upnp.cpp

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#include "vision-precomp.h"  // Precompiled headers
 
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/****************************************************************************************\
* Exhaustive Linearization for Robust Camera Pose and Focal Length Estimation.
* Contributed by Edgar Riba
\****************************************************************************************/
 
#include <iostream>
#include <mrpt/otherlibs/do_opencv_includes.h>
 
// Opencv 2.3 had a broken <opencv/eigen.h> in Ubuntu 14.04 Trusty => Disable
// PNP classes
#include <mrpt/config.h>
#if MRPT_HAS_OPENCV && MRPT_OPENCV_VERSION_NUM < 0x240
#undef MRPT_HAS_OPENCV
#define MRPT_HAS_OPENCV 0
#endif
 
#include <limits>
using namespace std;
#if MRPT_HAS_OPENCV
 
using namespace cv;
 
#include "upnp.h"
using namespace mrpt::vision::pnp;
 
upnp::upnp(const Mat& cameraMatrix, const Mat& opoints, const Mat& ipoints)
{
        if (cameraMatrix.depth() == CV_32F)
                init_camera_parameters<float>(cameraMatrix);
        else
                init_camera_parameters<double>(cameraMatrix);
 
        number_of_correspondences = std::max(
                opoints.checkVector(3, CV_32F), opoints.checkVector(3, CV_64F));
 
        pws.resize(3 * number_of_correspondences);
        us.resize(2 * number_of_correspondences);
 
        if (opoints.depth() == ipoints.depth())
        {
                if (opoints.depth() == CV_32F)
                        init_points<Point3f, Point2f>(opoints, ipoints);
                else
                        init_points<Point3d, Point2d>(opoints, ipoints);
        }
        else if (opoints.depth() == CV_32F)
                init_points<Point3f, Point2d>(opoints, ipoints);
        else
                init_points<Point3d, Point2f>(opoints, ipoints);
 
        alphas.resize(4 * number_of_correspondences);
        pcs.resize(3 * number_of_correspondences);
 
        max_nr = 0;
        A1 = nullptr;
        A2 = nullptr;
}
 
upnp::~upnp()
{
        if (A1) delete[] A1;
        if (A2) delete[] A2;
}
 
double upnp::compute_pose(Mat& R, Mat& t)
{
        choose_control_points();
        compute_alphas();
 
        Mat* M = new Mat(2 * number_of_correspondences, 12, CV_64F);
 
        for (int i = 0; i < number_of_correspondences; i++)
        {
                fill_M(M, 2 * i, &alphas[0] + 4 * i, us[2 * i], us[2 * i + 1]);
        }
 
        double mtm[12 * 12], d[12], ut[12 * 12], vt[12 * 12];
        Mat MtM = Mat(12, 12, CV_64F, mtm);
        Mat D = Mat(12, 1, CV_64F, d);
        Mat Ut = Mat(12, 12, CV_64F, ut);
        Mat Vt = Mat(12, 12, CV_64F, vt);
 
        MtM = M->t() * (*M);
        SVD::compute(MtM, D, Ut, Vt, SVD::MODIFY_A | SVD::FULL_UV);
        Mat(Ut.t()).copyTo(Ut);
        M->release();
        delete M;
 
        double l_6x12[6 * 12], rho[6];
        Mat L_6x12 = Mat(6, 12, CV_64F, l_6x12);
        Mat Rho = Mat(6, 1, CV_64F, rho);
 
        compute_L_6x12(ut, l_6x12);
        compute_rho(rho);
 
        double Betas[3][4], Efs[3][1], rep_errors[3];
        double Rs[3][3][3], ts[3][3];
 
        find_betas_and_focal_approx_1(&Ut, &Rho, Betas[1], Efs[1]);
        gauss_newton(&L_6x12, &Rho, Betas[1], Efs[1]);
        rep_errors[1] = compute_R_and_t(ut, Betas[1], Rs[1], ts[1]);
 
        find_betas_and_focal_approx_2(&Ut, &Rho, Betas[2], Efs[2]);
        gauss_newton(&L_6x12, &Rho, Betas[2], Efs[2]);
        rep_errors[2] = compute_R_and_t(ut, Betas[2], Rs[2], ts[2]);
 
        int N = 1;
        if (rep_errors[2] < rep_errors[1]) N = 2;
 
