dls.h

#ifndef DLS_H_
#define DLS_H_

#include "precomp.hpp"

#include <iostream>

using namespace std;
using namespace cv;

class dls
{
public:
    dls(const cv::Mat& opoints, const cv::Mat& ipoints);
    ~dls();

    bool compute_pose(cv::Mat& R, cv::Mat& t);

private:

    // initialisation
    template <typename OpointType, typename IpointType>
    void init_points(const cv::Mat& opoints, const cv::Mat& ipoints)
    {
        for(int i = 0; i < N; i++)
        {
            p.at<double>(0,i) = opoints.at<OpointType>(i).x;
            p.at<double>(1,i) = opoints.at<OpointType>(i).y;
            p.at<double>(2,i) = opoints.at<OpointType>(i).z;

            // compute mean of object points
            mn.at<double>(0) += p.at<double>(0,i);
            mn.at<double>(1) += p.at<double>(1,i);
            mn.at<double>(2) += p.at<double>(2,i);

            // make z into unit vectors from normalized pixel coords
            double sr = std::pow(ipoints.at<IpointType>(i).x, 2) +
                        std::pow(ipoints.at<IpointType>(i).y, 2) + (double)1;
                   sr = std::sqrt(sr);

            z.at<double>(0,i) = ipoints.at<IpointType>(i).x / sr;
            z.at<double>(1,i) = ipoints.at<IpointType>(i).y / sr;
            z.at<double>(2,i) = (double)1 / sr;
        }

        mn.at<double>(0) /= (double)N;
        mn.at<double>(1) /= (double)N;
        mn.at<double>(2) /= (double)N;
    }

    // main algorithm
    cv::Mat LeftMultVec(const cv::Mat& v);
    void run_kernel(const cv::Mat& pp);
    void build_coeff_matrix(const cv::Mat& pp, cv::Mat& Mtilde, cv::Mat& D);
    void compute_eigenvec(const cv::Mat& Mtilde, cv::Mat& eigenval_real, cv::Mat& eigenval_imag,
                                                 cv::Mat& eigenvec_real, cv::Mat& eigenvec_imag);
    void fill_coeff(const cv::Mat * D);

    // useful functions
    cv::Mat cayley_LS_M(const std::vector<double>& a, const std::vector<double>& b,
                        const std::vector<double>& c, const std::vector<double>& u);
    cv::Mat Hessian(const double s[]);
    cv::Mat cayley2rotbar(const cv::Mat& s);
    cv::Mat skewsymm(const cv::Mat * X1);

    // extra functions
    cv::Mat rotx(const double t);
    cv::Mat roty(const double t);
    cv::Mat rotz(const double t);
    cv::Mat mean(const cv::Mat& M);
    bool is_empty(const cv::Mat * v);
    bool positive_eigenvalues(const cv::Mat * eigenvalues);

    cv::Mat p, z, mn;        // object-image points
    int N;                // number of input points
    std::vector<double> f1coeff, f2coeff, f3coeff, cost_; // coefficient for coefficients matrix
    std::vector<cv::Mat> C_est_, t_est_;    // optimal candidates
    cv::Mat C_est__, t_est__;                // optimal found solution
    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;
                    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];
                            double vr = (d[i] - p) * (d[i] - p) + e[i] * e[i] - q * q;
                            double 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), cdivr(0), cdivi(0), d(0), e(0), ort(0), V(0), H(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