Ceres-Solver学习日志：自动求导使用样例与针孔成像器的应用

1.定义残差类

         基于自动求导的核心是定义残差类。

         说到要定义一个类，感觉是要实现很复杂的功能，而实际上该类就实现一个功能，即实现残差模型。

         必须在残差类中重截operator()实现残差模型，很多时候，残差类也就这么一个成员，无需再添加其它成员，就能实现残差模型。

         如何实现残差类取决于残差模型，反映在代码实现上，就是如何实现operator()，operator()的形式如下：

         template <typename tp> bool operator()(const tp * const param1,…, const tp * const param9, tp *residual) const {…}

         (1)以上给出的是十个参数的operator()，可根据实际情况选择参数个数，最少两个，最多十个。

         (2)最后一个参数residual是必须的，它用于保存计算的残差值，其维度等于残差项的个数。

         (3)前九个参数可根据实际需求裁剪，但至少一个。当被优化的所有参数可以归为一组时，就只需要一个；当被优化的参数需要被分为多组时，则需要多个。

　　  定义多少个残差类取决于具体业务模型，但不外乎也就以下几种情况：

         (1)定义一个残差类，这个残差类实现所有残差模型，耦合性强，不推荐。

         (2)定义多个残差类，某些残差类实现多个残差模型，耦合性强，不推荐。

         (3)定义多个残差类，每个残差类实现一个残差模型，每个残差模型实现所有时刻采集的数据，扩展性差，不推荐。

         (4)定义多个残差类，每个残差类实现一个残差模型，每个残差模型实现单个时刻采集的数据，扩展性好，最实用。 

         这里之所以还区分每个时刻，是因为对于某些业务，尽管残差模型一样，但在不同时刻采集数据，部分模型参数值(真值)不一样，所以给定的初值和被优化后的最优值不一样，如VSLAM中基于不同位置的观察数据来计算每次观察时的位姿和相机内参。

2.使用残差类

         在决定了operator()的形式后，也就决定了微分类CostFunction和Problem::AddResidualBlock的使用形式。

         以AutoDiffCostFunction为例，其形式如下：

         AutoDiffCostFunction<Functor, int M, int N0=0, int N1=0, int N2=0, int N3=0, int N4=0, int N5=0, …, int N9=0>

         (1)第一个模板参数Functor就是定义好的残差类。

         (2)第二个模板参数表示残差项的个数，若运行时才能确定则用ceres::DYNMIC。

         (3)第三个及之后的模板参数，表示被优化的每组参数中包含的参数个数，即与operator()中参数对应，假设param2包含k个参数，则N2=k。

         Problem::AddResidualBlock的形式如下：

         Problem::AddResidualBlock(costfunction, lossfunction, param1,…param9)

         Problem::AddResidualBlock(costfunction, lossfunction, vectorparam)

         (1)前两个参数分别是微分模型和损失函数模型。

         (2)之后的九个参数具体要多少个，与operator()对应，也可以用std::vector合为一个参数后传进去。

3.使用样例

         提供两个使用样例，封装为两个类：

         OptimizeRt：基于单次观察，优化此次观察的外参，定义了ProjectionModelRt类。

         OptimizeKDRt：基于多次观察，优化相机内参和所有观察的外参，定义了ProjectionModelKDRt类。

         其中类MotionSim用于生成仿真数据，使用说明参见《CV学习日志：CV开发之三维仿真器》。

         存在可能不收敛的测试结果，属于正常现象，可修改初值精度来增加收敛性。

         以下是详细代码，依赖于C++14、OpenCV4.x、Ceres和Spdlog。

#include <opencv2/opencv.hpp>
#include <opencv2/viz.hpp>
#include <spdlog/spdlog.h>
#include <ceres/ceres.h>
using namespace std;
using namespace cv;

class MotionSim
{
public:
    static void TestMe(int argc, char** argv)
    {
        MotionSim motionSim(false);
        motionSim.camFovX = 45;
        motionSim.camFovY = 30;
        motionSim.camRand = 10;
        motionSim.enableVerbose = false;
        motionSim.runMotion(false, false, 7);
        motionSim.visMotion();
    }

