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Implemented 'principle component analysis' utility in gtx, including tests

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SGrottel 2021-05-10 13:14:29 +02:00
parent 761a842a59
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glm/gtx/pca.hpp Normal file
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/// @ref gtx_pca
/// @file glm/gtx/pca.hpp
///
/// @see core (dependence)
/// @see ext_scalar_relational (dependence)
///
/// @defgroup gtx_pca GLM_GTX_pca
/// @ingroup gtx
///
/// Include <glm/gtx/pca.hpp> to use the features of this extension.
///
/// Implements functions required for fundamental 'princple component analysis' in 2D, 3D, and 4D:
/// 1) Computing a covariance matrics from a list of _relative_ position vectors
/// 2) Compute the eigenvalues and eigenvectors of the covariance matrics
/// This is useful, e.g., to compute an object-aligned bounding box from vertices of an object.
/// https://en.wikipedia.org/wiki/Principal_component_analysis
///
/// Example:
/// ```
/// std::vector<glm::dvec3> ptData;
/// // ... fill ptData with some point data, e.g. vertices
///
/// glm::dvec3 center = computeCenter(ptData);
///
/// glm::dmat3 covarMat = glm::computeCovarianceMatrix(ptData.data(), ptData.size(), center);
///
/// glm::dvec3 evals;
/// glm::dmat3 evecs;
/// int evcnt = glm::findEigenvaluesSymReal(covarMat, evals, evecs);
///
/// if(evcnt != 3)
/// // ... error handling
///
/// glm::sortEigenvalues(evals, evecs);
///
/// // ... now evecs[0] points in the direction (symmetric) of the largest spatial distribuion within ptData
/// ```
#pragma once
// Dependency:
#include "../glm.hpp"
#include "../ext/scalar_relational.hpp"
#if GLM_MESSAGES == GLM_ENABLE && !defined(GLM_EXT_INCLUDED)
# ifndef GLM_ENABLE_EXPERIMENTAL
# pragma message("GLM: GLM_GTX_pca is an experimental extension and may change in the future. Use #define GLM_ENABLE_EXPERIMENTAL before including it, if you really want to use it.")
# else
# pragma message("GLM: GLM_GTX_pca extension included")
# endif
#endif
namespace glm {
/// @addtogroup gtx_pca
/// @{
/// Compute a covariance matrix form an array of relative coordinates `v` (e.g., relative to the center of gravity of the object)
/// @param v Points to a memory holding `n` times vectors
template<length_t D, typename T, qualifier Q>
GLM_INLINE mat<D, D, T, Q> computeCovarianceMatrix(vec<D, T, Q> const* v, size_t n);
/// Compute a covariance matrix form an array of absolute coordinates `v` and a precomputed center of gravity `c`
/// @param v Points to a memory holding `n` times vectors
template<length_t D, typename T, qualifier Q>
GLM_INLINE mat<D, D, T, Q> computeCovarianceMatrix(vec<D, T, Q> const* v, size_t n, vec<D, T, Q> const& c);
/// Compute a covariance matrix form a pair of iterators `b` (begin) and `e` (end) of a container with relative coordinates (e.g., relative to the center of gravity of the object)
/// Dereferencing an iterator of type I must yield a `vec&lt;D, T, Q%gt;`
template<length_t D, typename T, qualifier Q, typename I>
GLM_FUNC_DECL mat<D, D, T, Q> computeCovarianceMatrix(I const& b, I const& e);
/// Compute a covariance matrix form a pair of iterators `b` (begin) and `e` (end) of a container with absolute coordinates and a precomputed center of gravity `c`
/// Dereferencing an iterator of type I must yield a `vec&lt;D, T, Q%gt;`
template<length_t D, typename T, qualifier Q, typename I>
GLM_FUNC_DECL mat<D, D, T, Q> computeCovarianceMatrix(I const& b, I const& e, vec<D, T, Q> const& c);
/// Assuming the provided covariance matrix `covarMat` is symmetric and real-valued, this function find the `D` Eigenvalues of the matrix, and also provides the corresponding Eigenvectors.
/// Note: the data in `outEigenvalues` and `outEigenvectors` are in matching order, i.e. `outEigenvector[i]` is the Eigenvector of the Eigenvalue `outEigenvalue[i]`.
/// This is a numeric implementation to find the Eigenvalues, using 'QL decomposition` (variant of QR decomposition: https://en.wikipedia.org/wiki/QR_decomposition).
/// @param covarMat A symmetric, real-valued covariance matrix, e.g. computed from `computeCovarianceMatrix`.
/// @param outEigenvalues Vector to receive the found eigenvalues
/// @param outEigenvectors Matrix to receive the found eigenvectors corresponding to the found eigenvalues, as column vectors
/// @return The number of eigenvalues found, usually D if the precondition of the covariance matrix is met.
template<length_t D, typename T, qualifier Q>
GLM_FUNC_DECL unsigned int findEigenvaluesSymReal
(
mat<D, D, T, Q> const& covarMat,
vec<D, T, Q>& outEigenvalues,
mat<D, D, T, Q>& outEigenvectors
);
/// Sorts a group of Eigenvalues&Eigenvectors, for largest Eigenvalue to smallest Eigenvalue.
/// The data in `outEigenvalues` and `outEigenvectors` are assumed to be matching order, i.e. `outEigenvector[i]` is the Eigenvector of the Eigenvalue `outEigenvalue[i]`.
template<typename T, qualifier Q>
GLM_INLINE void sortEigenvalues(vec<2, T, Q>& eigenvalues, mat<2, 2, T, Q>& eigenvectors);
/// Sorts a group of Eigenvalues&Eigenvectors, for largest Eigenvalue to smallest Eigenvalue.
