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diff --git a/thirdparty/thekla_atlas/nvmath/Fitting.cpp b/thirdparty/thekla_atlas/nvmath/Fitting.cpp
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+// This code is in the public domain -- Ignacio Castaño <castano@gmail.com>
+
+#include "Fitting.h"
+#include "Vector.inl"
+#include "Plane.inl"
+
+#include "nvcore/Array.inl"
+#include "nvcore/Utils.h" // max, swap
+
+#include <float.h> // FLT_MAX
+//#include <vector>
+#include <string.h>
+
+using namespace nv;
+
+// @@ Move to EigenSolver.h
+
+// @@ We should be able to do something cheaper...
+static Vector3 estimatePrincipalComponent(const float * __restrict matrix)
+{
+ const Vector3 row0(matrix[0], matrix[1], matrix[2]);
+ const Vector3 row1(matrix[1], matrix[3], matrix[4]);
+ const Vector3 row2(matrix[2], matrix[4], matrix[5]);
+
+ float r0 = lengthSquared(row0);
+ float r1 = lengthSquared(row1);
+ float r2 = lengthSquared(row2);
+
+ if (r0 > r1 && r0 > r2) return row0;
+ if (r1 > r2) return row1;
+ return row2;
+}
+
+
+static inline Vector3 firstEigenVector_PowerMethod(const float *__restrict matrix)
+{
+ if (matrix[0] == 0 && matrix[3] == 0 && matrix[5] == 0)
+ {
+ return Vector3(0.0f);
+ }
+
+ Vector3 v = estimatePrincipalComponent(matrix);
+
+ const int NUM = 8;
+ for (int i = 0; i < NUM; i++)
+ {
+ float x = v.x * matrix[0] + v.y * matrix[1] + v.z * matrix[2];
+ float y = v.x * matrix[1] + v.y * matrix[3] + v.z * matrix[4];
+ float z = v.x * matrix[2] + v.y * matrix[4] + v.z * matrix[5];
+
+ float norm = max(max(x, y), z);
+
+ v = Vector3(x, y, z) / norm;
+ }
+
+ return v;
+}
+
+
+Vector3 nv::Fit::computeCentroid(int n, const Vector3 *__restrict points)
+{
+ Vector3 centroid(0.0f);
+
+ for (int i = 0; i < n; i++)
+ {
+ centroid += points[i];
+ }
+ centroid /= float(n);
+
+ return centroid;
+}
+
+Vector3 nv::Fit::computeCentroid(int n, const Vector3 *__restrict points, const float *__restrict weights, Vector3::Arg metric)
+{
+ Vector3 centroid(0.0f);
+ float total = 0.0f;
+
+ for (int i = 0; i < n; i++)
+ {
+ total += weights[i];
+ centroid += weights[i]*points[i];
+ }
+ centroid /= total;
+
+ return centroid;
+}
+
+Vector4 nv::Fit::computeCentroid(int n, const Vector4 *__restrict points)
+{
+ Vector4 centroid(0.0f);
+
+ for (int i = 0; i < n; i++)
+ {
+ centroid += points[i];
+ }
+ centroid /= float(n);
+
+ return centroid;
+}
+
+Vector4 nv::Fit::computeCentroid(int n, const Vector4 *__restrict points, const float *__restrict weights, Vector4::Arg metric)
+{
+ Vector4 centroid(0.0f);
+ float total = 0.0f;
+
+ for (int i = 0; i < n; i++)
+ {
+ total += weights[i];
+ centroid += weights[i]*points[i];
+ }
+ centroid /= total;
+
+ return centroid;
+}
+
+
+
+Vector3 nv::Fit::computeCovariance(int n, const Vector3 *__restrict points, float *__restrict covariance)
+{
+ // compute the centroid
+ Vector3 centroid = computeCentroid(n, points);
+
+ // compute covariance matrix
+ for (int i = 0; i < 6; i++)
+ {
+ covariance[i] = 0.0f;
+ }
+
+ for (int i = 0; i < n; i++)
+ {
+ Vector3 v = points[i] - centroid;
+
+ covariance[0] += v.x * v.x;
+ covariance[1] += v.x * v.y;
+ covariance[2] += v.x * v.z;
+ covariance[3] += v.y * v.y;
+ covariance[4] += v.y * v.z;
+ covariance[5] += v.z * v.z;
+ }
+
+ return centroid;
+}
+
+Vector3 nv::Fit::computeCovariance(int n, const Vector3 *__restrict points, const float *__restrict weights, Vector3::Arg metric, float *__restrict covariance)
+{
+ // compute the centroid
+ Vector3 centroid = computeCentroid(n, points, weights, metric);
+
+ // compute covariance matrix
