opencv/modules/3d/src/ippe.cpp

// This file is part of OpenCV project.
// It is subject to the license terms in the LICENSE file found in the top-level directory
// of this distribution and at http://opencv.org/license.html

// This file is based on file issued with the following license:

/*============================================================================

Copyright 2017 Toby Collins
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#include "precomp.hpp"
#include "ippe.hpp"

namespace cv {
namespace IPPE {
PoseSolver::PoseSolver() : IPPE_SMALL(1e-3)
{
}

void PoseSolver::solveGeneric(InputArray _objectPoints, InputArray _imagePoints, OutputArray _rvec1, OutputArray _tvec1,
                              float& err1, OutputArray _rvec2, OutputArray _tvec2, float& err2)
{
    Mat normalizedImagePoints;
    if (_imagePoints.getMat().type() == CV_32FC2)
    {
        _imagePoints.getMat().convertTo(normalizedImagePoints, CV_64F);
    }
    else
    {
        normalizedImagePoints = _imagePoints.getMat();
    }

    //solve:
    Mat Ma, Mb;
    solveGeneric(_objectPoints, normalizedImagePoints, Ma, Mb);

    //the two poses computed by IPPE (sorted):
    Mat M1, M2;

    //sort poses by reprojection error:
    sortPosesByReprojError(_objectPoints, normalizedImagePoints, Ma, Mb, M1, M2, err1, err2);

    //fill outputs
    rot2vec(M1.colRange(0, 3).rowRange(0, 3), _rvec1);
    rot2vec(M2.colRange(0, 3).rowRange(0, 3), _rvec2);

    M1.colRange(3, 4).rowRange(0, 3).copyTo(_tvec1);
    M2.colRange(3, 4).rowRange(0, 3).copyTo(_tvec2);
}

void PoseSolver::solveGeneric(InputArray _objectPoints, InputArray _normalizedInputPoints,
                              OutputArray _Ma, OutputArray _Mb)
{
    //argument checking:
    size_t n = static_cast<size_t>(_normalizedInputPoints.rows()) * static_cast<size_t>(_normalizedInputPoints.cols()); //number of points
    int objType = _objectPoints.type();
    int type_input = _normalizedInputPoints.type();

    CV_CheckType(objType, objType == CV_32FC3 || objType == CV_64FC3,
                 "Type of _objectPoints must be CV_32FC3 or CV_64FC3" );
    CV_CheckType(type_input, type_input == CV_32FC2 || type_input == CV_64FC2,
                 "Type of _normalizedInputPoints must be CV_32FC2 or CV_64FC2" );
    CV_Assert(_objectPoints.rows() == 1 || _objectPoints.cols() == 1);
    CV_Assert(_objectPoints.rows() >= 4 || _objectPoints.cols() >= 4);
    CV_Assert(_normalizedInputPoints.rows() == 1 || _normalizedInputPoints.cols() == 1);
    CV_Assert(static_cast<size_t>(_objectPoints.rows()) * static_cast<size_t>(_objectPoints.cols()) == n);

    Mat normalizedInputPoints;
    if (type_input == CV_32FC2)
    {
        _normalizedInputPoints.getMat().convertTo(normalizedInputPoints, CV_64F);
    }
    else
    {
        normalizedInputPoints = _normalizedInputPoints.getMat();
    }

    Mat objectInputPoints;
    if (objType == CV_32FC3)
    {
        _objectPoints.getMat().convertTo(objectInputPoints, CV_64F);
    }
    else
    {
        objectInputPoints = _objectPoints.getMat();
    }

    Mat canonicalObjPoints;
    Mat MmodelPoints2Canonical;

    //transform object points to the canonical position (zero centred and on the plane z=0):
    makeCanonicalObjectPoints(objectInputPoints, canonicalObjPoints, MmodelPoints2Canonical);

    //compute the homography mapping the model's points to normalizedInputPoints
    Matx33d H;
    HomographyHO::homographyHO(canonicalObjPoints, _normalizedInputPoints, H);

    //now solve
    Mat MaCanon, MbCanon;
    solveCanonicalForm(canonicalObjPoints, normalizedInputPoints, H, MaCanon, MbCanon);

    //transform computed poses to account for canonical transform:
    Mat Ma = MaCanon * MmodelPoints2Canonical;
    Mat Mb = MbCanon * MmodelPoints2Canonical;

    //output poses:
    Ma.copyTo(_Ma);
    Mb.copyTo(_Mb);
}

void PoseSolver::solveCanonicalForm(InputArray _canonicalObjPoints, InputArray _normalizedInputPoints, const Matx33d& H,
                                    OutputArray _Ma, OutputArray _Mb)
{
    _Ma.create(4, 4, CV_64FC1);
    _Mb.create(4, 4, CV_64FC1);

    Mat Ma = _Ma.getMat();
    Mat Mb = _Mb.getMat();

    //initialise poses:
    Ma.setTo(0);
    Ma.at<double>(3, 3) = 1;
    Mb.setTo(0);
    Mb.at<double>(3, 3) = 1;

    //Compute the Jacobian J of the homography at (0,0):
    double j00 = H(0, 0) - H(2, 0) * H(0, 2);
    double j01 = H(0, 1) - H(2, 1) * H(0, 2);
    double j10 = H(1, 0) - H(2, 0) * H(1, 2);
    double j11 = H(1, 1) - H(2, 1) * H(1, 2);

    //Compute the transformation of (0,0) into the image:
    double v0 = H(0, 2);
    double v1 = H(1, 2);

    //compute the two rotation solutions:
    Mat Ra = Ma.colRange(0, 3).rowRange(0, 3);
    Mat Rb = Mb.colRange(0, 3).rowRange(0, 3);
    computeRotations(j00, j01, j10, j11, v0, v1, Ra, Rb);

