///////////////////////////////////////////////////////////////////////////
//
// Copyright (c) 2010, Industrial Light & Magic, a division of Lucas
// Digital Ltd. LLC
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///////////////////////////////////////////////////////////////////////////
#include "ImathMatrixAlgo.h"
#include "ImathRandom.h"
#include "ImathEuler.h"
#include <iostream>
#include <assert.h>
#include <cmath>
#include <vector>
#include <limits>
// Verify that if our transformation is already orthogonal, procrustes doesn't
// change that:
template <typename T>
void
testTranslationRotationMatrix (const IMATH_INTERNAL_NAMESPACE::M44d& mat)
{
std::cout << "Testing known translate/rotate matrix:\n " << mat;
typedef IMATH_INTERNAL_NAMESPACE::Vec3<T> Vec;
static IMATH_INTERNAL_NAMESPACE::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_INTERNAL_NAMESPACE::V3d a (rand.nextf(), rand.nextf(), rand.nextf());
IMATH_INTERNAL_NAMESPACE::V3d b = a * mat;
from.push_back (Vec(a));
to.push_back (Vec(b));
}
std::vector<T> weights (numPoints, T(1));
const IMATH_INTERNAL_NAMESPACE::M44d m1 = procrustesRotationAndTranslation (&from[0], &to[0], &weights[0], numPoints);
const IMATH_INTERNAL_NAMESPACE::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_INTERNAL_NAMESPACE::V3d a = from[i];
const IMATH_INTERNAL_NAMESPACE::V3d b = to[i];
const IMATH_INTERNAL_NAMESPACE::V3d b1 = a * m1;
const IMATH_INTERNAL_NAMESPACE::V3d b2 = a * m2;
assert ((b - b1).length() < eps);
assert ((b - b2).length() < eps);
}
std::cout << " OK\n";
}
// Test that if we pass in a matrix that we know consists only of translates,
// rotates, and uniform scale that we get an exact match.
template <typename T>
void testWithTranslateRotateAndScale (const IMATH_INTERNAL_NAMESPACE::M44d& m)
{
std::cout << "Testing with known translate/rotate/scale matrix\n" << m;
IMATH_INTERNAL_NAMESPACE::Rand48 rand(5376);
typedef IMATH_INTERNAL_NAMESPACE::Vec3<T> V3;
std::vector<V3> from;
std::vector<T> weights;
const float eps = 1e-4;
std::cout << "numPoints: " << std::flush;
for (size_t numPoints = 1; numPoints < 10; ++numPoints)
{
from.push_back (V3(rand.nextf(), rand.nextf(), rand.nextf()));
weights.push_back (rand.nextf());
std::cout << from.size() << " ";
std::vector<V3> to;
for (size_t i = 0; i < from.size(); ++i)
to.push_back (from[i] * m);
// weighted:
IMATH_INTERNAL_NAMESPACE::M44d res = IMATH_INTERNAL_NAMESPACE::procrustesRotationAndTranslation (&from[0], &to[0], &weights[0], from.size(), true);
for (size_t i = 0; i < from.size(); ++i)
assert ((from[i] * res - to[i]).length() < eps);
// unweighted:
res = IMATH_INTERNAL_NAMESPACE::procrustesRotationAndTranslation (&from[0], &to[0], from.size(), true);
for (size_t i = 0; i < from.size(); ++i)
assert ((from[i] * res - to[i]).length() < eps);
}
std::cout << " OK\n";
}
template <typename T>
double
procrustesError (const IMATH_INTERNAL_NAMESPACE::Vec3<T>* from,
const IMATH_INTERNAL_NAMESPACE::Vec3<T>* to,
const T* weights,
const size_t n,
const IMATH_INTERNAL_NAMESPACE::M44d& xform)