/// The data in `outEigenvalues` and `outEigenvectors` are assumed to be matching order, i.e. `outEigenvector[i]` is the Eigenvector of the Eigenvalue `outEigenvalue[i]`.
template<typename T, qualifier Q>
GLM_INLINE void sortEigenvalues(vec<3, T, Q>& eigenvalues, mat<3, 3, T, Q>& eigenvectors);
/// Sorts a group of Eigenvalues&Eigenvectors, for largest Eigenvalue to smallest Eigenvalue.
/// The data in `outEigenvalues` and `outEigenvectors` are assumed to be matching order, i.e. `outEigenvector[i]` is the Eigenvector of the Eigenvalue `outEigenvalue[i]`.
template<typename T, qualifier Q>
GLM_INLINE void sortEigenvalues(vec<4, T, Q>& eigenvalues, mat<4, 4, T, Q>& eigenvectors);
/// @}
}//namespace glm
#include "pca.inl"

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/// @ref gtx_pca
namespace glm {
template<length_t D, typename T, qualifier Q>
GLM_INLINE mat<D, D, T, Q> computeCovarianceMatrix(vec<D, T, Q> const* v, size_t n)
{
return std::move(computeCovarianceMatrix<D, T, Q, vec<D, T, Q> const*>(v, v + n));
}
template<length_t D, typename T, qualifier Q>
GLM_INLINE mat<D, D, T, Q> computeCovarianceMatrix(vec<D, T, Q> const* v, size_t n, vec<D, T, Q> const& c)
{
return std::move(computeCovarianceMatrix<D, T, Q, vec<D, T, Q> const*>(v, v + n, c));
}
template<length_t D, typename T, qualifier Q, typename I>
GLM_FUNC_DECL mat<D, D, T, Q> computeCovarianceMatrix(I const& b, I const& e)
{
glm::mat<D, D, T, Q> m{ 0 };
size_t cnt = 0;
for (I i = b; i != e; i++)
{
vec<D, T, Q> const& v = *i;
for (length_t x = 0; x < D; ++x)
for (length_t y = 0; y < D; ++y)
m[x][y] += static_cast<T>(v[x] * v[y]);
cnt++;
}
if (cnt > 0) m /= static_cast<T>(cnt);
return std::move(m);
}
template<length_t D, typename T, qualifier Q, typename I>
GLM_FUNC_DECL mat<D, D, T, Q> computeCovarianceMatrix(I const& b, I const& e, vec<D, T, Q> const& c)
{
glm::mat<D, D, T, Q> m{ 0 };
glm::vec<D, T, Q> v;
size_t cnt = 0;
for (I i = b; i != e; i++)
{
v = *i - c;
for (length_t x = 0; x < D; ++x)
for (length_t y = 0; y < D; ++y)
m[x][y] += static_cast<T>(v[x] * v[y]);
cnt++;
}
if (cnt > 0) m /= static_cast<T>(cnt);
return std::move(m);
}
template<length_t D, typename T, qualifier Q>
GLM_FUNC_DECL unsigned int findEigenvaluesSymReal
(
mat<D, D, T, Q> const& covarMat,
vec<D, T, Q>& outEigenvalues,
mat<D, D, T, Q>& outEigenvectors
)
{
T a[D * D]; // matrix -- input and workspace for algorithm (will be changed inplace)
auto A = [&](length_t const& r, length_t const& c) -> T& { return a[r * D + c]; };
T d[D]; // diagonal elements
T e[D]; // off-diagonal elements
for (length_t r = 0; r < D; r++) {
for (length_t c = 0; c < D; c++) {
A(r, c) = covarMat[c][r];
}
}
// 1. Householder reduction.
length_t l, k, j, i;
T scale, hh, h, g, f;
constexpr T epsilon = static_cast<T>(0.0000001);
for (i = D; i >= 2; i--) {
l = i - 1;
h = scale = 0;
if (l > 1) {
for (k = 1; k <= l; k++) {
scale += glm::abs(A(i - 1, k - 1));
}
if (glm::equal<T>(scale, 0, epsilon)) {
e[i - 1] = A(i - 1, l - 1);
} else {
for (k = 1; k <= l; k++) {
A(i - 1, k - 1) /= scale;
h += A(i - 1, k - 1) * A(i - 1, k - 1);
}
f = A(i - 1, l - 1);
g = ((f >= 0) ? -glm::sqrt(h) : glm::sqrt(h));
e[i - 1] = scale * g;
h -= f * g;
A(i - 1, l - 1) = f - g;
f = 0;
for (j = 1; j <= l; j++) {
A(j - 1, i - 1) = A(i - 1, j - 1) / h;
g = 0;
for (k = 1; k <= j; k++) {
g += A(j - 1, k - 1) * A(i - 1, k - 1);
}
for (k = j + 1; k <= l; k++) {
g += A(k - 1, j - 1) * A(i - 1, k - 1);
}
e[j - 1] = g / h;
f += e[j - 1] * A(i - 1, j - 1);
}
hh = f / (h + h);
for (j = 1; j <= l; j++) {
f = A(i - 1, j - 1);
e[j - 1] = g = e[j - 1] - hh * f;
for (k = 1; k <= j; k++) {
A(j - 1, k - 1) -= (f * e[k - 1] + g * A(i - 1, k - 1));
}
}
}
} else {
e[i - 1] = A(i - 1, l - 1);
}
d[i - 1] = h;
}
d[0] = 0;
e[0] = 0;
for (i = 1; i <= D; i++) {
l = i - 1;
if (!glm::equal<T>(d[i - 1], 0, epsilon)) {
for (j = 1; j <= l; j++) {
g = 0;
for (k = 1; k <= l; k++) {
g += A(i - 1, k - 1) * A(k - 1, j - 1);
}
for (k = 1; k <= l; k++) {
A(k - 1, j - 1) -= g * A(k - 1, i - 1);
}
}
}