    //for each rotation solution, compute the corresponding translation solution:
    Mat ta = Ma.colRange(3, 4).rowRange(0, 3);
    Mat tb = Mb.colRange(3, 4).rowRange(0, 3);
    computeTranslation(_canonicalObjPoints, _normalizedInputPoints, Ra, ta);
    computeTranslation(_canonicalObjPoints, _normalizedInputPoints, Rb, tb);
}

void PoseSolver::solveSquare(InputArray _objectPoints, InputArray _imagePoints, OutputArray _rvec1, OutputArray _tvec1,
                             float& err1, OutputArray _rvec2, OutputArray _tvec2, float& err2)
{
    //allocate outputs:
    _rvec1.create(3, 1, CV_64FC1);
    _tvec1.create(3, 1, CV_64FC1);
    _rvec2.create(3, 1, CV_64FC1);
    _tvec2.create(3, 1, CV_64FC1);

    Mat objectPoints2D;

    //generate the object points:
    objectPoints2D.create(1, 4, CV_64FC2);
    Mat objectPoints = _objectPoints.getMat();
    double squareLength;
    if (objectPoints.depth() == CV_32F)
    {
        objectPoints2D.ptr<Vec2d>(0)[0] = Vec2d(objectPoints.ptr<Vec3f>(0)[0](0), objectPoints.ptr<Vec3f>(0)[0](1));
        objectPoints2D.ptr<Vec2d>(0)[1] = Vec2d(objectPoints.ptr<Vec3f>(0)[1](0), objectPoints.ptr<Vec3f>(0)[1](1));
        objectPoints2D.ptr<Vec2d>(0)[2] = Vec2d(objectPoints.ptr<Vec3f>(0)[2](0), objectPoints.ptr<Vec3f>(0)[2](1));
        objectPoints2D.ptr<Vec2d>(0)[3] = Vec2d(objectPoints.ptr<Vec3f>(0)[3](0), objectPoints.ptr<Vec3f>(0)[3](1));

        squareLength = sqrt( (objectPoints.ptr<Vec3f>(0)[1](0) - objectPoints.ptr<Vec3f>(0)[0](0))*
                             (objectPoints.ptr<Vec3f>(0)[1](0) - objectPoints.ptr<Vec3f>(0)[0](0)) +
                             (objectPoints.ptr<Vec3f>(0)[1](1) - objectPoints.ptr<Vec3f>(0)[0](1))*
                             (objectPoints.ptr<Vec3f>(0)[1](1) - objectPoints.ptr<Vec3f>(0)[0](1)) );
    }
    else
    {
        objectPoints2D.ptr<Vec2d>(0)[0] = Vec2d(objectPoints.ptr<Vec3d>(0)[0](0), objectPoints.ptr<Vec3d>(0)[0](1));
        objectPoints2D.ptr<Vec2d>(0)[1] = Vec2d(objectPoints.ptr<Vec3d>(0)[1](0), objectPoints.ptr<Vec3d>(0)[1](1));
        objectPoints2D.ptr<Vec2d>(0)[2] = Vec2d(objectPoints.ptr<Vec3d>(0)[2](0), objectPoints.ptr<Vec3d>(0)[2](1));
        objectPoints2D.ptr<Vec2d>(0)[3] = Vec2d(objectPoints.ptr<Vec3d>(0)[3](0), objectPoints.ptr<Vec3d>(0)[3](1));

        squareLength = sqrt( (objectPoints.ptr<Vec3d>(0)[1](0) - objectPoints.ptr<Vec3d>(0)[0](0))*
                             (objectPoints.ptr<Vec3d>(0)[1](0) - objectPoints.ptr<Vec3d>(0)[0](0)) +
                             (objectPoints.ptr<Vec3d>(0)[1](1) - objectPoints.ptr<Vec3d>(0)[0](1))*
                             (objectPoints.ptr<Vec3d>(0)[1](1) - objectPoints.ptr<Vec3d>(0)[0](1)) );
    }

    Mat H; //homography from canonical object points to normalized pixels

    Mat normalizedInputPoints;
    if (_imagePoints.getMat().type() == CV_32FC2)
    {
        _imagePoints.getMat().convertTo(normalizedInputPoints, CV_64F);
    }
    else
    {
        normalizedInputPoints = _imagePoints.getMat();
    }

    //compute H
    homographyFromSquarePoints(normalizedInputPoints, squareLength / 2.0, H);

    //now solve
    Mat Ma, Mb;
    solveCanonicalForm(objectPoints2D, normalizedInputPoints, H, Ma, Mb);

    //sort poses according to reprojection error:
    Mat M1, M2;
    sortPosesByReprojError(_objectPoints, normalizedInputPoints, Ma, Mb, M1, M2, err1, err2);

    //fill outputs
    rot2vec(M1.colRange(0, 3).rowRange(0, 3), _rvec1);
    rot2vec(M2.colRange(0, 3).rowRange(0, 3), _rvec2);

    M1.colRange(3, 4).rowRange(0, 3).copyTo(_tvec1);
    M2.colRange(3, 4).rowRange(0, 3).copyTo(_tvec2);
}

void PoseSolver::generateSquareObjectCorners3D(double squareLength, OutputArray _objectPoints)
{
    _objectPoints.create(1, 4, CV_64FC3);
    Mat objectPoints = _objectPoints.getMat();
    objectPoints.ptr<Vec3d>(0)[0] = Vec3d(-squareLength / 2.0, squareLength / 2.0, 0.0);
    objectPoints.ptr<Vec3d>(0)[1] = Vec3d(squareLength / 2.0, squareLength / 2.0, 0.0);
    objectPoints.ptr<Vec3d>(0)[2] = Vec3d(squareLength / 2.0, -squareLength / 2.0, 0.0);
    objectPoints.ptr<Vec3d>(0)[3] = Vec3d(-squareLength / 2.0, -squareLength / 2.0, 0.0);
}

void PoseSolver::generateSquareObjectCorners2D(double squareLength, OutputArray _objectPoints)
{
    _objectPoints.create(1, 4, CV_64FC2);
    Mat objectPoints = _objectPoints.getMat();
    objectPoints.ptr<Vec2d>(0)[0] = Vec2d(-squareLength / 2.0, squareLength / 2.0);
    objectPoints.ptr<Vec2d>(0)[1] = Vec2d(squareLength / 2.0, squareLength / 2.0);
    objectPoints.ptr<Vec2d>(0)[2] = Vec2d(squareLength / 2.0, -squareLength / 2.0);
    objectPoints.ptr<Vec2d>(0)[3] = Vec2d(-squareLength / 2.0, -squareLength / 2.0);
}