        Mat(3, 1, CV_64F, ts[N]).copyTo(t);
        Mat(3, 3, CV_64F, Rs[N]).copyTo(R);
        fu = fv = Efs[N][0];
 
        return fu;
}
 
void upnp::copy_R_and_t(
        const double R_src[3][3], const double t_src[3], double R_dst[3][3],
        double t_dst[3])
{
        for (int i = 0; i < 3; i++)
        {
                for (int j = 0; j < 3; j++) R_dst[i][j] = R_src[i][j];
                t_dst[i] = t_src[i];
        }
}
 
void upnp::estimate_R_and_t(double R[3][3], double t[3])
{
        double pc0[3], pw0[3];
 
        pc0[0] = pc0[1] = pc0[2] = 0.0;
        pw0[0] = pw0[1] = pw0[2] = 0.0;
 
        for (int i = 0; i < number_of_correspondences; i++)
        {
                const double* pc = &pcs[3 * i];
                const double* pw = &pws[3 * i];
 
                for (int j = 0; j < 3; j++)
                {
                        pc0[j] += pc[j];
                        pw0[j] += pw[j];
                }
        }
        for (int j = 0; j < 3; j++)
        {
                pc0[j] /= number_of_correspondences;
                pw0[j] /= number_of_correspondences;
        }
 
        double abt[3 * 3], abt_d[3], abt_u[3 * 3], abt_v[3 * 3];
        Mat ABt = Mat(3, 3, CV_64F, abt);
        Mat ABt_D = Mat(3, 1, CV_64F, abt_d);
        Mat ABt_U = Mat(3, 3, CV_64F, abt_u);
        Mat ABt_V = Mat(3, 3, CV_64F, abt_v);
 
        ABt.setTo(0.0);
        for (int i = 0; i < number_of_correspondences; i++)
        {
                double* pc = &pcs[3 * i];
                double* pw = &pws[3 * i];
 
                for (int j = 0; j < 3; j++)
                {
                        abt[3 * j] += (pc[j] - pc0[j]) * (pw[0] - pw0[0]);
                        abt[3 * j + 1] += (pc[j] - pc0[j]) * (pw[1] - pw0[1]);
                        abt[3 * j + 2] += (pc[j] - pc0[j]) * (pw[2] - pw0[2]);
                }
        }
 
        SVD::compute(ABt, ABt_D, ABt_U, ABt_V, SVD::MODIFY_A);
        Mat(ABt_V.t()).copyTo(ABt_V);
 
        for (int i = 0; i < 3; i++)
                for (int j = 0; j < 3; j++) R[i][j] = dot(abt_u + 3 * i, abt_v + 3 * j);
 
        const double det =
                R[0][0] * R[1][1] * R[2][2] + R[0][1] * R[1][2] * R[2][0] +
                R[0][2] * R[1][0] * R[2][1] - R[0][2] * R[1][1] * R[2][0] -
                R[0][1] * R[1][0] * R[2][2] - R[0][0] * R[1][2] * R[2][1];
 
        if (det < 0)
        {
                R[2][0] = -R[2][0];
                R[2][1] = -R[2][1];
                R[2][2] = -R[2][2];
        }
 
        t[0] = pc0[0] - dot(R[0], pw0);
        t[1] = pc0[1] - dot(R[1], pw0);
        t[2] = pc0[2] - dot(R[2], pw0);
}
 
void upnp::solve_for_sign(void)
{
        if (pcs[2] < 0.0)
        {
                for (int i = 0; i < 4; i++)
                        for (int j = 0; j < 3; j++) ccs[i][j] = -ccs[i][j];
 
                for (int i = 0; i < number_of_correspondences; i++)
                {
                        pcs[3 * i] = -pcs[3 * i];
                        pcs[3 * i + 1] = -pcs[3 * i + 1];
                        pcs[3 * i + 2] = -pcs[3 * i + 2];
                }
        }
}
 
double upnp::compute_R_and_t(
        const double* ut, const double* betas, double R[3][3], double t[3])
{
        compute_ccs(betas, ut);
        compute_pcs();
 
        solve_for_sign();
 
        estimate_R_and_t(R, t);
 
        return reprojection_error(R, t);
}
 
double upnp::reprojection_error(const double R[3][3], const double t[3])
{
        double sum2 = 0.0;
 
        for (int i = 0; i < number_of_correspondences; i++)
        {
                double* pw = &pws[3 * i];
                double Xc = dot(R[0], pw) + t[0];
                double Yc = dot(R[1], pw) + t[1];
                double inv_Zc = 1.0 / (dot(R[2], pw) + t[2]);
                double ue = uc + fu * Xc * inv_Zc;
                double ve = vc + fv * Yc * inv_Zc;
                double u = us[2 * i], v = us[2 * i + 1];
 
                sum2 += sqrt((u - ue) * (u - ue) + (v - ve) * (v - ve));
        }
 
        return sum2 / number_of_correspondences;
}
 
void upnp::choose_control_points()
{
        for (int i = 0; i < 4; ++i) cws[i][0] = cws[i][1] = cws[i][2] = 0.0;
        cws[0][0] = cws[1][1] = cws[2][2] = 1.0;
}
 
void upnp::compute_alphas()
{
        Mat CC = Mat(4, 3, CV_64F, &cws);
        Mat PC = Mat(number_of_correspondences, 3, CV_64F, &pws[0]);
        Mat ALPHAS = Mat(number_of_correspondences, 4, CV_64F, &alphas[0]);
 