d[i - 1] = A(i - 1, i - 1);
A(i - 1, i - 1) = 1;
for (j = 1; j <= l; j++) {
A(j - 1, i - 1) = A(i - 1, j - 1) = 0;
}
}
// 2. Calculation of eigenvalues and eigenvectors (QL algorithm)
length_t m, iter;
T s, r, p, dd, c, b;
const length_t MAX_ITER = 30;
auto transferSign = [](T const& v, T const& s) { return ((s) >= 0 ? glm::abs(v) : -glm::abs(v)); };
auto pythag = [](T const& a, T const& b) {
constexpr T epsilon = static_cast<T>(0.0000001);
T absa = glm::abs(a);
T absb = glm::abs(b);
if (absa > absb) {
absb /= absa;
absb *= absb;
return absa * glm::sqrt(static_cast<T>(1) + absb);
}
if (glm::equal<T>(absb, 0, epsilon)) return static_cast<T>(0);
absa /= absb;
absa *= absa;
return absb * glm::sqrt(static_cast<T>(1) + absa);
};
for (i = 2; i <= D; i++) {
e[i - 2] = e[i - 1];
}
e[D - 1] = 0;
for (l = 1; l <= D; l++) {
iter = 0;
do {
for (m = l; m <= D - 1; m++) {
dd = glm::abs(d[m - 1]) + glm::abs(d[m - 1 + 1]);
if (glm::equal<T>(glm::abs(e[m - 1]) + dd, dd, epsilon)) break;
}
if (m != l) {
if (iter++ == MAX_ITER) {
return 0; // Too many iterations in FindEigenvalues
}
g = (d[l - 1 + 1] - d[l - 1]) / (2 * e[l - 1]);
r = pythag(g, 1);
g = d[m - 1] - d[l - 1] + e[l - 1] / (g + transferSign(r, g));
s = c = 1;
p = 0;
for (i = m - 1; i >= l; i--) {
f = s * e[i - 1];
b = c * e[i - 1];
e[i - 1 + 1] = r = pythag(f, g);
if (glm::equal<T>(r, 0, epsilon)) {
d[i - 1 + 1] -= p;
e[m - 1] = 0;
break;
}
s = f / r;
c = g / r;
g = d[i - 1 + 1] - p;
r = (d[i - 1] - g) * s + 2 * c * b;
d[i - 1 + 1] = g + (p = s * r);
g = c * r - b;
for (k = 1; k <= D; k++) {
f = A(k - 1, i - 1 + 1);
A(k - 1, i - 1 + 1) = s * A(k - 1, i - 1) + c * f;
A(k - 1, i - 1) = c * A(k - 1, i - 1) - s * f;
}
}
if (glm::equal<T>(r, 0, epsilon) && (i >= l)) continue;
d[l - 1] -= p;
e[l - 1] = g;
e[m - 1] = 0;
}
} while (m != l);
}
// 3. output
for(i = 0; i < D; i++)
outEigenvalues[i] = d[i];
for(i = 0; i < D; i++)
for(j = 0; j < D; j++)
outEigenvectors[i][j] = A(j, i);
return D;
}
template<typename T, qualifier Q>
GLM_INLINE void sortEigenvalues(vec<2, T, Q>& eigenvalues, mat<2, 2, T, Q>& eigenvectors)
{
if (eigenvalues[0] < eigenvalues[1])
{
std::swap(eigenvalues[0], eigenvalues[1]);
std::swap(eigenvectors[0], eigenvectors[1]);
}
}
template<typename T, qualifier Q>
GLM_INLINE void sortEigenvalues(vec<3, T, Q>& eigenvalues, mat<3, 3, T, Q>& eigenvectors)
{
if (eigenvalues[0] < eigenvalues[1])
{
std::swap(eigenvalues[0], eigenvalues[1]);
std::swap(eigenvectors[0], eigenvectors[1]);
}
if (eigenvalues[0] < eigenvalues[2])
{
std::swap(eigenvalues[0], eigenvalues[2]);
std::swap(eigenvectors[0], eigenvectors[2]);
}
if (eigenvalues[1] < eigenvalues[2])
{
std::swap(eigenvalues[1], eigenvalues[2]);
std::swap(eigenvectors[1], eigenvectors[2]);
}
}
template<typename T, qualifier Q>
GLM_INLINE void sortEigenvalues(vec<4, T, Q>& eigenvalues, mat<4, 4, T, Q>& eigenvectors)
{
if (eigenvalues[0] < eigenvalues[2])
{
std::swap(eigenvalues[0], eigenvalues[2]);
std::swap(eigenvectors[0], eigenvectors[2]);
}
if (eigenvalues[1] < eigenvalues[3])
{
std::swap(eigenvalues[1], eigenvalues[3]);
std::swap(eigenvectors[1], eigenvectors[3]);
}
if (eigenvalues[0] < eigenvalues[1])
{
std::swap(eigenvalues[0], eigenvalues[1]);
std::swap(eigenvectors[0], eigenvectors[1]);
}
if (eigenvalues[2] < eigenvalues[3])
{
std::swap(eigenvalues[2], eigenvalues[3]);
std::swap(eigenvectors[2], eigenvectors[3]);
}
if (eigenvalues[1] < eigenvalues[2])
{
std::swap(eigenvalues[1], eigenvalues[2]);
std::swap(eigenvectors[1], eigenvectors[2]);
}
}
}//namespace glm

1
test/gtx/CMakeLists.txt
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@ -37,6 +37,7 @@ glmCreateTestGTC(gtx_normalize_dot)
glmCreateTestGTC(gtx_number_precision) glmCreateTestGTC(gtx_number_precision)
glmCreateTestGTC(gtx_orthonormalize) glmCreateTestGTC(gtx_orthonormalize)
glmCreateTestGTC(gtx_optimum_pow) glmCreateTestGTC(gtx_optimum_pow)
glmCreateTestGTC(gtx_pca)
glmCreateTestGTC(gtx_perpendicular) glmCreateTestGTC(gtx_perpendicular)
glmCreateTestGTC(gtx_polar_coordinates) glmCreateTestGTC(gtx_polar_coordinates)