double PoseSolver::meanSceneDepth(InputArray _objectPoints, InputArray _rvec, InputArray _tvec)
{
    CV_CheckType(_objectPoints.type(), _objectPoints.type() == CV_64FC3,
                 "Type of _objectPoints must be CV_64FC3" );

    size_t n = static_cast<size_t>(_objectPoints.rows() * _objectPoints.cols());
    Mat R;
    Mat q;
    Rodrigues(_rvec, R);
    double zBar = 0;

    for (size_t i = 0; i < n; i++)
    {
        Mat p(_objectPoints.getMat().at<Point3d>(static_cast<int>(i)));
        q = R * p + _tvec.getMat();
        double z;
        if (q.depth() == CV_64F)
        {
            z = q.at<double>(2);
        }
        else
        {
            z = static_cast<double>(q.at<float>(2));
        }
        zBar += z;
    }
    return zBar / static_cast<double>(n);
}

void PoseSolver::rot2vec(InputArray _R, OutputArray _r)
{
    CV_CheckType(_R.type(), _R.type() == CV_64FC1,
                 "Type of _R must be CV_64FC1" );
    CV_Assert(_R.rows() == 3);
    CV_Assert(_R.cols() == 3);

    _r.create(3, 1, CV_64FC1);

    Mat R = _R.getMat();
    Mat rvec = _r.getMat();

    double trace = R.at<double>(0, 0) + R.at<double>(1, 1) + R.at<double>(2, 2);
    double w_norm = std::acos((trace - 1.0) / 2.0);
    double eps = std::numeric_limits<float>::epsilon();
    double d = 1 / (2 * std::sin(w_norm)) * w_norm;
    if (w_norm < eps) //rotation is the identity
    {
        rvec.setTo(0);
    }
    else
    {
        double c0 = R.at<double>(2, 1) - R.at<double>(1, 2);
        double c1 = R.at<double>(0, 2) - R.at<double>(2, 0);
        double c2 = R.at<double>(1, 0) - R.at<double>(0, 1);
        rvec.at<double>(0) = d * c0;
        rvec.at<double>(1) = d * c1;
        rvec.at<double>(2) = d * c2;
    }
}

void PoseSolver::computeTranslation(InputArray _objectPoints, InputArray _normalizedImgPoints, InputArray _R, OutputArray _t)
{
    //This is solved by building the linear system At = b, where t corresponds to the (unknown) translation.
    //This is then inverted with the associated normal equations to give t = inv(transpose(A)*A)*transpose(A)*b
    //For efficiency we only store the coefficients of (transpose(A)*A) and (transpose(A)*b)

    CV_CheckType(_objectPoints.type(), _objectPoints.type() == CV_64FC2,
                 "Type of _objectPoints must be CV_64FC2" );
    CV_CheckType(_normalizedImgPoints.type(), _normalizedImgPoints.type() == CV_64FC2,
                 "Type of _normalizedImgPoints must be CV_64FC2" );
    CV_CheckType(_R.type(), _R.type() == CV_64FC1,
                 "Type of _R must be CV_64FC1" );
    CV_Assert(_R.rows() == 3 && _R.cols() == 3);
    CV_Assert(_objectPoints.rows() == 1 || _objectPoints.cols() == 1);
    CV_Assert(_normalizedImgPoints.rows() == 1 || _normalizedImgPoints.cols() == 1);

    size_t n = static_cast<size_t>(_normalizedImgPoints.rows() * _normalizedImgPoints.cols());
    CV_Assert(n == static_cast<size_t>(_objectPoints.rows() * _objectPoints.cols()));

    Mat objectPoints = _objectPoints.getMat();
    Mat imgPoints = _normalizedImgPoints.getMat();

    _t.create(3, 1, CV_64FC1);

    Mat R = _R.getMat();

    //coefficients of (transpose(A)*A)
    double ATA00 = static_cast<double>(n);
    double ATA02 = 0;
    double ATA11 = static_cast<double>(n);
    double ATA12 = 0;
    double ATA20 = 0;
    double ATA21 = 0;
    double ATA22 = 0;

    //coefficients of (transpose(A)*b)
    double ATb0 = 0;
    double ATb1 = 0;
    double ATb2 = 0;

    //now loop through each point and increment the coefficients:
    for (int i = 0; i < static_cast<int>(n); i++)
    {
        const Vec2d& objPt = objectPoints.at<Vec2d>(i);
        double rx = R.at<double>(0, 0) * objPt(0) + R.at<double>(0, 1) * objPt(1);
        double ry = R.at<double>(1, 0) * objPt(0) + R.at<double>(1, 1) * objPt(1);
        double rz = R.at<double>(2, 0) * objPt(0) + R.at<double>(2, 1) * objPt(1);

        const Vec2d& imgPt = imgPoints.at<Vec2d>(i);
        double a2 = -imgPt(0);
        double b2 = -imgPt(1);

        ATA02 = ATA02 + a2;
        ATA12 = ATA12 + b2;
        ATA20 = ATA20 + a2;
        ATA21 = ATA21 + b2;
        ATA22 = ATA22 + a2 * a2 + b2 * b2;

        double bx = -a2 * rz - rx;
        double by = -b2 * rz - ry;

        ATb0 = ATb0 + bx;
        ATb1 = ATb1 + by;
        ATb2 = ATb2 + a2 * bx + b2 * by;
    }

    double detAInv = 1.0 / (ATA00 * ATA11 * ATA22 - ATA00 * ATA12 * ATA21 - ATA02 * ATA11 * ATA20);

    //S gives inv(transpose(A)*A)/det(A)^2
    //construct S:
    double S00 = ATA11 * ATA22 - ATA12 * ATA21;
    double S01 = ATA02 * ATA21;
    double S02 = -ATA02 * ATA11;
    double S10 = ATA12 * ATA20;
    double S11 = ATA00 * ATA22 - ATA02 * ATA20;
    double S12 = -ATA00 * ATA12;
    double S20 = -ATA11 * ATA20;
    double S21 = -ATA00 * ATA21;
    double S22 = ATA00 * ATA11;

