
// Handle degeneracy at the origin
if (sx == 0.0 && sy == 0.0) {
*dx = 0.0;
*dy = 0.0;
return;
}
if (sx >= -sy) {
if (sx > sy) {
// Handle first region of disk
r = sx;
if (sy > 0.0) theta = sy/r;
else theta = 8.0f + sy/r;
}
else {
// Handle second region of disk
r = sy;
theta = 2.0f - sx/r;
}
}
else {
if (sx <= sy) {
// Handle third region of disk
r = -sx;
theta = 4.0f - sy/r;
}
else {
// Handle fourth region of disk
r = -sy;
theta = 6.0f + sx/r;
}
}
theta *= M_PI / 4.f;
*dx = r * cosf(theta);
*dy = r * sinf(theta);
}作用：
（上图的b，均匀采样一个圆盘）
Cosine-Weighted-Hemisphere Sampling
1. Vector CosineSampleHemisphere(float u1, float u2)
inline Vector CosineSampleHemisphere(float u1, float u2) {
Vector ret;
ConcentricSampleDisk(u1, u2, &ret.x, &ret.y);
ret.z = sqrtf(max(0.f, 1.f - ret.x*ret.x - ret.y*ret.y));
return ret;
}
inline float CosineHemispherePdf(float costheta, float phi) {
return costheta * INV_PI;
}作用：
（采样一个半球的方向，得到的球顶部的方向的概率更加大，pdf 与 cos 成正比，这里的函数的做法就是，x，y 的采样就直接用 disk 的采样，而z 的采样就是 cos 的）
it is often useful to sample from a distribution
that has a shape similar to that of the integrand being estimated.For example, because
the scattering equation weights the product of the BSDF and the incident radiance with
a cosine term, it is useful to have a method that generates directions that are more likely
to be close to the top of the hemisphere, where the cosine term has a large value, than the
bottom, where the cosine term is small.
Mathematically, this means that we would like to sample directions ω from a PDF
Sampling A Triangle
1. void UniformSampleTriangle(float ud1, float ud2, float *u, float *v);
void UniformSampleTriangle(float u1, float u2, float *u, float *v) {
float su1 = sqrtf(u1);
*u = 1.f - su1;
*v = u2 * su1;
}作用：
（均匀地采样一个三角形）
Although uniformly sampling a triangle
Distribution2D
struct Distribution2D {
// Distribution2D Public Methods
Distribution2D(const float *data, int nu, int nv);
~Distribution2D();
void SampleContinuous(float u0, float u1, float uv[2],
float *pdf) const {
float pdfs[2];
int v;
uv[1] = pMarginal->SampleContinuous(u1, &pdfs[1], &v);
uv[0] = pConditionalV[v]->SampleContinuous(u0, &pdfs[0]);
*pdf = pdfs[0] * pdfs[1];
}
float Pdf(float u, float v) const {
int iu = Clamp(Float2Int(u * pConditionalV[0]->count), 0,
pConditionalV[0]->count-1);
int iv = Clamp(Float2Int(v * pMarginal->count), 0,
pMarginal->count-1);
if (pConditionalV[iv]->funcInt * pMarginal->funcInt == 0.f) return 0.f;
return (pConditionalV[iv]->func[iu] * pMarginal->func[iv]) /
(pConditionalV[iv]->funcInt * pMarginal->funcInt);
}
private:
// Distribution2D Private Data
vector<Distribution1D *> pConditionalV;
Distribution1D *pMarginal;
};
Distribution2D::Distribution2D(const float *func, int nu, int nv) {
pConditionalV.reserve(nv);
for (int v = 0; v < nv; ++v) {
// Compute conditional sampling distribution for $\tilde{v}$
pConditionalV.push_back(new Distribution1D(&func[v*nu], nu));
}
// Compute marginal sampling distribution $p[\tilde{v}]$
vector<float> marginalFunc;
marginalFunc.reserve(nv);
for (int v = 0; v < nv; ++v)
marginalFunc.push_back(pConditionalV[v]->funcInt);
pMarginal = new Distribution1D(&marginalFunc[0], nv);
}
Distribution2D::~Distribution2D() {
delete pMarginal;
for (uint32_t i = 0; i < pConditionalV.size(); ++i)
delete pConditionalV[i];
}作用：
（Distribution2D 利用 u,v,[0,1], 进行采样一个离散2D分布函数，例如 Texture, 内部主要的实现都是使用 Distribution1D）
Our final example will show how to sample from discrete 2D distributions. We will
consider the case of a 2D function defined over (u, v) ∈ [0, 1]2 by a 2D array of nu × nv
sample values. This case is particularly useful for generating samples from distributions
defined by texture maps and environment maps.