int main()
{
Imath::Rand48 rng;
float min = std::numeric_limits<float>::max();
float max = -std::numeric_limits<float>::max();
PerlinNoise::Ptr perlin(new PerlinNoise);
for (int i = 0; i < 1000000; i++) {
Vector p;
p.x = rng.nextf() * 100;
p.y = rng.nextf() * 100;
p.z = rng.nextf() * 100;
float noise = perlin->eval(p.x, p.y, p.z);
min = std::min(min, noise);
max = std::max(max, noise);
}
cout << "min: " << min << endl
<< "max: " << max << endl;
}
void
testTranslationRotationMatrix (const Imath::M44d& mat)
{
std::cout << "Testing known translate/rotate matrix:\n " << mat;
typedef Imath::Vec3<T> Vec;
static Imath::Rand48 rand (2047);
size_t numPoints = 7;
std::vector<Vec> from; from.reserve (numPoints);
std::vector<Vec> to; to.reserve (numPoints);
for (size_t i = 0; i < numPoints; ++i)
{
Imath::V3d a (rand.nextf(), rand.nextf(), rand.nextf());
Imath::V3d b = a * mat;
from.push_back (Vec(a));
to.push_back (Vec(b));
}
std::vector<T> weights (numPoints, T(1));
const Imath::M44d m1 = procrustesRotationAndTranslation (&from[0], &to[0], &weights[0], numPoints);
const Imath::M44d m2 = procrustesRotationAndTranslation (&from[0], &to[0], numPoints);
const T eps = sizeof(T) == 8 ? 1e-8 : 1e-4;
for (size_t i = 0; i < numPoints; ++i)
{
const Imath::V3d a = from[i];
const Imath::V3d b = to[i];
const Imath::V3d b1 = a * m1;
const Imath::V3d b2 = a * m2;
assert ((b - b1).length() < eps);
assert ((b - b2).length() < eps);
}
std::cout << " OK\n";
}
void
testProcrustesWithMatrix (const Imath::M44d& m)
{
std::cout << "Testing Procrustes algorithm with arbitrary matrix: \n" << m;
std::vector<Imath::Vec3<T> > fromPoints;
std::vector<Imath::Vec3<T> > toPoints;
Imath::Rand48 random (1209);
std::cout << " numPoints: ";
for (size_t numPoints = 1; numPoints < 10; ++numPoints)
{
std::cout << numPoints << " " << std::flush;
fromPoints.clear(); toPoints.clear();
for (size_t i = 0; i < numPoints; ++i)
{
const Imath::V3d fromPt (random.nextf(), random.nextf(), random.nextf());
const Imath::V3d toPt = fromPt * m;
fromPoints.push_back (Imath::Vec3<T>(fromPt));
toPoints.push_back (Imath::Vec3<T>(toPt));
}
verifyProcrustes (fromPoints, toPoints);
}
std::cout << "OK\n";
}
Imath::C3f radiance(Scene& scene, Ray& ray, Imath::Rand48& rnd, const int depth)
{
Imath::V3d intersection_point, intersection_normal;
int obj_id;
if (!scene.intersect(ray, intersection_point, intersection_normal, obj_id))
{
return scene.bgColor;
}
Geometry geom = scene.geometries[obj_id];
Imath::V3d orienting_normal = intersection_normal;
if ((intersection_normal ^ ray.dir) > 0.0)
{
orienting_normal *= -1.0;
}
if (depth > LIMIT_DEPTH)
{
return geom.emission;
}
float reflect_ratio = (5.0f - static_cast<float>(depth)) / 5.0f;
Imath::C3f inc_rad(0,0,0), weight(1,1,1);
switch (geom.reflection)
{
case Reflection::Diffuse:
{
Imath::V3d w, u, v;
w = orienting_normal;
if (fabs(w.x) > 0.0000009)
{
u = (Imath::V3d(0.0, 1.0, 0.0) % w).normalized();
}
else
{
u = (Imath::V3d(1.0, 0.0, 0.0) % w).normalized();
}
v = w % u;
double r1 = 2.0 * M_PI * rnd.nextf();
double r2 = rnd.nextf();
double rr2 = sqrt(r2);
ray.pos = intersection_point;
ray.dir = (u * cos(r1) * rr2 + v * sin(r1) * rr2 + w * sqrt(1.0 - r2)).normalized();
inc_rad = radiance(scene, ray, rnd, depth + 1);
weight = geom.color * reflect_ratio;
}
break;
case Reflection::Specular:
ray.pos = intersection_point;
ray.dir -= intersection_normal * 2.0 * (intersection_normal ^ ray.dir);
inc_rad = radiance(scene, ray, rnd, depth + 1);
weight = geom.color * reflect_ratio;
break;
case Reflection::Refraction:
bool into = (orienting_normal ^ intersection_normal) > 0.0;
double default_refraction = 1.0;
double object_refraction = 1.5;
double ray_refraction;
if (into)
{
ray_refraction = default_refraction / object_refraction;
}
else
{
ray_refraction = object_refraction / default_refraction;
}
double incident_dot = ray.dir ^ orienting_normal;