    //solve t:
    Mat t = _t.getMat();
    t.at<double>(0) = detAInv * (S00 * ATb0 + S01 * ATb1 + S02 * ATb2);
    t.at<double>(1) = detAInv * (S10 * ATb0 + S11 * ATb1 + S12 * ATb2);
    t.at<double>(2) = detAInv * (S20 * ATb0 + S21 * ATb1 + S22 * ATb2);
}

void PoseSolver::computeRotations(double j00, double j01, double j10, double j11, double p, double q, OutputArray _R1, OutputArray _R2)
{
    //This is fairly optimized code which makes it hard to understand. The matlab code is certainly easier to read.
    _R1.create(3, 3, CV_64FC1);
    _R2.create(3, 3, CV_64FC1);

    Matx33d Rv;
    Matx31d v(p, q, 1);
    rotateVec2ZAxis(v,Rv);
    Rv = Rv.t();

    //setup the 2x2 SVD decomposition:
    double rv00 = Rv(0,0);
    double rv01 = Rv(0,1);
    double rv02 = Rv(0,2);

    double rv10 = Rv(1,0);
    double rv11 = Rv(1,1);
    double rv12 = Rv(1,2);

    double rv20 = Rv(2,0);
    double rv21 = Rv(2,1);
    double rv22 = Rv(2,2);

    double b00 = rv00 - p * rv20;
    double b01 = rv01 - p * rv21;
    double b10 = rv10 - q * rv20;
    double b11 = rv11 - q * rv21;

    double dtinv = 1.0 / ((b00 * b11 - b01 * b10));

    double binv00 = dtinv * b11;
    double binv01 = -dtinv * b01;
    double binv10 = -dtinv * b10;
    double binv11 = dtinv * b00;

    double a00 = binv00 * j00 + binv01 * j10;
    double a01 = binv00 * j01 + binv01 * j11;
    double a10 = binv10 * j00 + binv11 * j10;
    double a11 = binv10 * j01 + binv11 * j11;

    //compute the largest singular value of A:
    double ata00 = a00 * a00 + a01 * a01;
    double ata01 = a00 * a10 + a01 * a11;
    double ata11 = a10 * a10 + a11 * a11;

    double gamma2 = 0.5 * (ata00 + ata11 + sqrt((ata00 - ata11) * (ata00 - ata11) + 4.0 * ata01 * ata01));
    if (gamma2 < 0)
        CV_Error(Error::StsNoConv, "gamma2 is negative.");

    double gamma = sqrt(gamma2);

    if (std::fabs(gamma) < std::numeric_limits<float>::epsilon())
        CV_Error(Error::StsNoConv, "gamma is zero.");

    //reconstruct the full rotation matrices:
    double rtilde00 = a00 / gamma;
    double rtilde01 = a01 / gamma;
    double rtilde10 = a10 / gamma;
    double rtilde11 = a11 / gamma;

    double rtilde00_2 = rtilde00 * rtilde00;
    double rtilde01_2 = rtilde01 * rtilde01;
    double rtilde10_2 = rtilde10 * rtilde10;
    double rtilde11_2 = rtilde11 * rtilde11;

    double b0 = sqrt(-rtilde00_2 - rtilde10_2 + 1);
    double b1 = sqrt(-rtilde01_2 - rtilde11_2 + 1);
    double sp = (-rtilde00 * rtilde01 - rtilde10 * rtilde11);

    if (sp < 0)
    {
        b1 = -b1;
    }

    //store results:
    Mat R1 = _R1.getMat();
    Mat R2 = _R2.getMat();

    R1.at<double>(0, 0) = (rtilde00)*rv00 + (rtilde10)*rv01 + (b0)*rv02;
    R1.at<double>(0, 1) = (rtilde01)*rv00 + (rtilde11)*rv01 + (b1)*rv02;
    R1.at<double>(0, 2) = (b1 * rtilde10 - b0 * rtilde11) * rv00 + (b0 * rtilde01 - b1 * rtilde00) * rv01 + (rtilde00 * rtilde11 - rtilde01 * rtilde10) * rv02;
    R1.at<double>(1, 0) = (rtilde00)*rv10 + (rtilde10)*rv11 + (b0)*rv12;
    R1.at<double>(1, 1) = (rtilde01)*rv10 + (rtilde11)*rv11 + (b1)*rv12;
    R1.at<double>(1, 2) = (b1 * rtilde10 - b0 * rtilde11) * rv10 + (b0 * rtilde01 - b1 * rtilde00) * rv11 + (rtilde00 * rtilde11 - rtilde01 * rtilde10) * rv12;
    R1.at<double>(2, 0) = (rtilde00)*rv20 + (rtilde10)*rv21 + (b0)*rv22;
    R1.at<double>(2, 1) = (rtilde01)*rv20 + (rtilde11)*rv21 + (b1)*rv22;
    R1.at<double>(2, 2) = (b1 * rtilde10 - b0 * rtilde11) * rv20 + (b0 * rtilde01 - b1 * rtilde00) * rv21 + (rtilde00 * rtilde11 - rtilde01 * rtilde10) * rv22;