glmCreateTestGTC(gtx_projection) glmCreateTestGTC(gtx_projection)

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test/gtx/gtx_pca.cpp Normal file
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#define GLM_ENABLE_EXPERIMENTAL
#include <glm/glm.hpp>
#include <glm/gtx/pca.hpp>
#include <vector>
#include <random>
// Test data: 1AGA 'agarose double helix'
// https://www.rcsb.org/structure/1aga
// The fourth coordinate is randomized
namespace _1aga
{
// Fills `outTestData` with hard-coded atom positions from 1AGA
// The fourth coordinate is randomized
template<typename vec>
void fillTestData(std::vector<vec>& outTestData)
{
// x,y,z coordinates copied from RCSB PDB file of 1AGA
// w coordinate randomized with standard normal distribution
constexpr double _1aga[] = {
3.219, -0.637, 19.462, 2.286,
4.519, 0.024, 18.980, -0.828,
4.163, 1.425, 18.481, -0.810,
3.190, 1.341, 17.330, -0.170,
1.962, 0.991, 18.165, 0.816,
2.093, 1.952, 19.331, 0.276,
5.119, -0.701, 17.908, -0.490,
3.517, 2.147, 19.514, -0.207,
2.970, 2.609, 16.719, 0.552,
2.107, -0.398, 18.564, 0.403,
2.847, 2.618, 15.335, 0.315,
1.457, 3.124, 14.979, 0.683,
1.316, 3.291, 13.473, 0.446,
2.447, 4.155, 12.931, 1.324,
3.795, 3.614, 13.394, 0.112,
4.956, 4.494, 12.982, 0.253,
0.483, 2.217, 15.479, 1.316,
0.021, 3.962, 13.166, 1.522,
2.311, 5.497, 13.395, 0.248,
3.830, 3.522, 14.827, 0.591,
5.150, 4.461, 11.576, 0.635,
-1.057, 3.106, 13.132, 0.191,
-2.280, 3.902, 12.650, 1.135,
-3.316, 2.893, 12.151, 0.794,
-2.756, 2.092, 11.000, 0.720,
-1.839, 1.204, 11.835, -1.172,
-2.737, 0.837, 13.001, -0.313,
-1.952, 4.784, 11.578, 2.082,
-3.617, 1.972, 13.184, 0.653,
-3.744, 1.267, 10.389, -0.413,
-0.709, 2.024, 12.234, -1.747,
-3.690, 1.156, 9.005, -1.275,
-3.434, -0.300, 8.649, 0.441,
-3.508, -0.506, 7.143, 0.237,
-4.822, 0.042, 6.601, -2.856,
-5.027, 1.480, 7.064, 0.985,
-6.370, 2.045, 6.652, 0.915,
-2.162, -0.690, 9.149, 1.100,
-3.442, -1.963, 6.836, -0.081,
-5.916, -0.747, 7.065, -2.345,
-4.965, 1.556, 8.497, 0.504,
-6.439, 2.230, 5.246, 1.451,
-2.161, -2.469, 6.802, -1.171,
-2.239, -3.925, 6.320, -1.434,
-0.847, -4.318, 5.821, 0.098,
-0.434, -3.433, 4.670, -1.446,
-0.123, -2.195, 5.505, 0.182,
0.644, -2.789, 6.671, 0.865,
-3.167, -4.083, 5.248, -0.098,
0.101, -4.119, 6.854, -0.001,
0.775, -3.876, 4.059, 1.061,
-1.398, -1.625, 5.904, 0.230,
0.844, -3.774, 2.675, 1.313,
1.977, -2.824, 2.319, -0.112,
2.192, -2.785, 0.813, -0.981,
2.375, -4.197, 0.271, -0.355,
1.232, -5.093, 0.734, 0.632,
1.414, -6.539, 0.322, 0.576,
1.678, -1.527, 2.819, -1.187,
3.421, -1.999, 0.496, -1.770,
3.605, -4.750, 0.735, 1.099,
1.135, -5.078, 2.167, 0.854,
1.289, -6.691, -1.084, -0.487,
-1.057, 3.106, 22.602, -1.297,
-2.280, 3.902, 22.120, 0.376,
-3.316, 2.893, 21.621, 0.932,
-2.756, 2.092, 20.470, 1.680,
-1.839, 1.204, 21.305, 0.615,
-2.737, 0.837, 22.471, 0.899,
-1.952, 4.784, 21.048, -0.521,
-3.617, 1.972, 22.654, 0.133,
-3.744, 1.267, 19.859, 0.081,
-0.709, 2.024, 21.704, 1.420,
-3.690, 1.156, 18.475, -0.850,
-3.434, -0.300, 18.119, -0.249,
-3.508, -0.506, 16.613, 1.434,
-4.822, 0.042, 16.071, -2.466,
-5.027, 1.480, 16.534, -1.045,
-6.370, 2.045, 16.122, 1.707,
-2.162, -0.690, 18.619, -2.023,
-3.442, -1.963, 16.336, -0.304,
-5.916, -0.747, 16.535, 0.979,
-4.965, 1.556, 17.967, -1.165,
-6.439, 2.230, 14.716, 0.929,
-2.161, -2.469, 16.302, -0.234,
-2.239, -3.925, 15.820, -0.228,
-0.847, -4.318, 15.321, 1.844,
-0.434, -3.433, 14.170, 1.132,
-0.123, -2.195, 15.005, 0.211,
0.644, -2.789, 16.171, -0.632,
-3.167, -4.083, 14.748, -0.519,
0.101, -4.119, 16.354, 0.173,
0.775, -3.876, 13.559, 1.243,
-1.398, -1.625, 15.404, -0.187,
0.844, -3.774, 12.175, -1.332,
1.977, -2.824, 11.819, -1.616,
2.192, -2.785, 10.313, 1.320,
2.375, -4.197, 9.771, 0.237,
1.232, -5.093, 10.234, 0.851,
1.414, -6.539, 9.822, 1.816,
1.678, -1.527, 12.319, -1.657,
3.421, -1.999, 10.036, 1.559,
3.605, -4.750, 10.235, 0.831,
1.135, -5.078, 11.667, 0.060,
1.289, -6.691, 8.416, 1.066,
3.219, -0.637, 10.002, 2.111,