+ for (int i = 0; i < 6; i++)
+ {
+ covariance[i] = 0.0f;
+ }
+
+ for (int i = 0; i < n; i++)
+ {
+ Vector3 a = (points[i] - centroid) * metric;
+ Vector3 b = weights[i]*a;
+
+ covariance[0] += a.x * b.x;
+ covariance[1] += a.x * b.y;
+ covariance[2] += a.x * b.z;
+ covariance[3] += a.y * b.y;
+ covariance[4] += a.y * b.z;
+ covariance[5] += a.z * b.z;
+ }
+
+ return centroid;
+}
+
+Vector4 nv::Fit::computeCovariance(int n, const Vector4 *__restrict points, float *__restrict covariance)
+{
+ // compute the centroid
+ Vector4 centroid = computeCentroid(n, points);
+
+ // compute covariance matrix
+ for (int i = 0; i < 10; i++)
+ {
+ covariance[i] = 0.0f;
+ }
+
+ for (int i = 0; i < n; i++)
+ {
+ Vector4 v = points[i] - centroid;
+
+ covariance[0] += v.x * v.x;
+ covariance[1] += v.x * v.y;
+ covariance[2] += v.x * v.z;
+ covariance[3] += v.x * v.w;
+
+ covariance[4] += v.y * v.y;
+ covariance[5] += v.y * v.z;
+ covariance[6] += v.y * v.w;
+
+ covariance[7] += v.z * v.z;
+ covariance[8] += v.z * v.w;
+
+ covariance[9] += v.w * v.w;
+ }
+
+ return centroid;
+}
+
+Vector4 nv::Fit::computeCovariance(int n, const Vector4 *__restrict points, const float *__restrict weights, Vector4::Arg metric, float *__restrict covariance)
+{
+ // compute the centroid
+ Vector4 centroid = computeCentroid(n, points, weights, metric);
+
+ // compute covariance matrix
+ for (int i = 0; i < 10; i++)
+ {
+ covariance[i] = 0.0f;
+ }
+
+ for (int i = 0; i < n; i++)
+ {
+ Vector4 a = (points[i] - centroid) * metric;
+ Vector4 b = weights[i]*a;
+
+ covariance[0] += a.x * b.x;
+ covariance[1] += a.x * b.y;
+ covariance[2] += a.x * b.z;
+ covariance[3] += a.x * b.w;
+
+ covariance[4] += a.y * b.y;
+ covariance[5] += a.y * b.z;
+ covariance[6] += a.y * b.w;
+
+ covariance[7] += a.z * b.z;
+ covariance[8] += a.z * b.w;
+
+ covariance[9] += a.w * b.w;
+ }
+
+ return centroid;
+}
+
+
+
+Vector3 nv::Fit::computePrincipalComponent_PowerMethod(int n, const Vector3 *__restrict points)
+{
+ float matrix[6];
+ computeCovariance(n, points, matrix);
+
+ return firstEigenVector_PowerMethod(matrix);
+}
+
+Vector3 nv::Fit::computePrincipalComponent_PowerMethod(int n, const Vector3 *__restrict points, const float *__restrict weights, Vector3::Arg metric)
+{
+ float matrix[6];
+ computeCovariance(n, points, weights, metric, matrix);
+
+ return firstEigenVector_PowerMethod(matrix);
+}
+
+
+
+static inline Vector3 firstEigenVector_EigenSolver3(const float *__restrict matrix)
+{
+ if (matrix[0] == 0 && matrix[3] == 0 && matrix[5] == 0)
+ {
+ return Vector3(0.0f);
+ }
+
+ float eigenValues[3];
+ Vector3 eigenVectors[3];
+ if (!nv::Fit::eigenSolveSymmetric3(matrix, eigenValues, eigenVectors))
+ {
+ return Vector3(0.0f);
+ }
+
+ return eigenVectors[0];
+}
+
+Vector3 nv::Fit::computePrincipalComponent_EigenSolver(int n, const Vector3 *__restrict points)
+{
+ float matrix[6];
+ computeCovariance(n, points, matrix);
+
+ return firstEigenVector_EigenSolver3(matrix);
+}
+
+Vector3 nv::Fit::computePrincipalComponent_EigenSolver(int n, const Vector3 *__restrict points, const float *__restrict weights, Vector3::Arg metric)
+{
+ float matrix[6];
+ computeCovariance(n, points, weights, metric, matrix);
+
+ return firstEigenVector_EigenSolver3(matrix);
+}
+
+
+
+static inline Vector4 firstEigenVector_EigenSolver4(const float *__restrict matrix)
+{
+ if (matrix[0] == 0 && matrix[4] == 0 && matrix[7] == 0&& matrix[9] == 0)
+ {
+ return Vector4(0.0f);
+ }
+
+ float eigenValues[4];
+ Vector4 eigenVectors[4];
+ if (!nv::Fit::eigenSolveSymmetric4(matrix, eigenValues, eigenVectors))
