#include "precomp.hpp"
using namespace std;
-
+using namespace cv;
class dls
{
double cost__; // optimal found solution
};
+
+class EigenvalueDecomposition {
+private:
+
+ // Holds the data dimension.
+ int n;
+
+ // Stores real/imag part of a complex division.
+ double cdivr, cdivi;
+
+ // Pointer to internal memory.
+ double *d, *e, *ort;
+ double **V, **H;
+
+ // Holds the computed eigenvalues.
+ Mat _eigenvalues;
+
+ // Holds the computed eigenvectors.
+ Mat _eigenvectors;
+
+ // Allocates memory.
+ template<typename _Tp>
+ _Tp *alloc_1d(int m) {
+ return new _Tp[m];
+ }
+
+ // Allocates memory.
+ template<typename _Tp>
+ _Tp *alloc_1d(int m, _Tp val) {
+ _Tp *arr = alloc_1d<_Tp> (m);
+ for (int i = 0; i < m; i++)
+ arr[i] = val;
+ return arr;
+ }
+
+ // Allocates memory.
+ template<typename _Tp>
+ _Tp **alloc_2d(int m, int _n) {
+ _Tp **arr = new _Tp*[m];
+ for (int i = 0; i < m; i++)
+ arr[i] = new _Tp[_n];
+ return arr;
+ }
+
+ // Allocates memory.
+ template<typename _Tp>
+ _Tp **alloc_2d(int m, int _n, _Tp val) {
+ _Tp **arr = alloc_2d<_Tp> (m, _n);
+ for (int i = 0; i < m; i++) {
+ for (int j = 0; j < _n; j++) {
+ arr[i][j] = val;
+ }
+ }
+ return arr;
+ }
+
+ void cdiv(double xr, double xi, double yr, double yi) {
+ double r, dv;
+ if (std::abs(yr) > std::abs(yi)) {
+ r = yi / yr;
+ dv = yr + r * yi;
+ cdivr = (xr + r * xi) / dv;
+ cdivi = (xi - r * xr) / dv;
+ } else {
+ r = yr / yi;
+ dv = yi + r * yr;
+ cdivr = (r * xr + xi) / dv;
+ cdivi = (r * xi - xr) / dv;
+ }
+ }
+
+ // Nonsymmetric reduction from Hessenberg to real Schur form.
+
+ void hqr2() {
+
+ // This is derived from the Algol procedure hqr2,
+ // by Martin and Wilkinson, Handbook for Auto. Comp.,
+ // Vol.ii-Linear Algebra, and the corresponding
+ // Fortran subroutine in EISPACK.
+
+ // Initialize
+ int nn = this->n;
+ int n1 = nn - 1;
+ int low = 0;
+ int high = nn - 1;
+ double eps = std::pow(2.0, -52.0);
+ double exshift = 0.0;
+ double p = 0, q = 0, r = 0, s = 0, z = 0, t, w, x, y;
+
+ // Store roots isolated by balanc and compute matrix norm
+
+ double norm = 0.0;
+ for (int i = 0; i < nn; i++) {
+ if (i < low || i > high) {
+ d[i] = H[i][i];
+ e[i] = 0.0;
+ }
+ for (int j = std::max(i - 1, 0); j < nn; j++) {
+ norm = norm + std::abs(H[i][j]);
+ }
+ }
+
+ // Outer loop over eigenvalue index
+ int iter = 0;
+ while (n1 >= low) {
+
+ // Look for single small sub-diagonal element
+ int l = n1;
+ while (l > low) {
+ s = std::abs(H[l - 1][l - 1]) + std::abs(H[l][l]);
+ if (s == 0.0) {
+ s = norm;
+ }
+ if (std::abs(H[l][l - 1]) < eps * s) {
+ break;
+ }
+ l--;
+ }
+
+ // Check for convergence
+ // One root found
+
+ if (l == n1) {
+ H[n1][n1] = H[n1][n1] + exshift;
+ d[n1] = H[n1][n1];
+ e[n1] = 0.0;
+ n1--;
+ iter = 0;
+
+ // Two roots found
+
+ } else if (l == n1 - 1) {
+ w = H[n1][n1 - 1] * H[n1 - 1][n1];
+ p = (H[n1 - 1][n1 - 1] - H[n1][n1]) / 2.0;