4.519, 0.024, 9.520, -0.874,
4.163, 1.425, 9.021, -1.012,
3.190, 1.341, 7.870, -0.250,
1.962, 0.991, 8.705, -1.359,
2.093, 1.952, 9.871, -0.126,
5.119, -0.701, 8.448, 0.995,
3.517, 2.147, 10.054, 0.941,
2.970, 2.609, 7.259, -0.562,
2.107, -0.398, 9.104, -0.038,
2.847, 2.618, 5.875, 0.398,
1.457, 3.124, 5.519, 0.481,
1.316, 3.291, 4.013, -0.187,
2.447, 4.155, 3.471, -0.429,
3.795, 3.614, 3.934, -0.432,
4.956, 4.494, 3.522, -0.788,
0.483, 2.217, 6.019, -0.923,
0.021, 3.962, 3.636, -0.316,
2.311, 5.497, 3.935, -1.917,
3.830, 3.522, 5.367, -0.302,
5.150, 4.461, 2.116, -1.615
};
constexpr size_t _1agaSize = sizeof(_1aga) / (4 * sizeof(double));
outTestData.resize(_1agaSize);
for(size_t i = 0; i < _1agaSize; ++i)
for(size_t d = 0; d < vec::length(); ++d)
outTestData[i][d] = static_cast<typename vec::value_type>(_1aga[i * 4 + d]);
}
void getExpectedCovarDataPtr(const double*& ptr)
{
static constexpr double _1agaCovar4x4d[] = {
9.624340680272107, -0.000066573696146, -4.293213765684049, 0.018793741874528,
-0.000066573696146, 9.624439378684805, 5.351138726379443, -0.115692591458806,
-4.293213765684049, 5.351138726379443, 35.628485496346691, 0.908742392542202,
0.018793741874528, -0.115692591458806, 0.908742392542202, 1.097059718568909
};
ptr = _1agaCovar4x4d;
}
void getExpectedCovarDataPtr(const float*& ptr)
{
// note: the value difference to `_1agaCovar4x4d` is due to the numeric error propagation during computation of the covariance matrix.
static constexpr float _1agaCovar4x4f[] = {
9.624336242675781f, -0.000066711785621f, -4.293214797973633f, 0.018793795257807f,
-0.000066711785621f, 9.624438285827637f, 5.351140022277832f, -0.115692682564259f,
-4.293214797973633f, 5.351140022277832f, 35.628479003906250f, 0.908742427825928f,
0.018793795257807f, -0.115692682564259f, 0.908742427825928f, 1.097059369087219f
};
ptr = _1agaCovar4x4f;
}
template<glm::length_t D, typename T, glm::qualifier Q>
int checkCovarMat(glm::mat<D, D, T, Q> const& covarMat)
{
const T* expectedCovarData = nullptr;
getExpectedCovarDataPtr(expectedCovarData);
for(size_t x = 0; x < D; ++x)
for(size_t y = 0; y < D; ++y)
if(!glm::equal(covarMat[y][x], expectedCovarData[x * 4 + y], static_cast<T>(0.000001)))
return 1;
return 0;
}
template<glm::length_t D, typename T> void getExpectedEigenvaluesEigenvectorsDataPtr(const T*& evals, const T*& evecs);
template<> void getExpectedEigenvaluesEigenvectorsDataPtr<2, float>(const float*& evals, const float*& evecs)
{
static constexpr float expectedEvals[] = {
9.624471664428711f, 9.624302864074707f
};
static constexpr float expectedEvecs[] = {
-0.443000972270966f, 0.896521151065826f,
0.896521151065826f, 0.443000972270966f
};
evals = expectedEvals;
evecs = expectedEvecs;
}
template<> void getExpectedEigenvaluesEigenvectorsDataPtr<2, double>(const double*& evals, const double*& evecs)
{
static constexpr double expectedEvals[] = {
9.624472899262972, 9.624307159693940
};
static constexpr double expectedEvecs[] = {
-0.449720461624363, 0.893169360421846,
0.893169360421846, 0.449720461624363
};
evals = expectedEvals;
evecs = expectedEvecs;
}
template<> void getExpectedEigenvaluesEigenvectorsDataPtr<3, float>(const float*& evals, const float*& evecs)
{
static constexpr float expectedEvals[] = {
37.327442169189453f, 9.624311447143555f, 7.925499439239502f
};
static constexpr float expectedEvecs[] = {
-0.150428697466850f, 0.187497511506081f, 0.970678031444550f,
0.779980957508087f, 0.625803351402283f, -0.000005212802080f,
0.607454538345337f, -0.757109522819519f, 0.240383237600327f
};
evals = expectedEvals;
evecs = expectedEvecs;
}
template<> void getExpectedEigenvaluesEigenvectorsDataPtr<3, double>(const double*& evals, const double*& evecs)
{
static constexpr double expectedEvals[] = {
37.327449427468345, 9.624314341614987, 7.925501786220276
};
static constexpr double expectedEvecs[] = {
-0.150428640509585, 0.187497426513576, 0.970678082149394,
0.779981605126846, 0.625802441381904, -0.000004919018357,
0.607453635908278, -0.757110308615089, 0.240383154173870
};