        Mat CC_ = CC.clone().t();
        Mat PC_ = PC.clone().t();
 
        Mat row14 = Mat::ones(1, 4, CV_64F);
        Mat row1n = Mat::ones(1, number_of_correspondences, CV_64F);
 
        CC_.push_back(row14);
        PC_.push_back(row1n);
 
        ALPHAS = Mat(CC_.inv() * PC_).t();
}
 
void upnp::fill_M(
        Mat* M, const int row, const double* as, const double u, const double v)
{
        double* M1 = M->ptr<double>(row);
        double* M2 = M1 + 12;
 
        for (int i = 0; i < 4; i++)
        {
                M1[3 * i] = as[i] * fu;
                M1[3 * i + 1] = 0.0;
                M1[3 * i + 2] = as[i] * (uc - u);
 
                M2[3 * i] = 0.0;
                M2[3 * i + 1] = as[i] * fv;
                M2[3 * i + 2] = as[i] * (vc - v);
        }
}
 
void upnp::compute_ccs(const double* betas, const double* ut)
{
        for (int i = 0; i < 4; ++i) ccs[i][0] = ccs[i][1] = ccs[i][2] = 0.0;
 
        int N = 4;
        for (int i = 0; i < N; ++i)
        {
                const double* v = ut + 12 * (9 + i);
                for (int j = 0; j < 4; ++j)
                        for (int k = 0; k < 3; ++k) ccs[j][k] += betas[i] * v[3 * j + k];
        }
 
        for (int i = 0; i < 4; ++i) ccs[i][2] *= fu;
}
 
void upnp::compute_pcs(void)
{
        for (int i = 0; i < number_of_correspondences; i++)
        {
                double* a = &alphas[0] + 4 * i;
                double* pc = &pcs[0] + 3 * i;
 
                for (int j = 0; j < 3; j++)
                        pc[j] = a[0] * ccs[0][j] + a[1] * ccs[1][j] + a[2] * ccs[2][j] +
                                        a[3] * ccs[3][j];
        }
}
 
void upnp::find_betas_and_focal_approx_1(
        Mat* Ut, Mat* Rho, double* betas, double* efs)
{
        Mat Kmf1 = Mat(12, 1, CV_64F, Ut->ptr<double>(11));
        Mat dsq = Mat(6, 1, CV_64F, Rho->ptr<double>(0));
 
        Mat D = compute_constraint_distance_2param_6eq_2unk_f_unk(Kmf1);
        Mat Dt = D.t();
 
        Mat A = Dt * D;
        Mat b = Dt * dsq;
 
        Mat x = Mat(2, 1, CV_64F);
        solve(A, b, x);
 
        betas[0] = sqrt(abs(x.at<double>(0)));
        betas[1] = betas[2] = betas[3] = 0.0;
 
        efs[0] = sqrt(abs(x.at<double>(1))) / betas[0];
}
 
void upnp::find_betas_and_focal_approx_2(
        Mat* Ut, Mat* Rho, double* betas, double* efs)
{
        double u[12 * 12];
        Mat U = Mat(12, 12, CV_64F, u);
        Ut->copyTo(U);
 
        Mat Kmf1 = Mat(12, 1, CV_64F, Ut->ptr<double>(10));
        Mat Kmf2 = Mat(12, 1, CV_64F, Ut->ptr<double>(11));
        Mat dsq = Mat(6, 1, CV_64F, Rho->ptr<double>(0));
 
        Mat D = compute_constraint_distance_3param_6eq_6unk_f_unk(Kmf1, Kmf2);
 
        Mat A = D;
        Mat b = dsq;
 
        double x[6];
        Mat X = Mat(6, 1, CV_64F, x);
 
        solve(A, b, X, DECOMP_QR);
 
        double solutions[18][3];
        generate_all_possible_solutions_for_f_unk(x, solutions);
 