+ {
+ return Vector4(0.0f);
+ }
+
+ return eigenVectors[0];
+}
+
+Vector4 nv::Fit::computePrincipalComponent_EigenSolver(int n, const Vector4 *__restrict points)
+{
+ float matrix[10];
+ computeCovariance(n, points, matrix);
+
+ return firstEigenVector_EigenSolver4(matrix);
+}
+
+Vector4 nv::Fit::computePrincipalComponent_EigenSolver(int n, const Vector4 *__restrict points, const float *__restrict weights, Vector4::Arg metric)
+{
+ float matrix[10];
+ computeCovariance(n, points, weights, metric, matrix);
+
+ return firstEigenVector_EigenSolver4(matrix);
+}
+
+
+
+void ArvoSVD(int rows, int cols, float * Q, float * diag, float * R);
+
+Vector3 nv::Fit::computePrincipalComponent_SVD(int n, const Vector3 *__restrict points)
+{
+ // Store the points in an n x n matrix
+ Array<float> Q; Q.resize(n*n, 0.0f);
+ for (int i = 0; i < n; ++i)
+ {
+ Q[i*n+0] = points[i].x;
+ Q[i*n+1] = points[i].y;
+ Q[i*n+2] = points[i].z;
+ }
+
+ // Alloc space for the SVD outputs
+ Array<float> diag; diag.resize(n, 0.0f);
+ Array<float> R; R.resize(n*n, 0.0f);
+
+ ArvoSVD(n, n, &Q[0], &diag[0], &R[0]);
+
+ // Get the principal component
+ return Vector3(R[0], R[1], R[2]);
+}
+
+Vector4 nv::Fit::computePrincipalComponent_SVD(int n, const Vector4 *__restrict points)
+{
+ // Store the points in an n x n matrix
+ Array<float> Q; Q.resize(n*n, 0.0f);
+ for (int i = 0; i < n; ++i)
+ {
+ Q[i*n+0] = points[i].x;
+ Q[i*n+1] = points[i].y;
+ Q[i*n+2] = points[i].z;
+ Q[i*n+3] = points[i].w;
+ }
+
+ // Alloc space for the SVD outputs
+ Array<float> diag; diag.resize(n, 0.0f);
+ Array<float> R; R.resize(n*n, 0.0f);
+
+ ArvoSVD(n, n, &Q[0], &diag[0], &R[0]);
+
+ // Get the principal component
+ return Vector4(R[0], R[1], R[2], R[3]);
+}
+
+
+
+Plane nv::Fit::bestPlane(int n, const Vector3 *__restrict points)
+{
+ // compute the centroid and covariance
+ float matrix[6];
+ Vector3 centroid = computeCovariance(n, points, matrix);
+
+ if (matrix[0] == 0 && matrix[3] == 0 && matrix[5] == 0)
+ {
+ // If no plane defined, then return a horizontal plane.
+ return Plane(Vector3(0, 0, 1), centroid);
+ }
+
+ float eigenValues[3];
+ Vector3 eigenVectors[3];
+ if (!eigenSolveSymmetric3(matrix, eigenValues, eigenVectors)) {
+ // If no plane defined, then return a horizontal plane.
+ return Plane(Vector3(0, 0, 1), centroid);
+ }
+
+ return Plane(eigenVectors[2], centroid);
+}
+
+bool nv::Fit::isPlanar(int n, const Vector3 * points, float epsilon/*=NV_EPSILON*/)
+{
+ // compute the centroid and covariance
+ float matrix[6];
+ computeCovariance(n, points, matrix);
+
+ float eigenValues[3];
+ Vector3 eigenVectors[3];
+ if (!eigenSolveSymmetric3(matrix, eigenValues, eigenVectors)) {
+ return false;
+ }
+
+ return eigenValues[2] < epsilon;
+}
+
+
+
+// Tridiagonal solver from Charles Bloom.
+// Householder transforms followed by QL decomposition.
+// Seems to be based on the code from Numerical Recipes in C.
+
+static void EigenSolver3_Tridiagonal(float mat[3][3], float * diag, float * subd);
+static bool EigenSolver3_QLAlgorithm(float mat[3][3], float * diag, float * subd);
+
+bool nv::Fit::eigenSolveSymmetric3(const float matrix[6], float eigenValues[3], Vector3 eigenVectors[3])
+{
+ nvDebugCheck(matrix != NULL && eigenValues != NULL && eigenVectors != NULL);
+
+ float subd[3];
+ float diag[3];
+ float work[3][3];
+
+ work[0][0] = matrix[0];
+ work[0][1] = work[1][0] = matrix[1];
+ work[0][2] = work[2][0] = matrix[2];
+ work[1][1] = matrix[3];
+ work[1][2] = work[2][1] = matrix[4];
+ work[2][2] = matrix[5];
+