+ q = p * p + w;
+ z = std::sqrt(std::abs(q));
+ H[n1][n1] = H[n1][n1] + exshift;
+ H[n1 - 1][n1 - 1] = H[n1 - 1][n1 - 1] + exshift;
+ x = H[n1][n1];
+
+ // Real pair
+
+ if (q >= 0) {
+ if (p >= 0) {
+ z = p + z;
+ } else {
+ z = p - z;
+ }
+ d[n1 - 1] = x + z;
+ d[n1] = d[n1 - 1];
+ if (z != 0.0) {
+ d[n1] = x - w / z;
+ }
+ e[n1 - 1] = 0.0;
+ e[n1] = 0.0;
+ x = H[n1][n1 - 1];
+ s = std::abs(x) + std::abs(z);
+ p = x / s;
+ q = z / s;
+ r = std::sqrt(p * p + q * q);
+ p = p / r;
+ q = q / r;
+
+ // Row modification
+
+ for (int j = n1 - 1; j < nn; j++) {
+ z = H[n1 - 1][j];
+ H[n1 - 1][j] = q * z + p * H[n1][j];
+ H[n1][j] = q * H[n1][j] - p * z;
+ }
+
+ // Column modification
+
+ for (int i = 0; i <= n1; i++) {
+ z = H[i][n1 - 1];
+ H[i][n1 - 1] = q * z + p * H[i][n1];
+ H[i][n1] = q * H[i][n1] - p * z;
+ }
+
+ // Accumulate transformations
+
+ for (int i = low; i <= high; i++) {
+ z = V[i][n1 - 1];
+ V[i][n1 - 1] = q * z + p * V[i][n1];
+ V[i][n1] = q * V[i][n1] - p * z;
+ }
+
+ // Complex pair
+
+ } else {
+ d[n1 - 1] = x + p;
+ d[n1] = x + p;
+ e[n1 - 1] = z;
+ e[n1] = -z;
+ }
+ n1 = n1 - 2;
+ iter = 0;
+
+ // No convergence yet
+
+ } else {
+
+ // Form shift
+
+ x = H[n1][n1];
+ y = 0.0;
+ w = 0.0;
+ if (l < n1) {
+ y = H[n1 - 1][n1 - 1];
+ w = H[n1][n1 - 1] * H[n1 - 1][n1];
+ }
+
+ // Wilkinson's original ad hoc shift
+
+ if (iter == 10) {
+ exshift += x;
+ for (int i = low; i <= n1; i++) {
+ H[i][i] -= x;
+ }
+ s = std::abs(H[n1][n1 - 1]) + std::abs(H[n1 - 1][n1 - 2]);
+ x = y = 0.75 * s;
+ w = -0.4375 * s * s;
+ }
+
+ // MATLAB's new ad hoc shift
+
+ if (iter == 30) {
+ s = (y - x) / 2.0;
+ s = s * s + w;
+ if (s > 0) {
+ s = std::sqrt(s);
+ if (y < x) {
+ s = -s;
+ }
+ s = x - w / ((y - x) / 2.0 + s);
+ for (int i = low; i <= n1; i++) {
+ H[i][i] -= s;
+ }
+ exshift += s;
+ x = y = w = 0.964;
+ }
+ }
+
+ iter = iter + 1; // (Could check iteration count here.)
+
+ // Look for two consecutive small sub-diagonal elements
+ int m = n1 - 2;
+ while (m >= l) {
+ z = H[m][m];
+ r = x - z;
+ s = y - z;
+ p = (r * s - w) / H[m + 1][m] + H[m][m + 1];
+ q = H[m + 1][m + 1] - z - r - s;
+ r = H[m + 2][m + 1];
+ s = std::abs(p) + std::abs(q) + std::abs(r);
+ p = p / s;
+ q = q / s;
+ r = r / s;
+ if (m == l) {
+ break;
+ }
+ if (std::abs(H[m][m - 1]) * (std::abs(q) + std::abs(r)) < eps * (std::abs(p)
+ * (std::abs(H[m - 1][m - 1]) + std::abs(z) + std::abs(
+ H[m + 1][m + 1])))) {
+ break;
+ }
+ m--;
+ }
+
+ for (int i = m + 2; i <= n1; i++) {
+ H[i][i - 2] = 0.0;
+ if (i > m + 2) {
+ H[i][i - 3] = 0.0;
+ }
+ }
+
+ // Double QR step involving rows l:n and columns m:n
+
+ for (int k = m; k <= n1 - 1; k++) {
+ bool notlast = (k != n1 - 1);
+ if (k != m) {
+ p = H[k][k - 1];
+ q = H[k + 1][k - 1];
+ r = (notlast ? H[k + 2][k - 1] : 0.0);
+ x = std::abs(p) + std::abs(q) + std::abs(r);
+ if (x != 0.0) {
+ p = p / x;
