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 <mrpt/3rdparty/do_opencv_includes.h>
#include <iostream>

#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()
{
    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)
{
    auto* 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()
{
    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(std::abs(x.at<double>(0)));
    betas[1] = betas[2] = betas[3] = 0.0;

    efs[0] = sqrt(std::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(std::abs(betas[combination[i][0] - 1]));
        independent_term[1] = log(std::abs(betas[combination[i][1] - 1]));
        independent_term[2] = log(std::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] = std::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;
        auto* 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
    auto* 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