    R2.at<double>(0, 0) = (rtilde00)*rv00 + (rtilde10)*rv01 + (-b0) * rv02;
    R2.at<double>(0, 1) = (rtilde01)*rv00 + (rtilde11)*rv01 + (-b1) * rv02;
    R2.at<double>(0, 2) = (b0 * rtilde11 - b1 * rtilde10) * rv00 + (b1 * rtilde00 - b0 * rtilde01) * rv01 + (rtilde00 * rtilde11 - rtilde01 * rtilde10) * rv02;
    R2.at<double>(1, 0) = (rtilde00)*rv10 + (rtilde10)*rv11 + (-b0) * rv12;
    R2.at<double>(1, 1) = (rtilde01)*rv10 + (rtilde11)*rv11 + (-b1) * rv12;
    R2.at<double>(1, 2) = (b0 * rtilde11 - b1 * rtilde10) * rv10 + (b1 * rtilde00 - b0 * rtilde01) * rv11 + (rtilde00 * rtilde11 - rtilde01 * rtilde10) * rv12;
    R2.at<double>(2, 0) = (rtilde00)*rv20 + (rtilde10)*rv21 + (-b0) * rv22;
    R2.at<double>(2, 1) = (rtilde01)*rv20 + (rtilde11)*rv21 + (-b1) * rv22;
    R2.at<double>(2, 2) = (b0 * rtilde11 - b1 * rtilde10) * rv20 + (b1 * rtilde00 - b0 * rtilde01) * rv21 + (rtilde00 * rtilde11 - rtilde01 * rtilde10) * rv22;
}

void PoseSolver::homographyFromSquarePoints(InputArray _targetPoints, double halfLength, OutputArray H_)
{
    CV_CheckType(_targetPoints.type(), _targetPoints.type() == CV_32FC2 || _targetPoints.type() == CV_64FC2,
                 "Type of _targetPoints must be CV_32FC2 or CV_64FC2" );

    Mat pts = _targetPoints.getMat();

    double p1x, p1y;
    double p2x, p2y;
    double p3x, p3y;
    double p4x, p4y;

    if (_targetPoints.type() == CV_32FC2)
    {
        p1x = -pts.at<Vec2f>(0)(0);
        p1y = -pts.at<Vec2f>(0)(1);

        p2x = -pts.at<Vec2f>(1)(0);
        p2y = -pts.at<Vec2f>(1)(1);

        p3x = -pts.at<Vec2f>(2)(0);
        p3y = -pts.at<Vec2f>(2)(1);

        p4x = -pts.at<Vec2f>(3)(0);
        p4y = -pts.at<Vec2f>(3)(1);
    }
    else
    {
        p1x = -pts.at<Vec2d>(0)(0);
        p1y = -pts.at<Vec2d>(0)(1);

        p2x = -pts.at<Vec2d>(1)(0);
        p2y = -pts.at<Vec2d>(1)(1);

        p3x = -pts.at<Vec2d>(2)(0);
        p3y = -pts.at<Vec2d>(2)(1);

        p4x = -pts.at<Vec2d>(3)(0);
        p4y = -pts.at<Vec2d>(3)(1);
    }

    //analytic solution:
    double det = (halfLength * (p1x * p2y - p2x * p1y - p1x * p4y + p2x * p3y - p3x * p2y + p4x * p1y + p3x * p4y - p4x * p3y));
    if (abs(det) < 1e-9)
        CV_Error(Error::StsNoConv, "Determinant is zero!");
    double detsInv = -1 / det;

    Matx33d H;
    H(0, 0) = detsInv * (p1x * p3x * p2y - p2x * p3x * p1y - p1x * p4x * p2y + p2x * p4x * p1y - p1x * p3x * p4y + p1x * p4x * p3y + p2x * p3x * p4y - p2x * p4x * p3y);
    H(0, 1) = detsInv * (p1x * p2x * p3y - p1x * p3x * p2y - p1x * p2x * p4y + p2x * p4x * p1y + p1x * p3x * p4y - p3x * p4x * p1y - p2x * p4x * p3y + p3x * p4x * p2y);
    H(0, 2) = detsInv * halfLength * (p1x * p2x * p3y - p2x * p3x * p1y - p1x * p2x * p4y + p1x * p4x * p2y - p1x * p4x * p3y + p3x * p4x * p1y + p2x * p3x * p4y - p3x * p4x * p2y);
    H(1, 0) = detsInv * (p1x * p2y * p3y - p2x * p1y * p3y - p1x * p2y * p4y + p2x * p1y * p4y - p3x * p1y * p4y + p4x * p1y * p3y + p3x * p2y * p4y - p4x * p2y * p3y);
    H(1, 1) = detsInv * (p2x * p1y * p3y - p3x * p1y * p2y - p1x * p2y * p4y + p4x * p1y * p2y + p1x * p3y * p4y - p4x * p1y * p3y - p2x * p3y * p4y + p3x * p2y * p4y);
    H(1, 2) = detsInv * halfLength * (p1x * p2y * p3y - p3x * p1y * p2y - p2x * p1y * p4y + p4x * p1y * p2y - p1x * p3y * p4y + p3x * p1y * p4y + p2x * p3y * p4y - p4x * p2y * p3y);
    H(2, 0) = -detsInv * (p1x * p3y - p3x * p1y - p1x * p4y - p2x * p3y + p3x * p2y + p4x * p1y + p2x * p4y - p4x * p2y);
    H(2, 1) = detsInv * (p1x * p2y - p2x * p1y - p1x * p3y + p3x * p1y + p2x * p4y - p4x * p2y - p3x * p4y + p4x * p3y);
    H(2, 2) = 1.0;

    Mat(H, false).copyTo(H_);
}

void PoseSolver::makeCanonicalObjectPoints(InputArray _objectPoints, OutputArray _canonicalObjPoints, OutputArray _MmodelPoints2Canonical)
{
    int objType = _objectPoints.type();
    CV_CheckType(objType, objType == CV_32FC3 || objType == CV_64FC3,
                 "Type of _objectPoints must be CV_32FC3 or CV_64FC3" );

    int n = _objectPoints.rows() * _objectPoints.cols();

    _canonicalObjPoints.create(1, n, CV_64FC2);

    Mat objectPoints = _objectPoints.getMat();
    Mat canonicalObjPoints = _canonicalObjPoints.getMat();

    Mat UZero(3, n, CV_64FC1);

    double xBar = 0;
    double yBar = 0;
    double zBar = 0;
    bool isOnZPlane = true;
    for (int i = 0; i < n; i++)
    {
        double x, y, z;
        if (objType == CV_32FC3)
        {
            x = static_cast<double>(objectPoints.at<Vec3f>(i)[0]);
            y = static_cast<double>(objectPoints.at<Vec3f>(i)[1]);
            z = static_cast<double>(objectPoints.at<Vec3f>(i)[2]);
        }
        else
        {
            x = objectPoints.at<Vec3d>(i)[0];
            y = objectPoints.at<Vec3d>(i)[1];
            z = objectPoints.at<Vec3d>(i)[2];
        }

        if (abs(z) > IPPE_SMALL)
        {
            isOnZPlane = false;
        }

        xBar += x;
        yBar += y;
        zBar += z;