        // find solution with minimum reprojection error
        double min_error = std::numeric_limits<double>::max();
        int min_sol = 0;
        for (int i = 0; i < 18; ++i)
        {
                betas[3] = solutions[i][0];
                betas[2] = solutions[i][1];
                betas[1] = betas[0] = 0.0;
                fu = fv = solutions[i][2];
 
                double Rs[3][3], ts[3];
                double error_i = compute_R_and_t(u, betas, Rs, ts);
 
                if (error_i < min_error)
                {
                        min_error = error_i;
                        min_sol = i;
                }
        }
 
        betas[0] = solutions[min_sol][0];
        betas[1] = solutions[min_sol][1];
        betas[2] = betas[3] = 0.0;
 
        efs[0] = solutions[min_sol][2];
}
 
Mat upnp::compute_constraint_distance_2param_6eq_2unk_f_unk(const Mat& M1)
{
        Mat P = Mat(6, 2, CV_64F);
 
        double m[13];
        for (int i = 1; i < 13; ++i) m[i] = *M1.ptr<double>(i - 1);
 
        double t1 = pow(m[4], 2);
        double t4 = pow(m[1], 2);
        double t5 = pow(m[5], 2);
        double t8 = pow(m[2], 2);
        double t10 = pow(m[6], 2);
        double t13 = pow(m[3], 2);
        double t15 = pow(m[7], 2);
        double t18 = pow(m[8], 2);
        double t22 = pow(m[9], 2);
        double t26 = pow(m[10], 2);
        double t29 = pow(m[11], 2);
        double t33 = pow(m[12], 2);
 
        *P.ptr<double>(0, 0) =
                t1 - 2 * m[4] * m[1] + t4 + t5 - 2 * m[5] * m[2] + t8;
        *P.ptr<double>(0, 1) = t10 - 2 * m[6] * m[3] + t13;
        *P.ptr<double>(1, 0) =
                t15 - 2 * m[7] * m[1] + t4 + t18 - 2 * m[8] * m[2] + t8;
        *P.ptr<double>(1, 1) = t22 - 2 * m[9] * m[3] + t13;
        *P.ptr<double>(2, 0) =
                t26 - 2 * m[10] * m[1] + t4 + t29 - 2 * m[11] * m[2] + t8;
        *P.ptr<double>(2, 1) = t33 - 2 * m[12] * m[3] + t13;
        *P.ptr<double>(3, 0) =
                t15 - 2 * m[7] * m[4] + t1 + t18 - 2 * m[8] * m[5] + t5;
        *P.ptr<double>(3, 1) = t22 - 2 * m[9] * m[6] + t10;
        *P.ptr<double>(4, 0) =
                t26 - 2 * m[10] * m[4] + t1 + t29 - 2 * m[11] * m[5] + t5;
        *P.ptr<double>(4, 1) = t33 - 2 * m[12] * m[6] + t10;
        *P.ptr<double>(5, 0) =
                t26 - 2 * m[10] * m[7] + t15 + t29 - 2 * m[11] * m[8] + t18;
        *P.ptr<double>(5, 1) = t33 - 2 * m[12] * m[9] + t22;
 
        return P;
}
 
Mat upnp::compute_constraint_distance_3param_6eq_6unk_f_unk(
        const Mat& M1, const Mat& M2)
{
        Mat P = Mat(6, 6, CV_64F);
 
        double m[3][13];
        for (int i = 1; i < 13; ++i)
        {
                m[1][i] = *M1.ptr<double>(i - 1);
                m[2][i] = *M2.ptr<double>(i - 1);
        }
 
        double t1 = pow(m[1][4], 2);
        double t2 = pow(m[1][1], 2);
        double t7 = pow(m[1][5], 2);
        double t8 = pow(m[1][2], 2);
        double t11 = m[1][1] * m[2][1];
        double t12 = m[1][5] * m[2][5];
        double t15 = m[1][2] * m[2][2];
        double t16 = m[1][4] * m[2][4];
        double t19 = pow(m[2][4], 2);
        double t22 = pow(m[2][2], 2);
        double t23 = pow(m[2][1], 2);
        double t24 = pow(m[2][5], 2);
        double t28 = pow(m[1][6], 2);
        double t29 = pow(m[1][3], 2);
        double t34 = pow(m[1][3], 2);
        double t36 = m[1][6] * m[2][6];
        double t40 = pow(m[2][6], 2);
        double t41 = pow(m[2][3], 2);
        double t47 = pow(m[1][7], 2);
        double t48 = pow(m[1][8], 2);
        double t52 = m[1][7] * m[2][7];
        double t55 = m[1][8] * m[2][8];
        double t59 = pow(m[2][8], 2);
        double t62 = pow(m[2][7], 2);
        double t64 = pow(m[1][9], 2);
        double t68 = m[1][9] * m[2][9];
        double t74 = pow(m[2][9], 2);
        double t78 = pow(m[1][10], 2);
        double t79 = pow(m[1][11], 2);
        double t84 = m[1][10] * m[2][10];
        double t87 = m[1][11] * m[2][11];
        double t90 = pow(m[2][10], 2);
        double t95 = pow(m[2][11], 2);
        double t99 = pow(m[1][12], 2);
        double t101 = m[1][12] * m[2][12];
        double t105 = pow(m[2][12], 2);
 