+ EigenSolver3_Tridiagonal(work, diag, subd);
+ if (!EigenSolver3_QLAlgorithm(work, diag, subd))
+ {
+ for (int i = 0; i < 3; i++) {
+ eigenValues[i] = 0;
+ eigenVectors[i] = Vector3(0);
+ }
+ return false;
+ }
+
+ for (int i = 0; i < 3; i++) {
+ eigenValues[i] = (float)diag[i];
+ }
+
+ // eigenvectors are the columns; make them the rows :
+
+ for (int i=0; i < 3; i++)
+ {
+ for (int j = 0; j < 3; j++)
+ {
+ eigenVectors[j].component[i] = (float) work[i][j];
+ }
+ }
+
+ // shuffle to sort by singular value :
+ if (eigenValues[2] > eigenValues[0] && eigenValues[2] > eigenValues[1])
+ {
+ swap(eigenValues[0], eigenValues[2]);
+ swap(eigenVectors[0], eigenVectors[2]);
+ }
+ if (eigenValues[1] > eigenValues[0])
+ {
+ swap(eigenValues[0], eigenValues[1]);
+ swap(eigenVectors[0], eigenVectors[1]);
+ }
+ if (eigenValues[2] > eigenValues[1])
+ {
+ swap(eigenValues[1], eigenValues[2]);
+ swap(eigenVectors[1], eigenVectors[2]);
+ }
+
+ nvDebugCheck(eigenValues[0] >= eigenValues[1] && eigenValues[0] >= eigenValues[2]);
+ nvDebugCheck(eigenValues[1] >= eigenValues[2]);
+
+ return true;
+}
+
+static void EigenSolver3_Tridiagonal(float mat[3][3], float * diag, float * subd)
+{
+ // Householder reduction T = Q^t M Q
+ // Input:
+ // mat, symmetric 3x3 matrix M
+ // Output:
+ // mat, orthogonal matrix Q
+ // diag, diagonal entries of T
+ // subd, subdiagonal entries of T (T is symmetric)
+ const float epsilon = 1e-08f;
+
+ float a = mat[0][0];
+ float b = mat[0][1];
+ float c = mat[0][2];
+ float d = mat[1][1];
+ float e = mat[1][2];
+ float f = mat[2][2];
+
+ diag[0] = a;
+ subd[2] = 0.f;
+ if (fabsf(c) >= epsilon)
+ {
+ const float ell = sqrtf(b*b+c*c);
+ b /= ell;
+ c /= ell;
+ const float q = 2*b*e+c*(f-d);
+ diag[1] = d+c*q;
+ diag[2] = f-c*q;
+ subd[0] = ell;
+ subd[1] = e-b*q;
+ mat[0][0] = 1; mat[0][1] = 0; mat[0][2] = 0;
+ mat[1][0] = 0; mat[1][1] = b; mat[1][2] = c;
+ mat[2][0] = 0; mat[2][1] = c; mat[2][2] = -b;
+ }
+ else
+ {
+ diag[1] = d;
+ diag[2] = f;
+ subd[0] = b;
+ subd[1] = e;
+ mat[0][0] = 1; mat[0][1] = 0; mat[0][2] = 0;
+ mat[1][0] = 0; mat[1][1] = 1; mat[1][2] = 0;
+ mat[2][0] = 0; mat[2][1] = 0; mat[2][2] = 1;
+ }
+}
+
+static bool EigenSolver3_QLAlgorithm(float mat[3][3], float * diag, float * subd)
+{
+ // QL iteration with implicit shifting to reduce matrix from tridiagonal
+ // to diagonal
+ const int maxiter = 32;
+
+ for (int ell = 0; ell < 3; ell++)
+ {
+ int iter;
+ for (iter = 0; iter < maxiter; iter++)
+ {
+ int m;
+ for (m = ell; m <= 1; m++)
+ {
+ float dd = fabsf(diag[m]) + fabsf(diag[m+1]);
+ if ( fabsf(subd[m]) + dd == dd )
+ break;
+ }
+ if ( m == ell )
+ break;
+
+ float g = (diag[ell+1]-diag[ell])/(2*subd[ell]);
+ float r = sqrtf(g*g+1);
+ if ( g < 0 )
+ g = diag[m]-diag[ell]+subd[ell]/(g-r);
+ else
+ g = diag[m]-diag[ell]+subd[ell]/(g+r);
+ float s = 1, c = 1, p = 0;
+ for (int i = m-1; i >= ell; i--)
+ {
+ float f = s*subd[i], b = c*subd[i];
+ if ( fabsf(f) >= fabsf(g) )
+ {
+ c = g/f;
+ r = sqrtf(c*c+1);
+ subd[i+1] = f*r;
+ c *= (s = 1/r);
+ }
+ else
+ {
+ s = f/g;
+ r = sqrtf(s*s+1);
+ subd[i+1] = g*r;
+ s *= (c = 1/r);
+ }
+ g = diag[i+1]-p;
+ r = (diag[i]-g)*s+2*b*c;
+ p = s*r;
+ diag[i+1] = g+p;
+ g = c*r-b;
+
+ for (int k = 0; k < 3; k++)
+ {
+ f = mat[k][i+1];
+ mat[k][i+1] = s*mat[k][i]+c*f;
+ mat[k][i] = c*mat[k][i]-s*f;
+ }
+ }
+ diag[ell] -= p;
+ subd[ell] = g;
+ subd[m] = 0;
+ }
+
+ if ( iter == maxiter )
+ // should not get here under normal circumstances
+ return false;
+ }
+
+ return true;
+}
+
+
+
+// Tridiagonal solver for 4x4 symmetric matrices.