        UZero.at<double>(0, i) = x;
        UZero.at<double>(1, i) = y;
        UZero.at<double>(2, i) = z;
    }
    xBar = xBar / static_cast<double>(n);
    yBar = yBar / static_cast<double>(n);
    zBar = zBar / static_cast<double>(n);

    for (int i = 0; i < n; i++)
    {
        UZero.at<double>(0, i) -= xBar;
        UZero.at<double>(1, i) -= yBar;
        UZero.at<double>(2, i) -= zBar;
    }

    Matx44d MCenter = Matx44d::eye();
    MCenter(0, 3) = -xBar;
    MCenter(1, 3) = -yBar;
    MCenter(2, 3) = -zBar;

    if (isOnZPlane)
    {
        //MmodelPoints2Canonical is given by MCenter
        Mat(MCenter, false).copyTo(_MmodelPoints2Canonical);
        for (int i = 0; i < n; i++)
        {
            canonicalObjPoints.at<Vec2d>(i)[0] = UZero.at<double>(0, i);
            canonicalObjPoints.at<Vec2d>(i)[1] = UZero.at<double>(1, i);
        }
    }
    else
    {
        Mat UZeroAligned(3, n, CV_64FC1);
        Matx33d R; //rotation that rotates objectPoints to the plane z=0

        if (!computeObjextSpaceR3Pts(objectPoints,R))
        {
            //we could not compute R, probably because there is a duplicate point in {objectPoints(0),objectPoints(1),objectPoints(2)}.
            //So we compute it with the SVD (which is slower):
            computeObjextSpaceRSvD(UZero,R);
        }

        UZeroAligned = R * UZero;

        for (int i = 0; i < n; i++)
        {
            canonicalObjPoints.at<Vec2d>(i)[0] = UZeroAligned.at<double>(0, i);
            canonicalObjPoints.at<Vec2d>(i)[1] = UZeroAligned.at<double>(1, i);
            if (abs(UZeroAligned.at<double>(2, i)) > IPPE_SMALL)
                CV_Error(Error::StsNoConv, "Cannot transform object points to the plane z=0!");
        }

        Matx44d MRot = Matx44d::zeros();
        MRot(3, 3) = 1;

        for (int i = 0; i < 3; i++)
        {
            for (int j = 0; j < 3; j++)
            {
                MRot(i,j) = R(i,j);
            }
        }
        Matx44d Mb = MRot * MCenter;
        Mat(Mb, false).copyTo(_MmodelPoints2Canonical);
    }
}

void PoseSolver::evalReprojError(InputArray _objectPoints, InputArray _imagePoints, InputArray _M, float& err)
{
    Mat projectedPoints;
    Mat imagePoints = _imagePoints.getMat();
    Mat r;
    rot2vec(_M.getMat().colRange(0, 3).rowRange(0, 3), r);

    Mat K = Mat::eye(3, 3, CV_64FC1);
    Mat dist;
    projectPoints(_objectPoints, r, _M.getMat().colRange(3, 4).rowRange(0, 3), K, dist, projectedPoints);

    err = 0;
    int n = _objectPoints.rows() * _objectPoints.cols();

    float dx, dy;
    const int projPtsDepth = projectedPoints.depth();
    for (int i = 0; i < n; i++)
    {
        if (projPtsDepth == CV_32F)
        {
            dx = projectedPoints.at<Vec2f>(i)[0] - static_cast<float>(imagePoints.at<Vec2d>(i)[0]);
            dy = projectedPoints.at<Vec2f>(i)[1] - static_cast<float>(imagePoints.at<Vec2d>(i)[1]);
        }
        else
        {
            dx = static_cast<float>(projectedPoints.at<Vec2d>(i)[0] - imagePoints.at<Vec2d>(i)[0]);
            dy = static_cast<float>(projectedPoints.at<Vec2d>(i)[1] - imagePoints.at<Vec2d>(i)[1]);
        }

        err += dx * dx + dy * dy;
    }
    err = sqrt(err / (2.0f * n));
}

void PoseSolver::sortPosesByReprojError(InputArray _objectPoints, InputArray _imagePoints, InputArray _Ma, InputArray _Mb,
                                        OutputArray _M1, OutputArray _M2, float& err1, float& err2)
{
    float erra, errb;
    evalReprojError(_objectPoints, _imagePoints, _Ma, erra);
    evalReprojError(_objectPoints, _imagePoints, _Mb, errb);
    if (erra < errb)
    {
        err1 = erra;
        _Ma.copyTo(_M1);

        err2 = errb;
        _Mb.copyTo(_M2);
    }
    else
    {
        err1 = errb;
        _Mb.copyTo(_M1);

        err2 = erra;
        _Ma.copyTo(_M2);
    }
}

void PoseSolver::rotateVec2ZAxis(const Matx31d& a, Matx33d& Ra)
{
    double ax = a(0);
    double ay = a(1);
    double az = a(2);

    double nrm = sqrt(ax*ax + ay*ay + az*az);
    ax = ax/nrm;
    ay = ay/nrm;
    az = az/nrm;

    double c = az;

    if (abs(1.0+c) < std::numeric_limits<float>::epsilon())
    {
        Ra = Matx33d::zeros();
        Ra(0,0) = 1.0;
        Ra(1,1) = 1.0;
        Ra(2,2) = -1.0;
    }
    else
    {
        double d = 1.0/(1.0+c);
        double ax2 = ax*ax;
        double ay2 = ay*ay;
        double axay = ax*ay;

        Ra(0,0) = -ax2*d + 1.0;
        Ra(0,1) = -axay*d;
        Ra(0,2) = -ax;

        Ra(1,0) = -axay*d;
        Ra(1,1) = -ay2*d + 1.0;
        Ra(1,2) = -ay;