{
double result = 0.0;
double residual = 0.0;
for (size_t i = 0; i < n; ++i)
{
IMATH_INTERNAL_NAMESPACE::V3d xformed = IMATH_INTERNAL_NAMESPACE::V3d(from[i]) * xform;
IMATH_INTERNAL_NAMESPACE::V3d diff = xformed - IMATH_INTERNAL_NAMESPACE::V3d(to[i]);
const double w = weights[i];
const double mag = w * diff.length2();
// Use Kahan summation for the heck of it:
const double y = mag - residual;
const double t = result + y;
residual = (t - result) - y;
result = t;
}
return result;
}
template <typename T>
void
verifyProcrustes (const std::vector<IMATH_INTERNAL_NAMESPACE::Vec3<T> >& from,
const std::vector<IMATH_INTERNAL_NAMESPACE::Vec3<T> >& to)
{
typedef IMATH_INTERNAL_NAMESPACE::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_INTERNAL_NAMESPACE::M44d m1 = procrustesRotationAndTranslation (&from[0], &to[0], n);
IMATH_INTERNAL_NAMESPACE::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_INTERNAL_NAMESPACE::M44d m = procrustesRotationAndTranslation (&from[0], &to[0], &weights[0], n);
// with scale:
IMATH_INTERNAL_NAMESPACE::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_INTERNAL_NAMESPACE::M33d upperLeft;
for (int i = 0; i < 3; ++i)
for (int j = 0; j < 3; ++j)
upperLeft[i][j] = m[i][j];
IMATH_INTERNAL_NAMESPACE::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_INTERNAL_NAMESPACE::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_INTERNAL_NAMESPACE::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_INTERNAL_NAMESPACE::V3d diffTrans (delta * rand.nextf(), delta * rand.nextf(), delta * rand.nextf());
IMATH_INTERNAL_NAMESPACE::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_INTERNAL_NAMESPACE::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_INTERNAL_NAMESPACE::V3d center (0, 0, 0);
IMATH_INTERNAL_NAMESPACE::V3d netForce(0);
IMATH_INTERNAL_NAMESPACE::V3d netTorque(0);
for (int iPoint = 0; iPoint < n; ++iPoint)
{
const IMATH_INTERNAL_NAMESPACE::V3d force = weights[iPoint] * (from[iPoint]*m - to[iPoint]);
netForce += force;
netTorque += to[iPoint].cross (force);
}
assert (netForce.length2() < eps);
assert (netTorque.length2() < eps);
}
}
template <typename T>
void
testProcrustesWithMatrix (const IMATH_INTERNAL_NAMESPACE::M44d& m)
{
std::cout << "Testing Procrustes algorithm with arbitrary matrix: \n" << m;
std::vector<IMATH_INTERNAL_NAMESPACE::Vec3<T> > fromPoints;
std::vector<IMATH_INTERNAL_NAMESPACE::Vec3<T> > toPoints;
IMATH_INTERNAL_NAMESPACE::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_INTERNAL_NAMESPACE::V3d fromPt (random.nextf(), random.nextf(), random.nextf());
const IMATH_INTERNAL_NAMESPACE::V3d toPt = fromPt * m;
fromPoints.push_back (IMATH_INTERNAL_NAMESPACE::Vec3<T>(fromPt));
toPoints.push_back (IMATH_INTERNAL_NAMESPACE::Vec3<T>(toPt));
}
verifyProcrustes (fromPoints, toPoints);
}
std::cout << "OK\n";
}
template <typename T>
void
testProcrustesImp ()
{
// Test the empty case:
IMATH_INTERNAL_NAMESPACE::M44d id =
procrustesRotationAndTranslation ((IMATH_INTERNAL_NAMESPACE::Vec3<T>*) 0,
(IMATH_INTERNAL_NAMESPACE::Vec3<T>*) 0,
(T*) 0,
0);