+
+static void EigenSolver4_Tridiagonal(float mat[4][4], float * diag, float * subd);
+static bool EigenSolver4_QLAlgorithm(float mat[4][4], float * diag, float * subd);
+
+bool nv::Fit::eigenSolveSymmetric4(const float matrix[10], float eigenValues[4], Vector4 eigenVectors[4])
+{
+ nvDebugCheck(matrix != NULL && eigenValues != NULL && eigenVectors != NULL);
+
+ float subd[4];
+ float diag[4];
+ float work[4][4];
+
+ work[0][0] = matrix[0];
+ work[0][1] = work[1][0] = matrix[1];
+ work[0][2] = work[2][0] = matrix[2];
+ work[0][3] = work[3][0] = matrix[3];
+ work[1][1] = matrix[4];
+ work[1][2] = work[2][1] = matrix[5];
+ work[1][3] = work[3][1] = matrix[6];
+ work[2][2] = matrix[7];
+ work[2][3] = work[3][2] = matrix[8];
+ work[3][3] = matrix[9];
+
+ EigenSolver4_Tridiagonal(work, diag, subd);
+ if (!EigenSolver4_QLAlgorithm(work, diag, subd))
+ {
+ for (int i = 0; i < 4; i++) {
+ eigenValues[i] = 0;
+ eigenVectors[i] = Vector4(0);
+ }
+ return false;
+ }
+
+ for (int i = 0; i < 4; i++) {
+ eigenValues[i] = (float)diag[i];
+ }
+
+ // eigenvectors are the columns; make them the rows
+
+ for (int i = 0; i < 4; i++)
+ {
+ for (int j = 0; j < 4; j++)
+ {
+ eigenVectors[j].component[i] = (float) work[i][j];
+ }
+ }
+
+ // sort by singular value
+
+ for (int i = 0; i < 3; ++i)
+ {
+ for (int j = i+1; j < 4; ++j)
+ {
+ if (eigenValues[j] > eigenValues[i])
+ {
+ swap(eigenValues[i], eigenValues[j]);
+ swap(eigenVectors[i], eigenVectors[j]);
+ }
+ }
+ }
+
+ nvDebugCheck(eigenValues[0] >= eigenValues[1] && eigenValues[0] >= eigenValues[2] && eigenValues[0] >= eigenValues[3]);
+ nvDebugCheck(eigenValues[1] >= eigenValues[2] && eigenValues[1] >= eigenValues[3]);
+ nvDebugCheck(eigenValues[2] >= eigenValues[2]);
+
+ return true;
+}
+
+#include "nvmath/Matrix.inl"
+
+inline float signNonzero(float x)
+{
+ return (x >= 0.0f) ? 1.0f : -1.0f;
+}
+
+static void EigenSolver4_Tridiagonal(float mat[4][4], float * diag, float * subd)
+{
+ // Householder reduction T = Q^t M Q
+ // Input:
+ // mat, symmetric 3x3 matrix M
+ // Output:
+ // mat, orthogonal matrix Q
+ // diag, diagonal entries of T
+ // subd, subdiagonal entries of T (T is symmetric)
+
+ static const int n = 4;
+
+ // Set epsilon relative to size of elements in matrix
+ static const float relEpsilon = 1e-6f;
+ float maxElement = FLT_MAX;
+ for (int i = 0; i < n; ++i)
+ for (int j = 0; j < n; ++j)
+ maxElement = max(maxElement, fabsf(mat[i][j]));
+ float epsilon = relEpsilon * maxElement;
+
+ // Iterative algorithm, works for any size of matrix but might be slower than
+ // a closed-form solution for symmetric 4x4 matrices. Based on this article:
+ // http://en.wikipedia.org/wiki/Householder_transformation#Tridiagonalization
+
+ Matrix A, Q(identity);
+ memcpy(&A, mat, sizeof(float)*n*n);
+
+ // We proceed from left to right, making the off-tridiagonal entries zero in
+ // one column of the matrix at a time.