        *P.ptr<double>(0, 0) =
                t1 + t2 - 2 * m[1][4] * m[1][1] - 2 * m[1][5] * m[1][2] + t7 + t8;
        *P.ptr<double>(0, 1) = -2 * m[2][4] * m[1][1] + 2 * t11 + 2 * t12 -
                                                   2 * m[1][4] * m[2][1] - 2 * m[2][5] * m[1][2] +
                                                   2 * t15 + 2 * t16 - 2 * m[1][5] * m[2][2];
        *P.ptr<double>(0, 2) =
                t19 - 2 * m[2][4] * m[2][1] + t22 + t23 + t24 - 2 * m[2][5] * m[2][2];
        *P.ptr<double>(0, 3) = t28 + t29 - 2 * m[1][6] * m[1][3];
        *P.ptr<double>(0, 4) =
                -2 * m[2][6] * m[1][3] + 2 * t34 - 2 * m[1][6] * m[2][3] + 2 * t36;
        *P.ptr<double>(0, 5) = -2 * m[2][6] * m[2][3] + t40 + t41;
 
        *P.ptr<double>(1, 0) =
                t8 - 2 * m[1][8] * m[1][2] - 2 * m[1][7] * m[1][1] + t47 + t48 + t2;
        *P.ptr<double>(1, 1) =
                2 * t15 - 2 * m[1][8] * m[2][2] - 2 * m[2][8] * m[1][2] + 2 * t52 -
                2 * m[1][7] * m[2][1] - 2 * m[2][7] * m[1][1] + 2 * t55 + 2 * t11;
        *P.ptr<double>(1, 2) =
                -2 * m[2][8] * m[2][2] + t22 + t23 + t59 - 2 * m[2][7] * m[2][1] + t62;
        *P.ptr<double>(1, 3) = t29 + t64 - 2 * m[1][9] * m[1][3];
        *P.ptr<double>(1, 4) =
                2 * t34 + 2 * t68 - 2 * m[2][9] * m[1][3] - 2 * m[1][9] * m[2][3];
        *P.ptr<double>(1, 5) = -2 * m[2][9] * m[2][3] + t74 + t41;
 
        *P.ptr<double>(2, 0) =
                -2 * m[1][11] * m[1][2] + t2 + t8 + t78 + t79 - 2 * m[1][10] * m[1][1];
        *P.ptr<double>(2, 1) = 2 * t15 - 2 * m[1][11] * m[2][2] + 2 * t84 -
                                                   2 * m[1][10] * m[2][1] - 2 * m[2][10] * m[1][1] +
                                                   2 * t87 - 2 * m[2][11] * m[1][2] + 2 * t11;
        *P.ptr<double>(2, 2) =
                t90 + t22 - 2 * m[2][10] * m[2][1] + t23 - 2 * m[2][11] * m[2][2] + t95;
        *P.ptr<double>(2, 3) = -2 * m[1][12] * m[1][3] + t99 + t29;
        *P.ptr<double>(2, 4) =
                2 * t34 + 2 * t101 - 2 * m[2][12] * m[1][3] - 2 * m[1][12] * m[2][3];
        *P.ptr<double>(2, 5) = t41 + t105 - 2 * m[2][12] * m[2][3];
 
        *P.ptr<double>(3, 0) =
                t48 + t1 - 2 * m[1][8] * m[1][5] + t7 - 2 * m[1][7] * m[1][4] + t47;
        *P.ptr<double>(3, 1) = 2 * t16 - 2 * m[1][7] * m[2][4] + 2 * t55 + 2 * t52 -
                                                   2 * m[1][8] * m[2][5] - 2 * m[2][8] * m[1][5] -
                                                   2 * m[2][7] * m[1][4] + 2 * t12;
        *P.ptr<double>(3, 2) =
                t24 - 2 * m[2][8] * m[2][5] + t19 - 2 * m[2][7] * m[2][4] + t62 + t59;
        *P.ptr<double>(3, 3) = -2 * m[1][9] * m[1][6] + t64 + t28;
        *P.ptr<double>(3, 4) =
                2 * t68 + 2 * t36 - 2 * m[2][9] * m[1][6] - 2 * m[1][9] * m[2][6];
        *P.ptr<double>(3, 5) = t40 + t74 - 2 * m[2][9] * m[2][6];
 