assert (id == IMATH_INTERNAL_NAMESPACE::M44d());
id = procrustesRotationAndTranslation ((IMATH_INTERNAL_NAMESPACE::Vec3<T>*) 0,
(IMATH_INTERNAL_NAMESPACE::Vec3<T>*) 0,
0);
assert (id == IMATH_INTERNAL_NAMESPACE::M44d());
// First we'll test with a bunch of known translation/rotation matrices
// to make sure we get back exactly the same points:
IMATH_INTERNAL_NAMESPACE::M44d m;
m.makeIdentity();
testTranslationRotationMatrix<T> (m);
m.translate (IMATH_INTERNAL_NAMESPACE::V3d(3.0, 5.0, -0.2));
testTranslationRotationMatrix<T> (m);
m.rotate (IMATH_INTERNAL_NAMESPACE::V3d(M_PI, 0, 0));
testTranslationRotationMatrix<T> (m);
m.rotate (IMATH_INTERNAL_NAMESPACE::V3d(0, M_PI/4.0, 0));
testTranslationRotationMatrix<T> (m);
m.rotate (IMATH_INTERNAL_NAMESPACE::V3d(0, 0, -3.0/4.0 * M_PI));
testTranslationRotationMatrix<T> (m);
m.makeIdentity();
testWithTranslateRotateAndScale<T> (m);
m.translate (IMATH_INTERNAL_NAMESPACE::V3d(0.4, 6.0, 10.0));
testWithTranslateRotateAndScale<T> (m);
m.rotate (IMATH_INTERNAL_NAMESPACE::V3d(M_PI, 0, 0));
testWithTranslateRotateAndScale<T> (m);
m.rotate (IMATH_INTERNAL_NAMESPACE::V3d(0, M_PI/4.0, 0));
testWithTranslateRotateAndScale<T> (m);
m.rotate (IMATH_INTERNAL_NAMESPACE::V3d(0, 0, -3.0/4.0 * M_PI));
testWithTranslateRotateAndScale<T> (m);
m.scale (IMATH_INTERNAL_NAMESPACE::V3d(2.0, 2.0, 2.0));
testWithTranslateRotateAndScale<T> (m);
m.scale (IMATH_INTERNAL_NAMESPACE::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_INTERNAL_NAMESPACE::Vec3<T> > fromPoints;
std::vector<IMATH_INTERNAL_NAMESPACE::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_INTERNAL_NAMESPACE::Vec3<T>(cos(theta), sin(theta), 0));
toPoints.push_back (IMATH_INTERNAL_NAMESPACE::Vec3<T>(cos(theta + M_PI/3.0), sin(theta + M_PI/3.0), 0));
}
verifyProcrustes (fromPoints, toPoints);
IMATH_INTERNAL_NAMESPACE::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_INTERNAL_NAMESPACE::Vec3<T>(random.nextf(), random.nextf(), random.nextf()));
toPoints.push_back (IMATH_INTERNAL_NAMESPACE::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_INTERNAL_NAMESPACE::Vec3<T>(3, 4, 1));
testProcrustesWithMatrix<T> (m);
m.translate (IMATH_INTERNAL_NAMESPACE::Vec3<T>(-10, 2, 1));
testProcrustesWithMatrix<T> (m);
IMATH_INTERNAL_NAMESPACE::Eulerd rot (M_PI/3.0, 3.0*M_PI/4.0, 0);
m = m * rot.toMatrix44();
testProcrustesWithMatrix<T> (m);
m.scale (IMATH_INTERNAL_NAMESPACE::Vec3<T>(1.5, 6.4, 2.0));
testProcrustesWithMatrix<T> (m);
IMATH_INTERNAL_NAMESPACE::Eulerd rot2 (1.0, M_PI, M_PI/3.0);
m = m * rot.toMatrix44();
m.scale (IMATH_INTERNAL_NAMESPACE::Vec3<T>(-1, 1, 1));
testProcrustesWithMatrix<T> (m);
m.scale (IMATH_INTERNAL_NAMESPACE::Vec3<T>(1, 0.001, 1));
testProcrustesWithMatrix<T> (m);
m.scale (IMATH_INTERNAL_NAMESPACE::Vec3<T>(1, 1, 0));
testProcrustesWithMatrix<T> (m);
}
void
testProcrustes ()
{
std::cout << "Testing Procrustes algorithms in single precision..." << std::endl;
testProcrustesImp<float>();
std::cout << "Testing Procrustes algorithms in double precision..." << std::endl;
testProcrustesImp<double>();
}