+ for (int k = 0; k < n - 2; ++k)
+ {
+ float sum = 0.0f;
+ for (int j = k+1; j < n; ++j)
+ sum += A(j,k)*A(j,k);
+ float alpha = -signNonzero(A(k+1,k)) * sqrtf(sum);
+ float r = sqrtf(0.5f * (alpha*alpha - A(k+1,k)*alpha));
+
+ // If r is zero, skip this column - already in tridiagonal form
+ if (fabsf(r) < epsilon)
+ continue;
+
+ float v[n] = {};
+ v[k+1] = 0.5f * (A(k+1,k) - alpha) / r;
+ for (int j = k+2; j < n; ++j)
+ v[j] = 0.5f * A(j,k) / r;
+
+ Matrix P(identity);
+ for (int i = 0; i < n; ++i)
+ for (int j = 0; j < n; ++j)
+ P(i,j) -= 2.0f * v[i] * v[j];
+
+ A = mul(mul(P, A), P);
+ Q = mul(Q, P);
+ }
+
+ nvDebugCheck(fabsf(A(2,0)) < epsilon);
+ nvDebugCheck(fabsf(A(0,2)) < epsilon);
+ nvDebugCheck(fabsf(A(3,0)) < epsilon);
+ nvDebugCheck(fabsf(A(0,3)) < epsilon);
+ nvDebugCheck(fabsf(A(3,1)) < epsilon);
+ nvDebugCheck(fabsf(A(1,3)) < epsilon);
+
+ for (int i = 0; i < n; ++i)
+ diag[i] = A(i,i);
+ for (int i = 0; i < n - 1; ++i)
+ subd[i] = A(i+1,i);
+ subd[n-1] = 0.0f;
+
+ memcpy(mat, &Q, sizeof(float)*n*n);
+}
+
+static bool EigenSolver4_QLAlgorithm(float mat[4][4], float * diag, float * subd)
+{
+ // QL iteration with implicit shifting to reduce matrix from tridiagonal
+ // to diagonal
+ const int maxiter = 32;
+
+ for (int ell = 0; ell < 4; ell++)
+ {
+ int iter;
+ for (iter = 0; iter < maxiter; iter++)
+ {
+ int m;
+ for (m = ell; m < 3; m++)
+ {
+ float dd = fabsf(diag[m]) + fabsf(diag[m+1]);
+ if ( fabsf(subd[m]) + dd == dd )
+ break;
+ }
+ if ( m == ell )
+ break;
+
+ float g = (diag[ell+1]-diag[ell])/(2*subd[ell]);
+ float r = sqrtf(g*g+1);
+ if ( g < 0 )
+ g = diag[m]-diag[ell]+subd[ell]/(g-r);
+ else
+ g = diag[m]-diag[ell]+subd[ell]/(g+r);
+ float s = 1, c = 1, p = 0;
+ for (int i = m-1; i >= ell; i--)
+ {
+ float f = s*subd[i], b = c*subd[i];
+ if ( fabsf(f) >= fabsf(g) )
+ {
+ c = g/f;
+ r = sqrtf(c*c+1);
+ subd[i+1] = f*r;
+ c *= (s = 1/r);
+ }
+ else
+ {
+ s = f/g;
+ r = sqrtf(s*s+1);
+ subd[i+1] = g*r;
+ s *= (c = 1/r);
+ }
+ g = diag[i+1]-p;
+ r = (diag[i]-g)*s+2*b*c;
+ p = s*r;
+ diag[i+1] = g+p;
+ g = c*r-b;
+
+ for (int k = 0; k < 4; k++)
+ {
+ f = mat[k][i+1];
+ mat[k][i+1] = s*mat[k][i]+c*f;
+ mat[k][i] = c*mat[k][i]-s*f;
+ }
+ }
+ diag[ell] -= p;
+ subd[ell] = g;
+ subd[m] = 0;
+ }
+
+ if ( iter == maxiter )
+ // should not get here under normal circumstances
+ return false;
+ }
+
+ return true;
+}
+
+
+
+int nv::Fit::compute4Means(int n, const Vector3 *__restrict points, const float *__restrict weights, Vector3::Arg metric, Vector3 *__restrict cluster)
+{
+ // Compute principal component.
+ float matrix[6];
+ Vector3 centroid = computeCovariance(n, points, weights, metric, matrix);
+ Vector3 principal = firstEigenVector_PowerMethod(matrix);
+
+ // Pick initial solution.
+ int mini, maxi;
+ mini = maxi = 0;
+
+ float mindps, maxdps;
+ mindps = maxdps = dot(points[0] - centroid, principal);
+
+ for (int i = 1; i < n; ++i)
+ {
+ float dps = dot(points[i] - centroid, principal);
+
+ if (dps < mindps) {
+ mindps = dps;
+ mini = i;
+ }
+ else {
+ maxdps = dps;
+ maxi = i;
+ }
+ }
+
+ cluster[0] = centroid + mindps * principal;
+ cluster[1] = centroid + maxdps * principal;
+ cluster[2] = (2.0f * cluster[0] + cluster[1]) / 3.0f;
+ cluster[3] = (2.0f * cluster[1] + cluster[0]) / 3.0f;
+
+ // Now we have to iteratively refine the clusters.
+ while (true)
+ {
+ Vector3 newCluster[4] = { Vector3(0.0f), Vector3(0.0f), Vector3(0.0f), Vector3(0.0f) };
+ float total[4] = {0, 0, 0, 0};
+
+ for (int i = 0; i < n; ++i)
+ {
+ // Find nearest cluster.
+ int nearest = 0;
+ float mindist = FLT_MAX;
+ for (int j = 0; j < 4; j++)
+ {
+ float dist = lengthSquared((cluster[j] - points[i]) * metric);
+ if (dist < mindist)
+ {
+ mindist = dist;
+ nearest = j;
+ }
+ }
+
+ newCluster[nearest] += weights[i] * points[i];
+ total[nearest] += weights[i];
+ }
+
+ for (int j = 0; j < 4; j++)
+ {
+ if (total[j] != 0)
+ newCluster[j] /= total[j];
+ }
+
+ if (equal(cluster[0], newCluster[0]) && equal(cluster[1], newCluster[1]) &&
+ equal(cluster[2], newCluster[2]) && equal(cluster[3], newCluster[3]))
+ {
+ return (total[0] != 0) + (total[1] != 0) + (total[2] != 0) + (total[3] != 0);
+ }
+
+ cluster[0] = newCluster[0];
+ cluster[1] = newCluster[1];
+ cluster[2] = newCluster[2];
+ cluster[3] = newCluster[3];
+
+ // Sort clusters by weight.