        *P.ptr<double>(4, 0) =
                t1 - 2 * m[1][10] * m[1][4] + t7 + t78 + t79 - 2 * m[1][11] * m[1][5];
        *P.ptr<double>(4, 1) =
                2 * t84 - 2 * m[1][11] * m[2][5] - 2 * m[1][10] * m[2][4] + 2 * t16 -
                2 * m[2][11] * m[1][5] + 2 * t87 - 2 * m[2][10] * m[1][4] + 2 * t12;
        *P.ptr<double>(4, 2) =
                t19 + t24 - 2 * m[2][10] * m[2][4] - 2 * m[2][11] * m[2][5] + t95 + t90;
        *P.ptr<double>(4, 3) = t28 - 2 * m[1][12] * m[1][6] + t99;
        *P.ptr<double>(4, 4) =
                2 * t101 + 2 * t36 - 2 * m[2][12] * m[1][6] - 2 * m[1][12] * m[2][6];
        *P.ptr<double>(4, 5) = t105 - 2 * m[2][12] * m[2][6] + t40;
 
        *P.ptr<double>(5, 0) = -2 * m[1][10] * m[1][7] + t47 + t48 + t78 + t79 -
                                                   2 * m[1][11] * m[1][8];
        *P.ptr<double>(5, 1) = 2 * t84 + 2 * t87 - 2 * m[2][11] * m[1][8] -
                                                   2 * m[1][10] * m[2][7] - 2 * m[2][10] * m[1][7] +
                                                   2 * t55 + 2 * t52 - 2 * m[1][11] * m[2][8];
        *P.ptr<double>(5, 2) = -2 * m[2][10] * m[2][7] - 2 * m[2][11] * m[2][8] +
                                                   t62 + t59 + t90 + t95;
        *P.ptr<double>(5, 3) = t64 - 2 * m[1][12] * m[1][9] + t99;
        *P.ptr<double>(5, 4) =
                2 * t68 - 2 * m[2][12] * m[1][9] - 2 * m[1][12] * m[2][9] + 2 * t101;
        *P.ptr<double>(5, 5) = t105 - 2 * m[2][12] * m[2][9] + t74;
 
        return P;
}
 
void upnp::generate_all_possible_solutions_for_f_unk(
        const double betas[5], double solutions[18][3])
{
        int matrix_to_resolve[18][9] = {
                {2, 0, 0, 1, 1, 0, 2, 0, 2}, {2, 0, 0, 1, 1, 0, 1, 1, 2},
                {2, 0, 0, 1, 1, 0, 0, 2, 2}, {2, 0, 0, 0, 2, 0, 2, 0, 2},
                {2, 0, 0, 0, 2, 0, 1, 1, 2}, {2, 0, 0, 0, 2, 0, 0, 2, 2},
                {2, 0, 0, 2, 0, 2, 1, 1, 2}, {2, 0, 0, 2, 0, 2, 0, 2, 2},
                {2, 0, 0, 1, 1, 2, 0, 2, 2}, {1, 1, 0, 0, 2, 0, 2, 0, 2},
                {1, 1, 0, 0, 2, 0, 1, 1, 2}, {1, 1, 0, 2, 0, 2, 0, 2, 2},
                {1, 1, 0, 2, 0, 2, 1, 1, 2}, {1, 1, 0, 2, 0, 2, 0, 2, 2},
                {1, 1, 0, 1, 1, 2, 0, 2, 2}, {0, 2, 0, 2, 0, 2, 1, 1, 2},
                {0, 2, 0, 2, 0, 2, 0, 2, 2}, {0, 2, 0, 1, 1, 2, 0, 2, 2}};
 
        int combination[18][3] = {
                {1, 2, 4}, {1, 2, 5}, {1, 2, 6}, {1, 3, 4}, {1, 3, 5}, {1, 3, 6},
                {1, 4, 5}, {1, 4, 6}, {1, 5, 6}, {2, 3, 4}, {2, 3, 5}, {2, 3, 6},
                {2, 4, 5}, {2, 4, 6}, {2, 5, 6}, {3, 4, 5}, {3, 4, 6}, {3, 5, 6}};
 
        for (int i = 0; i < 18; ++i)
        {
                double matrix[9], independent_term[3];
                Mat M = Mat(3, 3, CV_64F, matrix);
                Mat I = Mat(3, 1, CV_64F, independent_term);
                Mat S = Mat(1, 3, CV_64F);
 
                for (int j = 0; j < 9; ++j) matrix[j] = (double)matrix_to_resolve[i][j];
 
                independent_term[0] = log(abs(betas[combination[i][0] - 1]));
                independent_term[1] = log(abs(betas[combination[i][1] - 1]));
                independent_term[2] = log(abs(betas[combination[i][2] - 1]));
 
                exp(Mat(M.inv() * I), S);
 
                solutions[i][0] = S.at<double>(0);
                solutions[i][1] = S.at<double>(1) * sign(betas[1]);
                solutions[i][2] = abs(S.at<double>(2));
        }
}
 
void upnp::gauss_newton(
        const Mat* L_6x12, const Mat* Rho, double betas[4], double* f)
{
        const int iterations_number = 50;
 