+ q = q / x;
+ r = r / x;
+ }
+ }
+ if (x == 0.0) {
+ break;
+ }
+ s = std::sqrt(p * p + q * q + r * r);
+ if (p < 0) {
+ s = -s;
+ }
+ if (s != 0) {
+ if (k != m) {
+ H[k][k - 1] = -s * x;
+ } else if (l != m) {
+ H[k][k - 1] = -H[k][k - 1];
+ }
+ p = p + s;
+ x = p / s;
+ y = q / s;
+ z = r / s;
+ q = q / p;
+ r = r / p;
+
+ // Row modification
+
+ for (int j = k; j < nn; j++) {
+ p = H[k][j] + q * H[k + 1][j];
+ if (notlast) {
+ p = p + r * H[k + 2][j];
+ H[k + 2][j] = H[k + 2][j] - p * z;
+ }
+ H[k][j] = H[k][j] - p * x;
+ H[k + 1][j] = H[k + 1][j] - p * y;
+ }
+
+ // Column modification
+
+ for (int i = 0; i <= std::min(n1, k + 3); i++) {
+ p = x * H[i][k] + y * H[i][k + 1];
+ if (notlast) {
+ p = p + z * H[i][k + 2];
+ H[i][k + 2] = H[i][k + 2] - p * r;
+ }
+ H[i][k] = H[i][k] - p;
+ H[i][k + 1] = H[i][k + 1] - p * q;
+ }
+
+ // Accumulate transformations
+
+ for (int i = low; i <= high; i++) {
+ p = x * V[i][k] + y * V[i][k + 1];
+ if (notlast) {
+ p = p + z * V[i][k + 2];
+ V[i][k + 2] = V[i][k + 2] - p * r;
+ }
+ V[i][k] = V[i][k] - p;
+ V[i][k + 1] = V[i][k + 1] - p * q;
+ }
+ } // (s != 0)
+ } // k loop
+ } // check convergence
+ } // while (n1 >= low)
+
+ // Backsubstitute to find vectors of upper triangular form
+
+ if (norm == 0.0) {
+ return;
+ }
+
+ for (n1 = nn - 1; n1 >= 0; n1--) {
+ p = d[n1];
+ q = e[n1];
+
+ // Real vector
+
+ if (q == 0) {
+ int l = n1;
+ H[n1][n1] = 1.0;
+ for (int i = n1 - 1; i >= 0; i--) {
+ w = H[i][i] - p;
+ r = 0.0;
+ for (int j = l; j <= n1; j++) {
+ r = r + H[i][j] * H[j][n1];
+ }
+ if (e[i] < 0.0) {
+ z = w;
+ s = r;
+ } else {
+ l = i;
+ if (e[i] == 0.0) {
+ if (w != 0.0) {
+ H[i][n1] = -r / w;
+ } else {
+ H[i][n1] = -r / (eps * norm);
+ }
+
+ // Solve real equations
+
+ } else {
+ x = H[i][i + 1];
+ y = H[i + 1][i];
+ q = (d[i] - p) * (d[i] - p) + e[i] * e[i];
+ t = (x * s - z * r) / q;
+ H[i][n1] = t;
+ if (std::abs(x) > std::abs(z)) {
+ H[i + 1][n1] = (-r - w * t) / x;
+ } else {
+ H[i + 1][n1] = (-s - y * t) / z;
+ }
+ }
+
+ // Overflow control
+
+ t = std::abs(H[i][n1]);
+ if ((eps * t) * t > 1) {
+ for (int j = i; j <= n1; j++) {
+ H[j][n1] = H[j][n1] / t;
+ }
+ }
+ }
+ }
+ // Complex vector
+ } else if (q < 0) {
+ int l = n1 - 1;
+
+ // Last vector component imaginary so matrix is triangular
+
+ if (std::abs(H[n1][n1 - 1]) > std::abs(H[n1 - 1][n1])) {
+ H[n1 - 1][n1 - 1] = q / H[n1][n1 - 1];
+ H[n1 - 1][n1] = -(H[n1][n1] - p) / H[n1][n1 - 1];
+ } else {
+ cdiv(0.0, -H[n1 - 1][n1], H[n1 - 1][n1 - 1] - p, q);
+ H[n1 - 1][n1 - 1] = cdivr;
+ H[n1 - 1][n1] = cdivi;
+ }
+ H[n1][n1 - 1] = 0.0;
+ H[n1][n1] = 1.0;
+ for (int i = n1 - 2; i >= 0; i--) {
+ double ra, sa, vr, vi;
+ ra = 0.0;
+ sa = 0.0;
+ for (int j = l; j <= n1; j++) {
+ ra = ra + H[i][j] * H[j][n1 - 1];
+ sa = sa + H[i][j] * H[j][n1];
+ }
+ w = H[i][i] - p;
+
+ if (e[i] < 0.0) {
+ z = w;
+ r = ra;