evals = expectedEvals;
evecs = expectedEvecs;
}
template<> void getExpectedEigenvaluesEigenvectorsDataPtr<4, float>(const float*& evals, const float*& evecs)
{
static constexpr float expectedEvals[] = {
37.347740173339844f, 9.624703407287598f, 7.940164566040039f, 1.061712265014648f
};
static constexpr float expectedEvecs[] = {
-0.150269940495491f, 0.187220811843872f, 0.970467865467072f, 0.023652425035834f,
0.779159665107727f, 0.626788496971130f, -0.000105984276161f, -0.006797631736845f,
0.608242213726044f, -0.755563497543335f, 0.238818943500519f, 0.046158745884895f,
-0.019251370802522f, 0.034755907952785f, -0.034024771302938f, 0.998630762100220f,
};
evals = expectedEvals;
evecs = expectedEvecs;
}
template<> void getExpectedEigenvaluesEigenvectorsDataPtr<4, double>(const double*& evals, const double*& evecs)
{
static constexpr double expectedEvals[] = {
37.347738991879226, 9.624706889211053, 7.940170752816341, 1.061708639965897
};
static constexpr double expectedEvecs[] = {
-0.150269954805403, 0.187220917596058, 0.970467838469868, 0.023652551509145,
0.779159831346545, 0.626788431871120, -0.000105940250315, -0.006797622027466,
0.608241962267880, -0.755563776664248, 0.238818902950296, 0.046158707986616,
-0.019251317755512, 0.034755849578017, -0.034024915369495, 0.998630924225204,
};
evals = expectedEvals;
evecs = expectedEvecs;
}
template<glm::length_t D, typename T, glm::qualifier Q>
int checkEigenvaluesEigenvectors(
glm::vec<D, T, Q> const& evals,
glm::mat<D, D, T, Q> const& evecs)
{
const T* expectedEvals = nullptr;
const T* expectedEvecs = nullptr;
getExpectedEigenvaluesEigenvectorsDataPtr<D, T>(expectedEvals, expectedEvecs);
for(int i = 0; i < D; ++i)
if(!glm::equal(evals[i], expectedEvals[i], static_cast<T>(0.000001)))
return 1;
for (int i = 0; i < D; ++i)
for (int d = 0; d < D; ++d)
if (!glm::equal(evecs[i][d], expectedEvecs[i * D + d], static_cast<T>(0.000001)))
return 1;
return 0;
}
} // namespace _1aga
// Compute center of gravity
template<typename vec>
vec computeCenter(const std::vector<vec>& testData)
{
double c[vec::length()];
std::fill(c, c + vec::length(), 0.0);
for(vec const& v : testData)
for(size_t d = 0; d < vec::length(); ++d)
c[d] += static_cast<double>(v[d]);
vec cVec;
for(size_t d = 0; d < vec::length(); ++d)
cVec[d] = static_cast<typename vec::value_type>(c[d] / static_cast<double>(testData.size()));
return std::move(cVec);
}
// Test sorting of Eigenvalue&Eigenvector lists. Use exhaustive search.
template<glm::length_t D, typename T, glm::qualifier Q>
int testEigenvalueSort()
{
// Test input data: four arbitrary values
constexpr glm::vec<D, T, Q> refVal
{
glm::vec<4, T, Q>
{
10, 8, 6, 4
}
};
// Test input data: four arbitrary vectors, which can be matched to the above values
constexpr glm::mat<D, D, T, Q> refVec
{
glm::mat<4, 4, T, Q>
{
10, 20, 5, 40,
8, 16, 4, 32,
6, 12, 3, 24,
4, 8, 2, 16
}
};
// Permutations of test input data for exhaustive check, based on `D` (1 <= D <= 4)
constexpr int permutationCount[]
{
0,
1,
2,
6,
24
};
// The permutations t perform, based on `D` (1 <= D <= 4)
constexpr glm::ivec4 permutation[]
{
{ 0, 1, 2, 3 },
{ 1, 0, 2, 3 }, // last for D = 2
{ 0, 2, 1, 3 },
{ 1, 2, 0, 3 },
{ 2, 0, 1, 3 },
{ 2, 1, 0, 3 }, // last for D = 3
{ 0, 1, 3, 2 },
{ 1, 0, 3, 2 },
{ 0, 2, 3, 1 },
{ 1, 2, 3, 0 },
{ 2, 0, 3, 1 },
{ 2, 1, 3, 0 },
{ 0, 3, 1, 2 },
{ 1, 3, 0, 2 },
{ 0, 3, 2, 1 },
{ 1, 3, 2, 0 },
{ 2, 3, 0, 1 },
{ 2, 3, 1, 0 },
{ 3, 0, 1, 2 },
{ 3, 1, 0, 2 },
{ 3, 0, 2, 1 },
{ 3, 1, 2, 0 },
{ 3, 2, 0, 1 },
{ 3, 2, 1, 0 } // last for D = 4
};
// Lambda utility to check the result
auto checkResult = [&refVal,&refVec](glm::vec<D, T, Q> const& value, glm::mat<D, D, T, Q> const& vector)
{
constexpr T epsilon = static_cast<T>(0.0000001);
// check that values are ordered ascending
for(int i = 1; i < D; ++i)
{
if(value[0] < value[1])
return false;
}
// check that values and vectors are equal to the reference values
for(int i = 0; i < D; ++i)
{
if(!glm::equal<T>(refVal[i], value[i], epsilon))
return false;
for(int j = 0; j < D; ++j)