        double a[6 * 4], b[6], x[4];
        Mat* A = new Mat(6, 4, CV_64F, a);
        Mat* B = new Mat(6, 1, CV_64F, b);
        Mat* X = new Mat(4, 1, CV_64F, x);
 
        for (int k = 0; k < iterations_number; k++)
        {
                compute_A_and_b_gauss_newton(
                        L_6x12->ptr<double>(0), Rho->ptr<double>(0), betas, A, B, f[0]);
                qr_solve(A, B, X);
                for (int i = 0; i < 3; i++) betas[i] += x[i];
                f[0] += x[3];
        }
 
        if (f[0] < 0) f[0] = -f[0];
        fu = fv = f[0];
 
        A->release();
        delete A;
 
        B->release();
        delete B;
 
        X->release();
        delete X;
}
 
void upnp::compute_A_and_b_gauss_newton(
        const double* l_6x12, const double* rho, const double betas[4], Mat* A,
        Mat* b, double const f)
{
        for (int i = 0; i < 6; i++)
        {
                const double* rowL = l_6x12 + i * 12;
                double* rowA = A->ptr<double>(i);
 
                rowA[0] = 2 * rowL[0] * betas[0] + rowL[1] * betas[1] +
                                  rowL[2] * betas[2] +
                                  f * f * (2 * rowL[6] * betas[0] + rowL[7] * betas[1] +
                                                   rowL[8] * betas[2]);
                rowA[1] = rowL[1] * betas[0] + 2 * rowL[3] * betas[1] +
                                  rowL[4] * betas[2] +
                                  f * f * (rowL[7] * betas[0] + 2 * rowL[9] * betas[1] +
                                                   rowL[10] * betas[2]);
                rowA[2] = rowL[2] * betas[0] + rowL[4] * betas[1] +
                                  2 * rowL[5] * betas[2] +
                                  f * f * (rowL[8] * betas[0] + rowL[10] * betas[1] +
                                                   2 * rowL[11] * betas[2]);
                rowA[3] =
                        2 * f *
                        (rowL[6] * betas[0] * betas[0] + rowL[7] * betas[0] * betas[1] +
                         rowL[8] * betas[0] * betas[2] + rowL[9] * betas[1] * betas[1] +
                         rowL[10] * betas[1] * betas[2] + rowL[11] * betas[2] * betas[2]);
 
                *b->ptr<double>(i) =
                        rho[i] -
                        (rowL[0] * betas[0] * betas[0] + rowL[1] * betas[0] * betas[1] +
                         rowL[2] * betas[0] * betas[2] + rowL[3] * betas[1] * betas[1] +
                         rowL[4] * betas[1] * betas[2] + rowL[5] * betas[2] * betas[2] +
                         f * f * rowL[6] * betas[0] * betas[0] +
                         f * f * rowL[7] * betas[0] * betas[1] +
                         f * f * rowL[8] * betas[0] * betas[2] +
                         f * f * rowL[9] * betas[1] * betas[1] +
                         f * f * rowL[10] * betas[1] * betas[2] +
                         f * f * rowL[11] * betas[2] * betas[2]);
        }
}
 
void upnp::compute_L_6x12(const double* ut, double* l_6x12)
{
        const double* v[3];
 
        v[0] = ut + 12 * 9;
        v[1] = ut + 12 * 10;
        v[2] = ut + 12 * 11;
 
        double dv[3][6][3];
 
        for (int i = 0; i < 3; i++)
        {
                int a = 0, b = 1;
                for (int j = 0; j < 6; j++)
                {
                        dv[i][j][0] = v[i][3 * a] - v[i][3 * b];
                        dv[i][j][1] = v[i][3 * a + 1] - v[i][3 * b + 1];
                        dv[i][j][2] = v[i][3 * a + 2] - v[i][3 * b + 2];
 
                        b++;
                        if (b > 3)
                        {
                                a++;
                                b = a + 1;
                        }
                }
        }
 
        for (int i = 0; i < 6; i++)
        {
                double* row = l_6x12 + 12 * i;
 
                row[0] = dotXY(dv[0][i], dv[0][i]);
                row[1] = 2.0f * dotXY(dv[0][i], dv[1][i]);
                row[2] = dotXY(dv[1][i], dv[1][i]);
                row[3] = 2.0f * dotXY(dv[0][i], dv[2][i]);
                row[4] = 2.0f * dotXY(dv[1][i], dv[2][i]);
                row[5] = dotXY(dv[2][i], dv[2][i]);
 