+ s = sa;
+ } else {
+ l = i;
+ if (e[i] == 0) {
+ cdiv(-ra, -sa, w, q);
+ H[i][n1 - 1] = cdivr;
+ H[i][n1] = cdivi;
+ } else {
+
+ // Solve complex equations
+
+ x = H[i][i + 1];
+ y = H[i + 1][i];
+ vr = (d[i] - p) * (d[i] - p) + e[i] * e[i] - q * q;
+ vi = (d[i] - p) * 2.0 * q;
+ if (vr == 0.0 && vi == 0.0) {
+ vr = eps * norm * (std::abs(w) + std::abs(q) + std::abs(x)
+ + std::abs(y) + std::abs(z));
+ }
+ cdiv(x * r - z * ra + q * sa,
+ x * s - z * sa - q * ra, vr, vi);
+ H[i][n1 - 1] = cdivr;
+ H[i][n1] = cdivi;
+ if (std::abs(x) > (std::abs(z) + std::abs(q))) {
+ H[i + 1][n1 - 1] = (-ra - w * H[i][n1 - 1] + q
+ * H[i][n1]) / x;
+ H[i + 1][n1] = (-sa - w * H[i][n1] - q * H[i][n1
+ - 1]) / x;
+ } else {
+ cdiv(-r - y * H[i][n1 - 1], -s - y * H[i][n1], z,
+ q);
+ H[i + 1][n1 - 1] = cdivr;
+ H[i + 1][n1] = cdivi;
+ }
+ }
+
+ // Overflow control
+
+ t = std::max(std::abs(H[i][n1 - 1]), std::abs(H[i][n1]));
+ if ((eps * t) * t > 1) {
+ for (int j = i; j <= n1; j++) {
+ H[j][n1 - 1] = H[j][n1 - 1] / t;
+ H[j][n1] = H[j][n1] / t;
+ }
+ }
+ }
+ }
+ }
+ }
+
+ // Vectors of isolated roots
+
+ for (int i = 0; i < nn; i++) {
+ if (i < low || i > high) {
+ for (int j = i; j < nn; j++) {
+ V[i][j] = H[i][j];
+ }
+ }
+ }
+
+ // Back transformation to get eigenvectors of original matrix
+
+ for (int j = nn - 1; j >= low; j--) {
+ for (int i = low; i <= high; i++) {
+ z = 0.0;
+ for (int k = low; k <= std::min(j, high); k++) {
+ z = z + V[i][k] * H[k][j];
+ }
+ V[i][j] = z;
+ }
+ }
+ }
+
+ // Nonsymmetric reduction to Hessenberg form.
+ void orthes() {
+ // This is derived from the Algol procedures orthes and ortran,
+ // by Martin and Wilkinson, Handbook for Auto. Comp.,
+ // Vol.ii-Linear Algebra, and the corresponding
+ // Fortran subroutines in EISPACK.
+ int low = 0;
+ int high = n - 1;
+
+ for (int m = low + 1; m <= high - 1; m++) {
+
+ // Scale column.
+
+ double scale = 0.0;
+ for (int i = m; i <= high; i++) {
+ scale = scale + std::abs(H[i][m - 1]);
+ }
+ if (scale != 0.0) {
+
+ // Compute Householder transformation.
+
+ double h = 0.0;
+ for (int i = high; i >= m; i--) {
+ ort[i] = H[i][m - 1] / scale;
+ h += ort[i] * ort[i];
+ }
+ double g = std::sqrt(h);
+ if (ort[m] > 0) {
+ g = -g;
+ }
+ h = h - ort[m] * g;
+ ort[m] = ort[m] - g;
+
+ // Apply Householder similarity transformation
+ // H = (I-u*u'/h)*H*(I-u*u')/h)
+
+ for (int j = m; j < n; j++) {
+ double f = 0.0;
+ for (int i = high; i >= m; i--) {
+ f += ort[i] * H[i][j];
+ }
+ f = f / h;
+ for (int i = m; i <= high; i++) {
+ H[i][j] -= f * ort[i];
+ }
+ }
+
+ for (int i = 0; i <= high; i++) {
+ double f = 0.0;
+ for (int j = high; j >= m; j--) {
+ f += ort[j] * H[i][j];
+ }
+ f = f / h;
+ for (int j = m; j <= high; j++) {
+ H[i][j] -= f * ort[j];
+ }
+ }
+ ort[m] = scale * ort[m];
+ H[m][m - 1] = scale * g;
+ }
+ }
+
+ // Accumulate transformations (Algol's ortran).
