epnp.cpp

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/* +------------------------------------------------------------------------+
   |                     Mobile Robot Programming Toolkit (MRPT)            |
   |                          https://www.mrpt.org/                         |
   |                                                                        |
   | Copyright (c) 2005-2019, Individual contributors, see AUTHORS file     |
   | See: https://www.mrpt.org/Authors - All rights reserved.               |
   | Released under BSD License. See: https://www.mrpt.org/License          |
   +------------------------------------------------------------------------+ */

#include <mrpt/config.h>
#include <iostream>
#include "vision-precomp.h"  // Precompiled headers

#if MRPT_HAS_OPENCV
#include <mrpt/otherlibs/do_opencv_includes.h>
using namespace cv;

#include "epnp.h"

mrpt::vision::pnp::epnp::epnp(
    const cv::Mat& cameraMatrix, const cv::Mat& opoints, const cv::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<float, float>(opoints, ipoints);
        else
            init_points<double, double>(opoints, ipoints);
    }
    else if (opoints.depth() == CV_32F)
        init_points<float, double>(opoints, ipoints);
    else
        init_points<double, float>(opoints, ipoints);

    alphas.resize(4 * number_of_correspondences);
    pcs.resize(3 * number_of_correspondences);

    max_nr = 0;
    A1 = nullptr;
    A2 = nullptr;
}

mrpt::vision::pnp::epnp::~epnp()
{
    if (A1) delete[] A1;
    if (A2) delete[] A2;
}

void mrpt::vision::pnp::epnp::choose_control_points()
{
    // Take C0 as the reference points centroid:
    cws[0][0] = cws[0][1] = cws[0][2] = 0;
    for (int i = 0; i < number_of_correspondences; i++)
        for (int j = 0; j < 3; j++) cws[0][j] += pws[3 * i + j];

    for (int j = 0; j < 3; j++) cws[0][j] /= number_of_correspondences;

    // Take C1, C2, and C3 from PCA on the reference points:
    CvMat* PW0 = cvCreateMat(number_of_correspondences, 3, CV_64F);

    double pw0tpw0[3 * 3], dc[3], uct[3 * 3];
    CvMat PW0tPW0 = cvMat(3, 3, CV_64F, pw0tpw0);
    CvMat DC = cvMat(3, 1, CV_64F, dc);
    CvMat UCt = cvMat(3, 3, CV_64F, uct);

    for (int i = 0; i < number_of_correspondences; i++)
        for (int j = 0; j < 3; j++)
            PW0->data.db[3 * i + j] = pws[3 * i + j] - cws[0][j];

    cvMulTransposed(PW0, &PW0tPW0, 1);
    cvSVD(&PW0tPW0, &DC, &UCt, nullptr, CV_SVD_MODIFY_A | CV_SVD_U_T);

    cvReleaseMat(&PW0);

    for (int i = 1; i < 4; i++)
    {
        double k = sqrt(dc[i - 1] / number_of_correspondences);
        for (int j = 0; j < 3; j++)
            cws[i][j] = cws[0][j] + k * uct[3 * (i - 1) + j];
    }
}

void mrpt::vision::pnp::epnp::compute_barycentric_coordinates()
{
    double cc[3 * 3], cc_inv[3 * 3];
    CvMat CC = cvMat(3, 3, CV_64F, cc);
    CvMat CC_inv = cvMat(3, 3, CV_64F, cc_inv);

    for (int i = 0; i < 3; i++)
        for (int j = 1; j < 4; j++) cc[3 * i + j - 1] = cws[j][i] - cws[0][i];

    cvInvert(&CC, &CC_inv, CV_SVD);
    double* ci = cc_inv;
    for (int i = 0; i < number_of_correspondences; i++)
    {
        double* pi = &pws[0] + 3 * i;
        double* a = &alphas[0] + 4 * i;

        for (int j = 0; j < 3; j++)
            a[1 + j] = ci[3 * j] * (pi[0] - cws[0][0]) +
                       ci[3 * j + 1] * (pi[1] - cws[0][1]) +
                       ci[3 * j + 2] * (pi[2] - cws[0][2]);
        a[0] = 1.0f - a[1] - a[2] - a[3];
    }
}