{
if(!glm::equal<T>(refVec[i][j], vector[i][j], epsilon))
return false;
}
}
return true; // all matched
};
// initial sanity check
if(!checkResult(refVal, refVec))
return 1;
// Exhaustive search through all permutations
for(int p = 0; p < permutationCount[D]; ++p)
{
glm::vec<D, T, Q> testVal;
glm::mat<D, D, T, Q> testVec;
for(int i = 0; i < D; ++i)
{
testVal[i] = refVal[permutation[p][i]];
testVec[i] = refVec[permutation[p][i]];
}
glm::sortEigenvalues(testVal, testVec);
if(!checkResult(testVal, testVec))
return 2 + p;
}
return 0;
}
// Test covariance matrix creation functions
template<glm::length_t D, typename T, glm::qualifier Q>
int testCovar(unsigned int dataSize, unsigned int randomEngineSeed)
{
typedef glm::vec<D, T, Q> vec;
typedef glm::mat<D, D, T, Q> mat;
// #1: test expected result with fixed data set
std::vector<vec> testData;
_1aga::fillTestData(testData);
// compute center of gravity
vec center = computeCenter(testData);
mat covarMat = glm::computeCovarianceMatrix(testData.data(), testData.size(), center);
if(_1aga::checkCovarMat(covarMat))
return 1;
// #2: test function variant consitency with random data
std::default_random_engine rndEng{ randomEngineSeed };
std::normal_distribution<T> normalDist;
testData.resize(dataSize);
// some common offset of all data
T offset[D];
for(size_t d = 0; d < D; ++d)
offset[d] = normalDist(rndEng);
// init data
for(size_t i = 0; i < dataSize; ++i)
for(size_t d = 0; d < D; ++d)
testData[i][d] = offset[d] + normalDist(rndEng);
center = computeCenter(testData);
std::vector<vec> centeredTestData;
centeredTestData.reserve(testData.size());
for(vec const& v : testData)
centeredTestData.push_back(v - center);
mat c1 = glm::computeCovarianceMatrix(centeredTestData.data(), centeredTestData.size());
mat c2 = glm::computeCovarianceMatrix<D, T, Q>(centeredTestData.begin(), centeredTestData.end());
mat c3 = glm::computeCovarianceMatrix(testData.data(), testData.size(), center);
mat c4 = glm::computeCovarianceMatrix<D, T, Q>(testData.rbegin(), testData.rend(), center);
if(c1 != c2)
return 1;
if(c1 != c3)
return 1;
if(c1 != c4)
return 1;
return 0;
}
template<glm::length_t D, typename T, glm::qualifier Q>
int testEigenvectors()
{
typedef glm::vec<D, T, Q> vec;
typedef glm::mat<D, D, T, Q> mat;
// test expected result with fixed data set
std::vector<vec> testData;
_1aga::fillTestData(testData);
vec center = computeCenter(testData);
mat covarMat = glm::computeCovarianceMatrix(testData.data(), testData.size(), center);
vec eigenvalues;
mat eigenvectors;
unsigned int c = glm::findEigenvaluesSymReal(covarMat, eigenvalues, eigenvectors);
if(c != D)
return 1;
glm::sortEigenvalues(eigenvalues, eigenvectors);
if(_1aga::checkEigenvaluesEigenvectors(eigenvalues, eigenvectors) != 0)
return 1;
return 0;
}
/// A simple small smoke test:
/// - a uniformly sampled block
/// - reconstruct main axes
/// - check order of eigenvalues equals order of extends of block in direction of main axes
int smokeTest()
{
using glm::vec3;
using glm::mat3;
std::vector<vec3> pts;
pts.reserve(11 * 15 * 7);
for(int x = -5; x <= 5; ++x)
for(int y = -7; y <= 7; ++y)
for(int z = -3; z <= 3; ++z)
pts.push_back(vec3{ x, y, z });
mat3 covar = glm::computeCovarianceMatrix(pts.data(), pts.size());
mat3 eVec;
vec3 eVal;
int eCnt = glm::findEigenvaluesSymReal(covar, eVal, eVec);
if(eCnt != 3)
return 1;
// sort eVec by decending eVal
if(eVal[0] < eVal[1])
{
std::swap(eVal[0], eVal[1]);
std::swap(eVec[0], eVec[1]);
}
if(eVal[0] < eVal[2])
{
std::swap(eVal[0], eVal[2]);
std::swap(eVec[0], eVec[2]);
}
if(eVal[1] < eVal[2])
{
std::swap(eVal[1], eVal[2]);
std::swap(eVec[1], eVec[2]);
}
if(!glm::all(glm::equal(glm::abs(eVec[0]), vec3{ 0, 1, 0 })))
return 2;
if(!glm::all(glm::equal(glm::abs(eVec[1]), vec3{ 1, 0, 0 })))
return 3;
if(!glm::all(glm::equal(glm::abs(eVec[2]), vec3{ 0, 0, 1 })))
return 4;
return 0;
}
int rndTest(unsigned int randomEngineSeed)
{
std::default_random_engine rndEng{ randomEngineSeed };
std::normal_distribution<double> normalDist;