+
+ for (int i = 0; i < n; i++) {
+ for (int j = 0; j < n; j++) {
+ V[i][j] = (i == j ? 1.0 : 0.0);
+ }
+ }
+
+ for (int m = high - 1; m >= low + 1; m--) {
+ if (H[m][m - 1] != 0.0) {
+ for (int i = m + 1; i <= high; i++) {
+ ort[i] = H[i][m - 1];
+ }
+ for (int j = m; j <= high; j++) {
+ double g = 0.0;
+ for (int i = m; i <= high; i++) {
+ g += ort[i] * V[i][j];
+ }
+ // Double division avoids possible underflow
+ g = (g / ort[m]) / H[m][m - 1];
+ for (int i = m; i <= high; i++) {
+ V[i][j] += g * ort[i];
+ }
+ }
+ }
+ }
+ }
+
+ // Releases all internal working memory.
+ void release() {
+ // releases the working data
+ delete[] d;
+ delete[] e;
+ delete[] ort;
+ for (int i = 0; i < n; i++) {
+ delete[] H[i];
+ delete[] V[i];
+ }
+ delete[] H;
+ delete[] V;
+ }
+
+ // Computes the Eigenvalue Decomposition for a matrix given in H.
+ void compute() {
+ // Allocate memory for the working data.
+ V = alloc_2d<double> (n, n, 0.0);
+ d = alloc_1d<double> (n);
+ e = alloc_1d<double> (n);
+ ort = alloc_1d<double> (n);
+ // Reduce to Hessenberg form.
+ orthes();
+ // Reduce Hessenberg to real Schur form.
+ hqr2();
+ // Copy eigenvalues to OpenCV Matrix.
+ _eigenvalues.create(1, n, CV_64FC1);
+ for (int i = 0; i < n; i++) {
+ _eigenvalues.at<double> (0, i) = d[i];
+ }
+ // Copy eigenvectors to OpenCV Matrix.
+ _eigenvectors.create(n, n, CV_64FC1);
+ for (int i = 0; i < n; i++)
+ for (int j = 0; j < n; j++)
+ _eigenvectors.at<double> (i, j) = V[i][j];
+ // Deallocate the memory by releasing all internal working data.
+ release();
+ }
+
+public:
+ EigenvalueDecomposition()
+ : n(0) { }
+
+ // Initializes & computes the Eigenvalue Decomposition for a general matrix
+ // given in src. This function is a port of the EigenvalueSolver in JAMA,
+ // which has been released to public domain by The MathWorks and the
+ // National Institute of Standards and Technology (NIST).
+ EigenvalueDecomposition(InputArray src) {
+ compute(src);
+ }
+
+ // This function computes the Eigenvalue Decomposition for a general matrix
+ // given in src. This function is a port of the EigenvalueSolver in JAMA,
+ // which has been released to public domain by The MathWorks and the
+ // National Institute of Standards and Technology (NIST).
+ void compute(InputArray src)
+ {
+ /*if(isSymmetric(src)) {
+ // Fall back to OpenCV for a symmetric matrix!
+ cv::eigen(src, _eigenvalues, _eigenvectors);
+ } else {*/
+ Mat tmp;
+ // Convert the given input matrix to double. Is there any way to
+ // prevent allocating the temporary memory? Only used for copying
+ // into working memory and deallocated after.
+ src.getMat().convertTo(tmp, CV_64FC1);
+ // Get dimension of the matrix.
+ this->n = tmp.cols;
+ // Allocate the matrix data to work on.
+ this->H = alloc_2d<double> (n, n);
+ // Now safely copy the data.
+ for (int i = 0; i < tmp.rows; i++) {
+ for (int j = 0; j < tmp.cols; j++) {
+ this->H[i][j] = tmp.at<double>(i, j);
+ }
+ }
+ // Deallocates the temporary matrix before computing.
+ tmp.release();
+ // Performs the eigenvalue decomposition of H.
+ compute();
+ // }
+ }
+
+ ~EigenvalueDecomposition() {}
+
+ // Returns the eigenvalues of the Eigenvalue Decomposition.
+ Mat eigenvalues() { return _eigenvalues; }
+ // Returns the eigenvectors of the Eigenvalue Decomposition.
+ Mat eigenvectors() { return _eigenvectors; }
+};
+
#endif // DLS_H
projects / platform / upstream / opencv.git / commitdiff