        Ra(2,0) = ax;
        Ra(2,1) = ay;
        Ra(2,2) = 1.0 - (ax2 + ay2)*d;
    }
}

bool PoseSolver::computeObjextSpaceR3Pts(InputArray _objectPoints, Matx33d& R)
{
    bool ret; //return argument
    double p1x,p1y,p1z;
    double p2x,p2y,p2z;
    double p3x,p3y,p3z;

    Mat objectPoints = _objectPoints.getMat();
    if (objectPoints.type() == CV_32FC3)
    {
        p1x = objectPoints.at<Vec3f>(0)[0];
        p1y = objectPoints.at<Vec3f>(0)[1];
        p1z = objectPoints.at<Vec3f>(0)[2];

        p2x = objectPoints.at<Vec3f>(1)[0];
        p2y = objectPoints.at<Vec3f>(1)[1];
        p2z = objectPoints.at<Vec3f>(1)[2];

        p3x = objectPoints.at<Vec3f>(2)[0];
        p3y = objectPoints.at<Vec3f>(2)[1];
        p3z = objectPoints.at<Vec3f>(2)[2];
    }
    else
    {
        p1x = objectPoints.at<Vec3d>(0)[0];
        p1y = objectPoints.at<Vec3d>(0)[1];
        p1z = objectPoints.at<Vec3d>(0)[2];

        p2x = objectPoints.at<Vec3d>(1)[0];
        p2y = objectPoints.at<Vec3d>(1)[1];
        p2z = objectPoints.at<Vec3d>(1)[2];

        p3x = objectPoints.at<Vec3d>(2)[0];
        p3y = objectPoints.at<Vec3d>(2)[1];
        p3z = objectPoints.at<Vec3d>(2)[2];
    }

    double nx = (p1y - p2y)*(p1z - p3z) - (p1y - p3y)*(p1z - p2z);
    double ny = (p1x - p3x)*(p1z - p2z) - (p1x - p2x)*(p1z - p3z);
    double nz = (p1x - p2x)*(p1y - p3y) - (p1x - p3x)*(p1y - p2y);

    double nrm = sqrt(nx*nx+ ny*ny + nz*nz);
    if (nrm > IPPE_SMALL)
    {
        nx = nx/nrm;
        ny = ny/nrm;
        nz = nz/nrm;
        Matx31d v(nx, ny, nz);
        rotateVec2ZAxis(v,R);
        ret = true;
    }
    else
    {
        ret = false;
    }
    return ret;
}

void PoseSolver::computeObjextSpaceRSvD(InputArray _objectPointsZeroMean, OutputArray _R)
{
    _R.create(3, 3, CV_64FC1);
    Mat R = _R.getMat();

    //we could not compute R with the first three points, so lets use the SVD
    SVD s;
    Mat W, U, VT;
    s.compute(_objectPointsZeroMean.getMat() * _objectPointsZeroMean.getMat().t(), W, U, VT);
    double s3 = W.at<double>(2);
    double s2 = W.at<double>(1);

    //check if points are coplanar:
    CV_Assert(s3 / s2 < IPPE_SMALL);

    R = U.t();
    if (determinant(R) < 0)
    {
        //this ensures R is a rotation matrix and not a general unitary matrix:
        R.at<double>(2, 0) = -R.at<double>(2, 0);
        R.at<double>(2, 1) = -R.at<double>(2, 1);
        R.at<double>(2, 2) = -R.at<double>(2, 2);
    }
}
} //namespace IPPE

namespace HomographyHO {
void normalizeDataIsotropic(InputArray _Data, OutputArray _DataN, OutputArray _T, OutputArray _Ti)
{
    Mat Data = _Data.getMat();
    int numPoints = Data.rows * Data.cols;
    CV_Assert(Data.rows == 1 || Data.cols == 1);
    CV_Assert(Data.channels() == 2 || Data.channels() == 3);
    CV_Assert(numPoints >= 4);

    int dataType = _Data.type();
    CV_CheckType(dataType, dataType == CV_32FC2 || dataType == CV_32FC3 || dataType == CV_64FC2 || dataType == CV_64FC3,
                 "Type of _Data must be one of CV_32FC2, CV_32FC3, CV_64FC2, CV_64FC3");

    _DataN.create(2, numPoints, CV_64FC1);

    _T.create(3, 3, CV_64FC1);
    _Ti.create(3, 3, CV_64FC1);

    Mat DataN = _DataN.getMat();
    Mat T = _T.getMat();
    Mat Ti = _Ti.getMat();

    _T.setTo(0);
    _Ti.setTo(0);

    int numChannels = Data.channels();
    double xm = 0;
    double ym = 0;
    for (int i = 0; i < numPoints; i++)
    {
        if (numChannels == 2)
        {
            if (dataType == CV_32FC2)
            {
                xm = xm + Data.at<Vec2f>(i)[0];
                ym = ym + Data.at<Vec2f>(i)[1];
            }
            else
            {
                xm = xm + Data.at<Vec2d>(i)[0];
                ym = ym + Data.at<Vec2d>(i)[1];
            }
        }
        else
        {
            if (dataType == CV_32FC3)
            {
                xm = xm + Data.at<Vec3f>(i)[0];
                ym = ym + Data.at<Vec3f>(i)[1];
            }
            else
            {
                xm = xm + Data.at<Vec3d>(i)[0];
                ym = ym + Data.at<Vec3d>(i)[1];
            }
        }
    }
    xm = xm / static_cast<double>(numPoints);
    ym = ym / static_cast<double>(numPoints);

    double kappa = 0;
    double xh, yh;

    for (int i = 0; i < numPoints; i++)
    {

        if (numChannels == 2)
        {
            if (dataType == CV_32FC2)
            {
                xh = Data.at<Vec2f>(i)[0] - xm;
                yh = Data.at<Vec2f>(i)[1] - ym;
            }
            else
            {
                xh = Data.at<Vec2d>(i)[0] - xm;
                yh = Data.at<Vec2d>(i)[1] - ym;
            }
        }
        else
        {
            if (dataType == CV_32FC3)
            {
                xh = Data.at<Vec3f>(i)[0] - xm;
                yh = Data.at<Vec3f>(i)[1] - ym;
            }
            else
            {
                xh = Data.at<Vec3d>(i)[0] - xm;
                yh = Data.at<Vec3d>(i)[1] - ym;
            }
        }