double critical_factor = 1.0 - pow(ray_refraction, 2) * (1.0 - pow(incident_dot,2));
Ray reflection_ray(intersection_point, ray.dir - intersection_normal * 2.0 * (intersection_normal ^ ray.dir));
Ray refraction_ray(intersection_point, (ray.dir * ray_refraction - intersection_normal * (into ? 1.0 : -1.0) * (incident_dot * ray_refraction + sqrt(critical_factor))).normalized());
// total reflection
if (critical_factor < 0.0)
{
inc_rad = radiance(scene, reflection_ray, rnd, depth + 1);
weight = geom.color * reflect_ratio;
break;
}
double a = object_refraction - default_refraction;
double b = object_refraction + default_refraction;
double vertical_incidence_factor = pow(a, 2) / pow(b, 2);
double c = 1.0 - (into ? -1.0 * incident_dot : (refraction_ray.dir ^ -1.0 * orienting_normal));
double fresnel_incidence_factor = vertical_incidence_factor + (1.0 - vertical_incidence_factor) * pow(c,5);
double radiance_scale = pow(ray_refraction, 2.0);
double refraction_factor = (1.0 - fresnel_incidence_factor) * radiance_scale;
double probability = 0.75 + fresnel_incidence_factor;
if (depth > 2)
{
if (rnd.nextf() < probability)
{
inc_rad = radiance(scene, reflection_ray, rnd, depth + 1) * fresnel_incidence_factor;
weight = geom.color * reflect_ratio;
}
else
{
inc_rad = radiance(scene, refraction_ray, rnd, depth + 1) * refraction_factor;
weight = geom.color * reflect_ratio;
}
}
else
{
inc_rad =
radiance(scene, reflection_ray, rnd, depth + 1) * fresnel_incidence_factor +
radiance(scene, refraction_ray, rnd, depth + 1) * refraction_factor;
weight = geom.color * reflect_ratio;
}
break;
}
return geom.emission + weight * inc_rad;
}
void
testProcrustesImp ()
{
// Test the empty case:
Imath::M44d id =
procrustesRotationAndTranslation ((Imath::Vec3<T>*) 0,
(Imath::Vec3<T>*) 0,
(T*) 0,
0);
assert (id == Imath::M44d());
id = procrustesRotationAndTranslation ((Imath::Vec3<T>*) 0,
(Imath::Vec3<T>*) 0,
0);
assert (id == Imath::M44d());
// First we'll test with a bunch of known translation/rotation matrices
// to make sure we get back exactly the same points:
Imath::M44d m;
m.makeIdentity();
testTranslationRotationMatrix<T> (m);
m.translate (Imath::V3d(3.0, 5.0, -0.2));
testTranslationRotationMatrix<T> (m);
m.rotate (Imath::V3d(M_PI, 0, 0));
testTranslationRotationMatrix<T> (m);
m.rotate (Imath::V3d(0, M_PI/4.0, 0));
testTranslationRotationMatrix<T> (m);
m.rotate (Imath::V3d(0, 0, -3.0/4.0 * M_PI));
testTranslationRotationMatrix<T> (m);
m.makeIdentity();
testWithTranslateRotateAndScale<T> (m);
m.translate (Imath::V3d(0.4, 6.0, 10.0));
testWithTranslateRotateAndScale<T> (m);
m.rotate (Imath::V3d(M_PI, 0, 0));
testWithTranslateRotateAndScale<T> (m);
m.rotate (Imath::V3d(0, M_PI/4.0, 0));
testWithTranslateRotateAndScale<T> (m);
m.rotate (Imath::V3d(0, 0, -3.0/4.0 * M_PI));
testWithTranslateRotateAndScale<T> (m);
m.scale (Imath::V3d(2.0, 2.0, 2.0));
testWithTranslateRotateAndScale<T> (m);
m.scale (Imath::V3d(0.01, 0.01, 0.01));
testWithTranslateRotateAndScale<T> (m);
// Now we'll test with some random point sets and verify
// the various Procrustes properties:
std::vector<Imath::Vec3<T> > fromPoints;
std::vector<Imath::Vec3<T> > toPoints;
fromPoints.clear(); toPoints.clear();
for (size_t i = 0; i < 4; ++i)
{
const T theta = T(2*i) / T(M_PI);
fromPoints.push_back (Imath::Vec3<T>(cos(theta), sin(theta), 0));
toPoints.push_back (Imath::Vec3<T>(cos(theta + M_PI/3.0), sin(theta + M_PI/3.0), 0));
}
verifyProcrustes (fromPoints, toPoints);
Imath::Rand48 random (1209);
for (size_t numPoints = 1; numPoints < 10; ++numPoints)
{
fromPoints.clear(); toPoints.clear();
for (size_t i = 0; i < numPoints; ++i)
{
fromPoints.push_back (Imath::Vec3<T>(random.nextf(), random.nextf(), random.nextf()));
toPoints.push_back (Imath::Vec3<T>(random.nextf(), random.nextf(), random.nextf()));
}
}
verifyProcrustes (fromPoints, toPoints);
// Test with some known matrices of varying degrees of quality:
testProcrustesWithMatrix<T> (m);