void mrpt::vision::pnp::epnp::fill_M(
    CvMat* M, const int row, const double* as, const double u, const double v)
{
    double* M1 = M->data.db + row * 12;
    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 mrpt::vision::pnp::epnp::compute_ccs(const double* betas, const double* ut)
{
    for (int i = 0; i < 4; i++) ccs[i][0] = ccs[i][1] = ccs[i][2] = 0.0f;

    for (int i = 0; i < 4; i++)
    {
        const double* v = ut + 12 * (11 - i);
        for (int j = 0; j < 4; j++)
            for (int k = 0; k < 3; k++) ccs[j][k] += betas[i] * v[3 * j + k];
    }
}

void mrpt::vision::pnp::epnp::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 mrpt::vision::pnp::epnp::compute_pose(cv::Mat& R, cv::Mat& t)
{
    choose_control_points();
    compute_barycentric_coordinates();

    CvMat* M = cvCreateMat(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];
    CvMat MtM = cvMat(12, 12, CV_64F, mtm);
    CvMat D = cvMat(12, 1, CV_64F, d);
    CvMat Ut = cvMat(12, 12, CV_64F, ut);

    cvMulTransposed(M, &MtM, 1);
    cvSVD(&MtM, &D, &Ut, nullptr, CV_SVD_MODIFY_A | CV_SVD_U_T);
    cvReleaseMat(&M);

    double l_6x10[6 * 10], rho[6];
    CvMat L_6x10 = cvMat(6, 10, CV_64F, l_6x10);
    CvMat Rho = cvMat(6, 1, CV_64F, rho);

    compute_L_6x10(ut, l_6x10);
    compute_rho(rho);

    double Betas[4][4], rep_errors[4];
    double Rs[4][3][3], ts[4][3];

    find_betas_approx_1(&L_6x10, &Rho, Betas[1]);
    gauss_newton(&L_6x10, &Rho, Betas[1]);
    rep_errors[1] = compute_R_and_t(ut, Betas[1], Rs[1], ts[1]);

    find_betas_approx_2(&L_6x10, &Rho, Betas[2]);
    gauss_newton(&L_6x10, &Rho, Betas[2]);
    rep_errors[2] = compute_R_and_t(ut, Betas[2], Rs[2], ts[2]);

    find_betas_approx_3(&L_6x10, &Rho, Betas[3]);
    gauss_newton(&L_6x10, &Rho, Betas[3]);
    rep_errors[3] = compute_R_and_t(ut, Betas[3], Rs[3], ts[3]);

    int N = 1;
    if (rep_errors[2] < rep_errors[1]) N = 2;
    if (rep_errors[3] < rep_errors[N]) N = 3;

    cv::Mat(3, 1, CV_64F, ts[N]).copyTo(t);
    cv::Mat(3, 3, CV_64F, Rs[N]).copyTo(R);
}

void mrpt::vision::pnp::epnp::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];
    }
}

double mrpt::vision::pnp::epnp::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 mrpt::vision::pnp::epnp::dot(const double* v1, const double* v2)
{
    return v1[0] * v2[0] + v1[1] * v2[1] + v1[2] * v2[2];
}

void mrpt::vision::pnp::epnp::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];
    CvMat ABt = cvMat(3, 3, CV_64F, abt);
    CvMat ABt_D = cvMat(3, 1, CV_64F, abt_d);
    CvMat ABt_U = cvMat(3, 3, CV_64F, abt_u);
    CvMat ABt_V = cvMat(3, 3, CV_64F, abt_v);

    cvSetZero(&ABt);
    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]);
        }
    }

    cvSVD(&ABt, &ABt_D, &ABt_U, &ABt_V, CV_SVD_MODIFY_A);

    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 mrpt::vision::pnp::epnp::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 mrpt::vision::pnp::epnp::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 mrpt::vision::pnp::epnp::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;
}