+ for (int i = 0; i < 4; i++)
+ {
+ for (int j = i; j > 0 && total[j] > total[j - 1]; j--)
+ {
+ swap( total[j], total[j - 1] );
+ swap( cluster[j], cluster[j - 1] );
+ }
+ }
+ }
+}
+
+
+
+// Adaptation of James Arvo's SVD code, as found in ZOH.
+
+inline float Sqr(float x) { return x*x; }
+
+inline float svd_pythag( float a, float b )
+{
+ float at = fabsf(a);
+ float bt = fabsf(b);
+ if( at > bt )
+ return at * sqrtf( 1.0f + Sqr( bt / at ) );
+ else if( bt > 0.0f )
+ return bt * sqrtf( 1.0f + Sqr( at / bt ) );
+ else return 0.0f;
+}
+
+inline float SameSign( float a, float b )
+{
+ float t;
+ if( b >= 0.0f ) t = fabsf( a );
+ else t = -fabsf( a );
+ return t;
+}
+
+void ArvoSVD(int rows, int cols, float * Q, float * diag, float * R)
+{
+ static const int MaxIterations = 30;
+
+ int i, j, k, l, p, q, iter;
+ float c, f, h, s, x, y, z;
+ float norm = 0.0f;
+ float g = 0.0f;
+ float scale = 0.0f;
+
+ Array<float> temp; temp.resize(cols, 0.0f);
+
+ for( i = 0; i < cols; i++ )
+ {
+ temp[i] = scale * g;
+ scale = 0.0f;
+ g = 0.0f;
+ s = 0.0f;
+ l = i + 1;
+
+ if( i < rows )
+ {
+ for( k = i; k < rows; k++ ) scale += fabsf( Q[k*cols+i] );
+ if( scale != 0.0f )
+ {
+ for( k = i; k < rows; k++ )
+ {
+ Q[k*cols+i] /= scale;
+ s += Sqr( Q[k*cols+i] );
+ }
+ f = Q[i*cols+i];
+ g = -SameSign( sqrtf(s), f );
+ h = f * g - s;
+ Q[i*cols+i] = f - g;
+ if( i != cols - 1 )
+ {
+ for( j = l; j < cols; j++ )
+ {
+ s = 0.0f;
+ for( k = i; k < rows; k++ ) s += Q[k*cols+i] * Q[k*cols+j];
+ f = s / h;
+ for( k = i; k < rows; k++ ) Q[k*cols+j] += f * Q[k*cols+i];
+ }
+ }
+ for( k = i; k < rows; k++ ) Q[k*cols+i] *= scale;
+ }
+ }
+
+ diag[i] = scale * g;
+ g = 0.0f;
+ s = 0.0f;
+ scale = 0.0f;
+
+ if( i < rows && i != cols - 1 )
+ {
+ for( k = l; k < cols; k++ ) scale += fabsf( Q[i*cols+k] );
+ if( scale != 0.0f )
+ {
+ for( k = l; k < cols; k++ )
+ {
+ Q[i*cols+k] /= scale;
+ s += Sqr( Q[i*cols+k] );
+ }
+ f = Q[i*cols+l];
+ g = -SameSign( sqrtf(s), f );
+ h = f * g - s;
+ Q[i*cols+l] = f - g;
+ for( k = l; k < cols; k++ ) temp[k] = Q[i*cols+k] / h;
+ if( i != rows - 1 )
+ {
+ for( j = l; j < rows; j++ )
+ {
+ s = 0.0f;
+ for( k = l; k < cols; k++ ) s += Q[j*cols+k] * Q[i*cols+k];
+ for( k = l; k < cols; k++ ) Q[j*cols+k] += s * temp[k];
+ }
+ }
+ for( k = l; k < cols; k++ ) Q[i*cols+k] *= scale;
+ }
+ }
+ norm = max( norm, fabsf( diag[i] ) + fabsf( temp[i] ) );
+ }
+
+
+ for( i = cols - 1; i >= 0; i-- )
+ {
+ if( i < cols - 1 )
+ {
+ if( g != 0.0f )
+ {
+ for( j = l; j < cols; j++ ) R[i*cols+j] = ( Q[i*cols+j] / Q[i*cols+l] ) / g;
+ for( j = l; j < cols; j++ )
+ {
+ s = 0.0f;
+ for( k = l; k < cols; k++ ) s += Q[i*cols+k] * R[j*cols+k];
+ for( k = l; k < cols; k++ ) R[j*cols+k] += s * R[i*cols+k];
+ }
+ }
+ for( j = l; j < cols; j++ )
+ {
+ R[i*cols+j] = 0.0f;
+ R[j*cols+i] = 0.0f;
+ }
+ }
+ R[i*cols+i] = 1.0f;
+ g = temp[i];
+ l = i;
+ }
+
+
+ for( i = cols - 1; i >= 0; i-- )
+ {
+ l = i + 1;
+ g = diag[i];
+ if( i < cols - 1 ) for( j = l; j < cols; j++ ) Q[i*cols+j] = 0.0f;