m.translate (Imath::Vec3<T>(3, 4, 1));
testProcrustesWithMatrix<T> (m);
m.translate (Imath::Vec3<T>(-10, 2, 1));
testProcrustesWithMatrix<T> (m);
Imath::Eulerd rot (M_PI/3.0, 3.0*M_PI/4.0, 0);
m = m * rot.toMatrix44();
testProcrustesWithMatrix<T> (m);
m.scale (Imath::Vec3<T>(1.5, 6.4, 2.0));
testProcrustesWithMatrix<T> (m);
Imath::Eulerd rot2 (1.0, M_PI, M_PI/3.0);
m = m * rot.toMatrix44();
m.scale (Imath::Vec3<T>(-1, 1, 1));
testProcrustesWithMatrix<T> (m);
m.scale (Imath::Vec3<T>(1, 0.001, 1));
testProcrustesWithMatrix<T> (m);
m.scale (Imath::Vec3<T>(1, 1, 0));
testProcrustesWithMatrix<T> (m);
}
void
verifyProcrustes (const std::vector<Imath::Vec3<T> >& from,
const std::vector<Imath::Vec3<T> >& to)
{
typedef Imath::Vec3<T> V3;
const T eps = std::sqrt(std::numeric_limits<T>::epsilon());
const size_t n = from.size();
// Validate that passing in uniform weights gives the same answer as
// passing in no weights:
std::vector<T> weights (from.size());
for (size_t i = 0; i < weights.size(); ++i)
weights[i] = 1;
Imath::M44d m1 = procrustesRotationAndTranslation (&from[0], &to[0], n);
Imath::M44d m2 = procrustesRotationAndTranslation (&from[0], &to[0], &weights[0], n);
for (int i = 0; i < 4; ++i)
for (int j = 0; j < 4; ++j)
assert (std::abs(m1[i][j] - m2[i][j]) < eps);
// Now try the weighted version:
for (size_t i = 0; i < weights.size(); ++i)
weights[i] = i+1;
Imath::M44d m = procrustesRotationAndTranslation (&from[0], &to[0], &weights[0], n);
// with scale:
Imath::M44d ms = procrustesRotationAndTranslation (&from[0], &to[0], &weights[0], n, true);
// Verify that it's orthonormal w/ positive determinant.
const T det = m.determinant();
assert (std::abs(det - T(1)) < eps);
// Verify orthonormal:
Imath::M33d upperLeft;
for (int i = 0; i < 3; ++i)
for (int j = 0; j < 3; ++j)
upperLeft[i][j] = m[i][j];
Imath::M33d product = upperLeft * upperLeft.transposed();
for (int i = 0; i < 3; ++i)
{
for (int j = 0; j < 3; ++j)
{
const double expected = (i == j ? 1.0 : 0.0);
assert (std::abs(product[i][j] - expected) < eps);
}
}
// Verify that nearby transforms are worse:
const size_t numTries = 10;
Imath::Rand48 rand (1056);
const double delta = 1e-3;
for (size_t i = 0; i < numTries; ++i)
{
// Construct an orthogonal rotation matrix using Euler angles:
Imath::Eulerd diffRot (delta * rand.nextf(), delta * rand.nextf(), delta * rand.nextf());
assert (procrustesError (&from[0], &to[0], &weights[0], n, m * diffRot.toMatrix44()) >
procrustesError (&from[0], &to[0], &weights[0], n, m));
// Try a small translation:
Imath::V3d diffTrans (delta * rand.nextf(), delta * rand.nextf(), delta * rand.nextf());
Imath::M44d translateMatrix;
translateMatrix.translate (diffTrans);
assert (procrustesError (&from[0], &to[0], &weights[0], n, m * translateMatrix) >
procrustesError (&from[0], &to[0], &weights[0], n, m));
}
// Try a small scale:
Imath::M44d newMat = ms;
const double scaleDiff = delta;
for (size_t i = 0; i < 3; ++i)
for (size_t j = 0; j < 3; ++j)
newMat[i][j] = ms[i][j] * (1.0 + scaleDiff);
assert (procrustesError (&from[0], &to[0], &weights[0], n, newMat) >
procrustesError (&from[0], &to[0], &weights[0], n, ms));
for (size_t i = 0; i < 3; ++i)
for (size_t j = 0; j < 3; ++j)
newMat[i][j] = ms[i][j] * (1.0 - scaleDiff);
assert (procrustesError (&from[0], &to[0], &weights[0], n, newMat) >
procrustesError (&from[0], &to[0], &weights[0], n, ms));
//
// Verify the magical property that makes shape springs work:
// when the displacements Q*A-B, times the weights,
// are applied as forces at B,
// there is zero net force and zero net torque.
//
{
Imath::V3d center (0, 0, 0);
Imath::V3d netForce(0);
Imath::V3d netTorque(0);
for (int iPoint = 0; iPoint < n; ++iPoint)
{
const Imath::V3d force = weights[iPoint] * (from[iPoint]*m - to[iPoint]);
netForce += force;
netTorque += to[iPoint].cross (force);
}
assert (netForce.length2() < eps);
assert (netTorque.length2() < eps);
}
}