// construct orthonormal system
glm::dvec3 x{ normalDist(rndEng), normalDist(rndEng), normalDist(rndEng) };
double l = glm::length(x);
while(l < 0.000001)
x = glm::dvec3{ normalDist(rndEng), normalDist(rndEng), normalDist(rndEng) };
x = glm::normalize(x);
glm::dvec3 y{ normalDist(rndEng), normalDist(rndEng), normalDist(rndEng) };
l = glm::length(y);
while(l < 0.000001)
y = glm::dvec3{ normalDist(rndEng), normalDist(rndEng), normalDist(rndEng) };
while(glm::abs(glm::dot(x, y)) < 0.000001)
{
y = glm::dvec3{ normalDist(rndEng), normalDist(rndEng), normalDist(rndEng) };
while(l < 0.000001)
y = glm::dvec3{ normalDist(rndEng), normalDist(rndEng), normalDist(rndEng) };
}
y = glm::normalize(y);
glm::dvec3 z = glm::normalize(glm::cross(x, y));
y = glm::normalize(glm::cross(z, x));
//printf("\n");
//printf("x: %.10lf, %.10lf, %.10lf\n", x.x, x.y, x.z);
//printf("y: %.10lf, %.10lf, %.10lf\n", y.x, y.y, y.z);
//printf("z: %.10lf, %.10lf, %.10lf\n", z.x, z.y, z.z);
// generate input point data
std::vector<glm::dvec3> ptData;
constexpr int patters[] = {
8, 0, 0,
4, 1, 2,
0, 2, 0,
0, 0, 4
};
glm::dvec3 offset{ normalDist(rndEng), normalDist(rndEng), normalDist(rndEng) };
for(int p = 0; p < 4; ++p)
for(int xs = 1; xs >= -1; xs -= 2)
for(int ys = 1; ys >= -1; ys -= 2)
for(int zs = 1; zs >= -1; zs -= 2)
ptData.push_back(
offset
+ x * static_cast<double>(patters[p * 3 + 0] * xs)
+ y * static_cast<double>(patters[p * 3 + 1] * ys)
+ z * static_cast<double>(patters[p * 3 + 2] * zs));
// perform PCA:
glm::dvec3 center = computeCenter(ptData);
glm::dmat3 covarMat = glm::computeCovarianceMatrix(ptData.data(), ptData.size(), center);
glm::dvec3 evals;
glm::dmat3 evecs;
int evcnt = glm::findEigenvaluesSymReal(covarMat, evals, evecs);
if(evcnt != 3)
return 1;
glm::sortEigenvalues(evals, evecs);
//printf("\n");
//printf("evec0: %.10lf, %.10lf, %.10lf\n", evecs[0].x, evecs[0].y, evecs[0].z);
//printf("evec2: %.10lf, %.10lf, %.10lf\n", evecs[2].x, evecs[2].y, evecs[2].z);
//printf("evec1: %.10lf, %.10lf, %.10lf\n", evecs[1].x, evecs[1].y, evecs[1].z);
if(glm::length(glm::abs(x) - glm::abs(evecs[0])) > 0.000001)
return 1;
if(glm::length(glm::abs(y) - glm::abs(evecs[2])) > 0.000001)
return 1;
if(glm::length(glm::abs(z) - glm::abs(evecs[1])) > 0.000001)
return 1;
return 0;
}
int main()
{
// A small smoke test to fail early with most problems
if(smokeTest())
return __LINE__;
// test sorting utility.
if(testEigenvalueSort<2, float, glm::defaultp>() != 0)
return __LINE__;
if(testEigenvalueSort<2, double, glm::defaultp>() != 0)
return __LINE__;
if(testEigenvalueSort<3, float, glm::defaultp>() != 0)
return __LINE__;
if(testEigenvalueSort<3, double, glm::defaultp>() != 0)
return __LINE__;
if(testEigenvalueSort<4, float, glm::defaultp>() != 0)
return __LINE__;
if(testEigenvalueSort<4, double, glm::defaultp>() != 0)
return __LINE__;
// Note: the random engine uses a fixed seed to create consistent and reproducible test data
// test covariance matrix computation from different data sources
if(testCovar<2, float, glm::defaultp>(100, 12345) != 0)
return __LINE__;
if(testCovar<2, double, glm::defaultp>(100, 42) != 0)
return __LINE__;
if(testCovar<3, float, glm::defaultp>(100, 2021) != 0)
return __LINE__;
if(testCovar<3, double, glm::defaultp>(100, 815) != 0)
return __LINE__;
if(testCovar<4, float, glm::defaultp>(100, 3141) != 0)
return __LINE__;
if(testCovar<4, double, glm::defaultp>(100, 174) != 0)
return __LINE__;
// test PCA eigen vector reconstruction
if(testEigenvectors<2, float, glm::defaultp>() != 0)
return __LINE__;
if(testEigenvectors<2, double, glm::defaultp>() != 0)
return __LINE__;
if(testEigenvectors<3, float, glm::defaultp>() != 0)
return __LINE__;
if(testEigenvectors<3, double, glm::defaultp>() != 0)
return __LINE__;
if (testEigenvectors<4, float, glm::defaultp>() != 0)
return __LINE__;
if (testEigenvectors<4, double, glm::defaultp>() != 0)
return __LINE__;
// Final tests with randomized data
if(rndTest(12345) != 0)
return __LINE__;
if(rndTest(42) != 0)
return __LINE__;
return 0;
}