+ if( g != 0.0f )
+ {
+ g = 1.0f / g;
+ if( i != cols - 1 )
+ {
+ for( j = l; j < cols; j++ )
+ {
+ s = 0.0f;
+ for( k = l; k < rows; k++ ) s += Q[k*cols+i] * Q[k*cols+j];
+ f = ( s / Q[i*cols+i] ) * g;
+ for( k = i; k < rows; k++ ) Q[k*cols+j] += f * Q[k*cols+i];
+ }
+ }
+ for( j = i; j < rows; j++ ) Q[j*cols+i] *= g;
+ }
+ else
+ {
+ for( j = i; j < rows; j++ ) Q[j*cols+i] = 0.0f;
+ }
+ Q[i*cols+i] += 1.0f;
+ }
+
+
+ for( k = cols - 1; k >= 0; k-- )
+ {
+ for( iter = 1; iter <= MaxIterations; iter++ )
+ {
+ int jump = 0;
+
+ for( l = k; l >= 0; l-- )
+ {
+ q = l - 1;
+ if( fabsf( temp[l] ) + norm == norm ) { jump = 1; break; }
+ if( fabsf( diag[q] ) + norm == norm ) { jump = 0; break; }
+ }
+
+ if( !jump )
+ {
+ c = 0.0f;
+ s = 1.0f;
+ for( i = l; i <= k; i++ )
+ {
+ f = s * temp[i];
+ temp[i] *= c;
+ if( fabsf( f ) + norm == norm ) break;
+ g = diag[i];
+ h = svd_pythag( f, g );
+ diag[i] = h;
+ h = 1.0f / h;
+ c = g * h;
+ s = -f * h;
+ for( j = 0; j < rows; j++ )
+ {
+ y = Q[j*cols+q];
+ z = Q[j*cols+i];
+ Q[j*cols+q] = y * c + z * s;
+ Q[j*cols+i] = z * c - y * s;
+ }
+ }
+ }
+
+ z = diag[k];
+ if( l == k )
+ {
+ if( z < 0.0f )
+ {
+ diag[k] = -z;
+ for( j = 0; j < cols; j++ ) R[k*cols+j] *= -1.0f;
+ }
+ break;
+ }
+ if( iter >= MaxIterations ) return;
+ x = diag[l];
+ q = k - 1;
+ y = diag[q];
+ g = temp[q];
+ h = temp[k];
+ f = ( ( y - z ) * ( y + z ) + ( g - h ) * ( g + h ) ) / ( 2.0f * h * y );
+ g = svd_pythag( f, 1.0f );
+ f = ( ( x - z ) * ( x + z ) + h * ( ( y / ( f + SameSign( g, f ) ) ) - h ) ) / x;
+ c = 1.0f;
+ s = 1.0f;
+ for( j = l; j <= q; j++ )
+ {
+ i = j + 1;
+ g = temp[i];
+ y = diag[i];
+ h = s * g;
+ g = c * g;
+ z = svd_pythag( f, h );
+ temp[j] = z;
+ c = f / z;
+ s = h / z;
+ f = x * c + g * s;
+ g = g * c - x * s;
+ h = y * s;
+ y = y * c;
+ for( p = 0; p < cols; p++ )
+ {
+ x = R[j*cols+p];
+ z = R[i*cols+p];
+ R[j*cols+p] = x * c + z * s;
+ R[i*cols+p] = z * c - x * s;
+ }
+ z = svd_pythag( f, h );
+ diag[j] = z;
+ if( z != 0.0f )
+ {
+ z = 1.0f / z;
+ c = f * z;
+ s = h * z;
+ }
+ f = c * g + s * y;
+ x = c * y - s * g;
+ for( p = 0; p < rows; p++ )
+ {
+ y = Q[p*cols+j];
+ z = Q[p*cols+i];
+ Q[p*cols+j] = y * c + z * s;
+ Q[p*cols+i] = z * c - y * s;
+ }
+ }
+ temp[l] = 0.0f;
+ temp[k] = f;
+ diag[k] = x;
+ }
+ }
+
+ // Sort the singular values into descending order.
+
+ for( i = 0; i < cols - 1; i++ )
+ {
+ float biggest = diag[i]; // Biggest singular value so far.
+ int bindex = i; // The row/col it occurred in.
+ for( j = i + 1; j < cols; j++ )
+ {
+ if( diag[j] > biggest )
+ {
+ biggest = diag[j];
+ bindex = j;
+ }
+ }
+ if( bindex != i ) // Need to swap rows and columns.
+ {
+ // Swap columns in Q.
+ for (int j = 0; j < rows; ++j)
+ swap(Q[j*cols+i], Q[j*cols+bindex]);
+
+ // Swap rows in R.
+ for (int j = 0; j < rows; ++j)
+ swap(R[i*cols+j], R[bindex*cols+j]);
+
+ // Swap elements in diag.
+ swap(diag[i], diag[bindex]);
+ }
+ }
+}
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