                row[6] = dotZ(dv[0][i], dv[0][i]);
                row[7] = 2.0f * dotZ(dv[0][i], dv[1][i]);
                row[8] = 2.0f * dotZ(dv[0][i], dv[2][i]);
                row[9] = dotZ(dv[1][i], dv[1][i]);
                row[10] = 2.0f * dotZ(dv[1][i], dv[2][i]);
                row[11] = dotZ(dv[2][i], dv[2][i]);
        }
}
 
void upnp::compute_rho(double* rho)
{
        rho[0] = dist2(cws[0], cws[1]);
        rho[1] = dist2(cws[0], cws[2]);
        rho[2] = dist2(cws[0], cws[3]);
        rho[3] = dist2(cws[1], cws[2]);
        rho[4] = dist2(cws[1], cws[3]);
        rho[5] = dist2(cws[2], cws[3]);
}
 
double upnp::dist2(const double* p1, const double* p2)
{
        return (p1[0] - p2[0]) * (p1[0] - p2[0]) +
                   (p1[1] - p2[1]) * (p1[1] - p2[1]) +
                   (p1[2] - p2[2]) * (p1[2] - p2[2]);
}
 
double upnp::dot(const double* v1, const double* v2)
{
        return v1[0] * v2[0] + v1[1] * v2[1] + v1[2] * v2[2];
}
 
double upnp::dotXY(const double* v1, const double* v2)
{
        return v1[0] * v2[0] + v1[1] * v2[1];
}
 
double upnp::dotZ(const double* v1, const double* v2) { return v1[2] * v2[2]; }
double upnp::sign(const double v)
{
        return (v < 0.0) ? -1.0 : (v > 0.0) ? 1.0 : 0.0;
}
 
void upnp::qr_solve(Mat* A, Mat* b, Mat* X)
{
        const int nr = A->rows;
        const int nc = A->cols;
 
        if (max_nr != 0 && max_nr < nr)
        {
                delete[] A1;
                delete[] A2;
        }
        if (max_nr < nr)
        {
                max_nr = nr;
                A1 = new double[nr];
                A2 = new double[nr];
        }
 
        double *pA = A->ptr<double>(0), *ppAkk = pA;
        for (int k = 0; k < nc; k++)
        {
                double *ppAik1 = ppAkk, eta = fabs(*ppAik1);
                for (int i = k + 1; i < nr; i++)
                {
                        double elt = fabs(*ppAik1);
                        if (eta < elt) eta = elt;
                        ppAik1 += nc;
                }
                if (eta == 0)
                {
                        A1[k] = A2[k] = 0.0;
                        // cerr << "God damnit, A is singular, this shouldn't happen." <<
                        // endl;
                        return;
                }
                else
                {
                        double *ppAik2 = ppAkk, sum2 = 0.0, inv_eta = 1. / eta;
                        for (int i = k; i < nr; i++)
                        {
                                *ppAik2 *= inv_eta;
                                sum2 += *ppAik2 * *ppAik2;
                                ppAik2 += nc;
                        }
                        double sigma = sqrt(sum2);
                        if (*ppAkk < 0) sigma = -sigma;
                        *ppAkk += sigma;
                        A1[k] = sigma * *ppAkk;
                        A2[k] = -eta * sigma;
                        for (int j = k + 1; j < nc; j++)
                        {
                                double *ppAik = ppAkk, sum = 0;
                                for (int i = k; i < nr; i++)
                                {
                                        sum += *ppAik * ppAik[j - k];
                                        ppAik += nc;
                                }
                                double tau = sum / A1[k];
                                ppAik = ppAkk;
                                for (int i = k; i < nr; i++)
                                {
                                        ppAik[j - k] -= tau * *ppAik;
                                        ppAik += nc;
                                }
                        }
                }
                ppAkk += nc + 1;
        }
 
        // b <- Qt b
        double *ppAjj = pA, *pb = b->ptr<double>(0);
        for (int j = 0; j < nc; j++)
        {
                double *ppAij = ppAjj, tau = 0;
                for (int i = j; i < nr; i++)
                {
                        tau += *ppAij * pb[i];
                        ppAij += nc;
                }
                tau /= A1[j];
                ppAij = ppAjj;
                for (int i = j; i < nr; i++)
                {
                        pb[i] -= tau * *ppAij;
                        ppAij += nc;
                }
                ppAjj += nc + 1;
        }
 
        // X = R-1 b
        double* pX = X->ptr<double>(0);
        pX[nc - 1] = pb[nc - 1] / A2[nc - 1];
        for (int i = nc - 2; i >= 0; i--)
        {
                double *ppAij = pA + i * nc + (i + 1), sum = 0;
 
                for (int j = i + 1; j < nc; j++)
                {
                        sum += *ppAij * pX[j];
                        ppAij++;
                }
                pX[i] = (pb[i] - sum) / A2[i];
        }
}
#endif