        DataN.at<double>(0, i) = xh;
        DataN.at<double>(1, i) = yh;
        kappa = kappa + xh * xh + yh * yh;
    }
    double beta = sqrt(2 * numPoints / kappa);
    DataN = DataN * beta;

    T.at<double>(0, 0) = 1.0 / beta;
    T.at<double>(1, 1) = 1.0 / beta;

    T.at<double>(0, 2) = xm;
    T.at<double>(1, 2) = ym;

    T.at<double>(2, 2) = 1;

    Ti.at<double>(0, 0) = beta;
    Ti.at<double>(1, 1) = beta;

    Ti.at<double>(0, 2) = -beta * xm;
    Ti.at<double>(1, 2) = -beta * ym;

    Ti.at<double>(2, 2) = 1;
}

void homographyHO(InputArray _srcPoints, InputArray _targPoints, Matx33d& H)
{
    Mat DataA, DataB, TA, TAi, TB, TBi;

    HomographyHO::normalizeDataIsotropic(_srcPoints, DataA, TA, TAi);
    HomographyHO::normalizeDataIsotropic(_targPoints, DataB, TB, TBi);

    int n = DataA.cols;
    CV_Assert(n == DataB.cols);

    Mat C1(1, n, CV_64FC1);
    Mat C2(1, n, CV_64FC1);
    Mat C3(1, n, CV_64FC1);
    Mat C4(1, n, CV_64FC1);

    double mC1 = 0, mC2 = 0, mC3 = 0, mC4 = 0;

    for (int i = 0; i < n; i++)
    {
        C1.at<double>(0, i) = -DataB.at<double>(0, i) * DataA.at<double>(0, i);
        C2.at<double>(0, i) = -DataB.at<double>(0, i) * DataA.at<double>(1, i);
        C3.at<double>(0, i) = -DataB.at<double>(1, i) * DataA.at<double>(0, i);
        C4.at<double>(0, i) = -DataB.at<double>(1, i) * DataA.at<double>(1, i);

        mC1 += C1.at<double>(0, i);
        mC2 += C2.at<double>(0, i);
        mC3 += C3.at<double>(0, i);
        mC4 += C4.at<double>(0, i);
    }

    mC1 /= n;
    mC2 /= n;
    mC3 /= n;
    mC4 /= n;

    Mat Mx(n, 3, CV_64FC1);
    Mat My(n, 3, CV_64FC1);

    for (int i = 0; i < n; i++)
    {
        Mx.at<double>(i, 0) = C1.at<double>(0, i) - mC1;
        Mx.at<double>(i, 1) = C2.at<double>(0, i) - mC2;
        Mx.at<double>(i, 2) = -DataB.at<double>(0, i);

        My.at<double>(i, 0) = C3.at<double>(0, i) - mC3;
        My.at<double>(i, 1) = C4.at<double>(0, i) - mC4;
        My.at<double>(i, 2) = -DataB.at<double>(1, i);
    }

    Mat DataAT, DataADataAT;

    transpose(DataA, DataAT);
    DataADataAT = DataA * DataAT;
    double dt = DataADataAT.at<double>(0, 0) * DataADataAT.at<double>(1, 1) - DataADataAT.at<double>(0, 1) * DataADataAT.at<double>(1, 0);

    Mat DataADataATi(2, 2, CV_64FC1);
    DataADataATi.at<double>(0, 0) = DataADataAT.at<double>(1, 1) / dt;
    DataADataATi.at<double>(0, 1) = -DataADataAT.at<double>(0, 1) / dt;
    DataADataATi.at<double>(1, 0) = -DataADataAT.at<double>(1, 0) / dt;
    DataADataATi.at<double>(1, 1) = DataADataAT.at<double>(0, 0) / dt;

    Mat Pp = DataADataATi * DataA;

    Mat Bx = Pp * Mx;
    Mat By = Pp * My;

    Mat Ex = DataAT * Bx;
    Mat Ey = DataAT * By;

    Mat D(2 * n, 3, CV_64FC1);

    for (int i = 0; i < n; i++)
    {
        D.at<double>(i, 0) = Mx.at<double>(i, 0) - Ex.at<double>(i, 0);
        D.at<double>(i, 1) = Mx.at<double>(i, 1) - Ex.at<double>(i, 1);
        D.at<double>(i, 2) = Mx.at<double>(i, 2) - Ex.at<double>(i, 2);

        D.at<double>(i + n, 0) = My.at<double>(i, 0) - Ey.at<double>(i, 0);
        D.at<double>(i + n, 1) = My.at<double>(i, 1) - Ey.at<double>(i, 1);
        D.at<double>(i + n, 2) = My.at<double>(i, 2) - Ey.at<double>(i, 2);
    }

    Mat DT, DDT;
    transpose(D, DT);
    DDT = DT * D;

    Mat S, U;
    eigen(DDT, S, U);

    Mat h789(3, 1, CV_64FC1);
    h789.at<double>(0, 0) = U.at<double>(2, 0);
    h789.at<double>(1, 0) = U.at<double>(2, 1);
    h789.at<double>(2, 0) = U.at<double>(2, 2);

    Mat h12 = -Bx * h789;
    Mat h45 = -By * h789;

    double h3 = -(mC1 * h789.at<double>(0, 0) + mC2 * h789.at<double>(1, 0));
    double h6 = -(mC3 * h789.at<double>(0, 0) + mC4 * h789.at<double>(1, 0));

    H(0, 0) = h12.at<double>(0, 0);
    H(0, 1) = h12.at<double>(1, 0);
    H(0, 2) = h3;

    H(1, 0) = h45.at<double>(0, 0);
    H(1, 1) = h45.at<double>(1, 0);
    H(1, 2) = h6;

    H(2, 0) = h789.at<double>(0, 0);
    H(2, 1) = h789.at<double>(1, 0);
    H(2, 2) = h789.at<double>(2, 0);

    H = Mat(TB * H * TAi);
    double h22_inv = 1 / H(2, 2);
    H = H * h22_inv;
}
}
} //namespace cv::_3d