// betas10        = [B11 B12 B22 B13 B23 B33 B14 B24 B34 B44]
// betas_approx_1 = [B11 B12     B13         B14]

void mrpt::vision::pnp::epnp::find_betas_approx_1(
    const CvMat* L_6x10, const CvMat* Rho, double* betas)
{
    double l_6x4[6 * 4], b4[4];
    CvMat L_6x4 = cvMat(6, 4, CV_64F, l_6x4);
    CvMat B4 = cvMat(4, 1, CV_64F, b4);

    for (int i = 0; i < 6; i++)
    {
        cvmSet(&L_6x4, i, 0, cvmGet(L_6x10, i, 0));
        cvmSet(&L_6x4, i, 1, cvmGet(L_6x10, i, 1));
        cvmSet(&L_6x4, i, 2, cvmGet(L_6x10, i, 3));
        cvmSet(&L_6x4, i, 3, cvmGet(L_6x10, i, 6));
    }

    cvSolve(&L_6x4, Rho, &B4, CV_SVD);

    if (b4[0] < 0)
    {
        betas[0] = sqrt(-b4[0]);
        betas[1] = -b4[1] / betas[0];
        betas[2] = -b4[2] / betas[0];
        betas[3] = -b4[3] / betas[0];
    }
    else
    {
        betas[0] = sqrt(b4[0]);
        betas[1] = b4[1] / betas[0];
        betas[2] = b4[2] / betas[0];
        betas[3] = b4[3] / betas[0];
    }
}

// betas10        = [B11 B12 B22 B13 B23 B33 B14 B24 B34 B44]
// betas_approx_2 = [B11 B12 B22                            ]

void mrpt::vision::pnp::epnp::find_betas_approx_2(
    const CvMat* L_6x10, const CvMat* Rho, double* betas)
{
    double l_6x3[6 * 3], b3[3];
    CvMat L_6x3 = cvMat(6, 3, CV_64F, l_6x3);
    CvMat B3 = cvMat(3, 1, CV_64F, b3);

    for (int i = 0; i < 6; i++)
    {
        cvmSet(&L_6x3, i, 0, cvmGet(L_6x10, i, 0));
        cvmSet(&L_6x3, i, 1, cvmGet(L_6x10, i, 1));
        cvmSet(&L_6x3, i, 2, cvmGet(L_6x10, i, 2));
    }

    cvSolve(&L_6x3, Rho, &B3, CV_SVD);

    if (b3[0] < 0)
    {
        betas[0] = sqrt(-b3[0]);
        betas[1] = (b3[2] < 0) ? sqrt(-b3[2]) : 0.0;
    }
    else
    {
        betas[0] = sqrt(b3[0]);
        betas[1] = (b3[2] > 0) ? sqrt(b3[2]) : 0.0;
    }

    if (b3[1] < 0) betas[0] = -betas[0];

    betas[2] = 0.0;
    betas[3] = 0.0;
}

// betas10        = [B11 B12 B22 B13 B23 B33 B14 B24 B34 B44]
// betas_approx_3 = [B11 B12 B22 B13 B23                    ]

void mrpt::vision::pnp::epnp::find_betas_approx_3(
    const CvMat* L_6x10, const CvMat* Rho, double* betas)
{
    double l_6x5[6 * 5], b5[5];
    CvMat L_6x5 = cvMat(6, 5, CV_64F, l_6x5);
    CvMat B5 = cvMat(5, 1, CV_64F, b5);

    for (int i = 0; i < 6; i++)
    {
        cvmSet(&L_6x5, i, 0, cvmGet(L_6x10, i, 0));
        cvmSet(&L_6x5, i, 1, cvmGet(L_6x10, i, 1));
        cvmSet(&L_6x5, i, 2, cvmGet(L_6x10, i, 2));
        cvmSet(&L_6x5, i, 3, cvmGet(L_6x10, i, 3));
        cvmSet(&L_6x5, i, 4, cvmGet(L_6x10, i, 4));
    }

    cvSolve(&L_6x5, Rho, &B5, CV_SVD);

    if (b5[0] < 0)
    {
        betas[0] = sqrt(-b5[0]);
        betas[1] = (b5[2] < 0) ? sqrt(-b5[2]) : 0.0;
    }
    else
    {
        betas[0] = sqrt(b5[0]);
        betas[1] = (b5[2] > 0) ? sqrt(b5[2]) : 0.0;
    }
    if (b5[1] < 0) betas[0] = -betas[0];
    betas[2] = b5[3] / betas[0];
    betas[3] = 0.0;
}

void mrpt::vision::pnp::epnp::compute_L_6x10(const double* ut, double* l_6x10)
{
    const double* v[4];

    v[0] = ut + 12 * 11;
    v[1] = ut + 12 * 10;
    v[2] = ut + 12 * 9;
    v[3] = ut + 12 * 8;

    double dv[4][6][3];

    for (int i = 0; i < 4; 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_6x10 + 10 * i;

        row[0] = dot(dv[0][i], dv[0][i]);
        row[1] = 2.0f * dot(dv[0][i], dv[1][i]);
        row[2] = dot(dv[1][i], dv[1][i]);
        row[3] = 2.0f * dot(dv[0][i], dv[2][i]);
        row[4] = 2.0f * dot(dv[1][i], dv[2][i]);
        row[5] = dot(dv[2][i], dv[2][i]);
        row[6] = 2.0f * dot(dv[0][i], dv[3][i]);
        row[7] = 2.0f * dot(dv[1][i], dv[3][i]);
        row[8] = 2.0f * dot(dv[2][i], dv[3][i]);
        row[9] = dot(dv[3][i], dv[3][i]);
    }
}

void mrpt::vision::pnp::epnp::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]);
}

void mrpt::vision::pnp::epnp::compute_A_and_b_gauss_newton(
    const double* l_6x10, const double* rho, const double betas[4], CvMat* A,
    CvMat* b)
{
    for (int i = 0; i < 6; i++)
    {
        const double* rowL = l_6x10 + i * 10;
        double* rowA = A->data.db + i * 4;

        rowA[0] = 2 * rowL[0] * betas[0] + rowL[1] * betas[1] +
                  rowL[3] * betas[2] + rowL[6] * betas[3];
        rowA[1] = rowL[1] * betas[0] + 2 * rowL[2] * betas[1] +
                  rowL[4] * betas[2] + rowL[7] * betas[3];
        rowA[2] = rowL[3] * betas[0] + rowL[4] * betas[1] +
                  2 * rowL[5] * betas[2] + rowL[8] * betas[3];
        rowA[3] = rowL[6] * betas[0] + rowL[7] * betas[1] + rowL[8] * betas[2] +
                  2 * rowL[9] * betas[3];

        cvmSet(
            b, i, 0,
            rho[i] -
                (rowL[0] * betas[0] * betas[0] + rowL[1] * betas[0] * betas[1] +
                 rowL[2] * betas[1] * betas[1] + rowL[3] * betas[0] * betas[2] +
                 rowL[4] * betas[1] * betas[2] + rowL[5] * betas[2] * betas[2] +
                 rowL[6] * betas[0] * betas[3] + rowL[7] * betas[1] * betas[3] +
                 rowL[8] * betas[2] * betas[3] +
                 rowL[9] * betas[3] * betas[3]));
    }
}

void mrpt::vision::pnp::epnp::gauss_newton(
    const CvMat* L_6x10, const CvMat* Rho, double betas[4])
{
    const int iterations_number = 5;

    double a[6 * 4], b[6], x[4];
    CvMat A = cvMat(6, 4, CV_64F, a);
    CvMat B = cvMat(6, 1, CV_64F, b);
    CvMat X = cvMat(4, 1, CV_64F, x);

    for (int k = 0; k < iterations_number; k++)
    {
        compute_A_and_b_gauss_newton(
            L_6x10->data.db, Rho->data.db, betas, &A, &B);
        qr_solve(&A, &B, &X);
        for (int i = 0; i < 4; i++) betas[i] += x[i];
    }
}

void mrpt::vision::pnp::epnp::qr_solve(CvMat* A, CvMat* b, CvMat* 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->data.db, *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->data.db;
    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->data.db;
    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