public:
    struct MotionView
    {
        Mat_<double> r = Mat_<double>(3, 1);
        Mat_<double> t = Mat_<double>(3, 1);
        Mat_<double> q = Mat_<double>(4, 1);
        Mat_<double> rt = Mat_<double>(6, 1);
        Mat_<double> radian = Mat_<double>(3, 1);
        Mat_<double> degree = Mat_<double>(3, 1);
        Mat_<double> R = Mat_<double>(3, 3);
        Mat_<double> T = Mat_<double>(3, 4);
        Mat_<double> K;
        Mat_<double> D;
        Mat_<Vec3d> point3D;
        Mat_<Vec2d> point2D;
        Mat_<int> point3DIds;
        string print(string savePath = "")
        {
            string str;
            str += fmt::format("r: {}
", cvarr2str(r.t()));
            str += fmt::format("t: {}
", cvarr2str(t.t()));
            str += fmt::format("q: {}
", cvarr2str(q.t()));
            str += fmt::format("rt: {}
", cvarr2str(rt.t()));
            str += fmt::format("radian: {}
", cvarr2str(radian.t()));
            str += fmt::format("degree: {}
", cvarr2str(degree.t()));
            str += fmt::format("R: {}
", cvarr2str(R));
            str += fmt::format("T: {}
", cvarr2str(T));
            str += fmt::format("K: {}
", cvarr2str(K));
            str += fmt::format("D: {}
", cvarr2str(D.t()));
            if (savePath.empty() == false) { FILE* out = fopen(savePath.c_str(), "w"); fprintf(out, str.c_str()); fclose(out); }
            return str;
        }
    };
    static string cvarr2str(InputArray v)
    {
        Ptr<Formatted> fmtd = cv::format(v, Formatter::FMT_DEFAULT);
        string dst; fmtd->reset();
        for (const char* str = fmtd->next(); str; str = fmtd->next()) dst += string(str);
        return dst;
    }
    static void euler2matrix(double e[3], double R[9], bool forward = true, int argc = 0, char** argv = 0)
    {
        if (argc > 0)
        {
            int N = 999;
            for (int k = 0; k < N; ++k)//OpenCV not better than DIY
            {
                //1.GenerateData
                Matx31d radian0 = radian0.randu(-3.14159265358979323846, 3.14159265358979323846);
                Matx33d R; euler2matrix(radian0.val, R.val, true);
                const double deg2rad = 3.14159265358979323846 * 0.0055555555555555556;
                const double rad2deg = 180 * 0.3183098861837906715;

                //2.CalcByOpenCV
                Matx31d radian1 = cv::RQDecomp3x3(R, Matx33d(), Matx33d()) * deg2rad;

                //3.CalcByDIY
                Matx31d radian2; euler2matrix(R.val, radian2.val, false);

                //4.AnalyzeError
                double infRadian0Radian1 = norm(radian0, radian1, NORM_INF);
                double infRadian1Radian2 = norm(radian1, radian2, NORM_INF);

                //5.PrintError
                cout << endl << "LoopCount: " << k << endl;
                if (infRadian0Radian1 > 0 || infRadian1Radian2 > 0)
                {
                    cout << endl << "5.1PrintError" << endl;
                    cout << endl << "infRadian0Radian1: " << infRadian0Radian1 << endl;
                    cout << endl << "infRadian1Radian2: " << infRadian1Radian2 << endl;
                    if (0)
                    {
                        cout << endl << "5.2PrintDiff" << endl;
                        cout << endl << "radian0-degree0:" << endl << radian0.t() << endl << radian0.t() * rad2deg << endl;
                        cout << endl << "radian1-degree1:" << endl << radian1.t() << endl << radian1.t() * rad2deg << endl;
                        cout << endl << "radian2-degree2:" << endl << radian2.t() << endl << radian2.t() * rad2deg << endl;
                        cout << endl << "5.3PrintOthers" << endl;
                        cout << endl << "R:" << endl << R << endl;
                    }
                    cout << endl << "Press any key to continue" << endl; std::getchar();
                }
            }
            return;
        }
        if (forward)//check with 3D Rotation Converter
        {
            double sinR = std::sin(e[0]);
            double sinP = std::sin(e[1]);
            double sinY = std::sin(e[2]);
            double cosR = std::cos(e[0]);
            double cosP = std::cos(e[1]);
            double cosY = std::cos(e[2]);

            //RPY indicates: first Yaw aroundZ, second Pitch aroundY, third Roll aroundX
            R[0] = cosY * cosP; R[1] = cosY * sinP * sinR - sinY * cosR; R[2] = cosY * sinP * cosR + sinY * sinR;
            R[3] = sinY * cosP; R[4] = sinY * sinP * sinR + cosY * cosR; R[5] = sinY * sinP * cosR - cosY * sinR;
            R[6] = -sinP;       R[7] = cosP * sinR;                      R[8] = cosP * cosR;
        }
        else
        {
            double vs1 = std::abs(R[6] - 1.);
            double vs_1 = std::abs(R[6] + 1.);
            if (vs1 > 1E-9 && vs_1 > 1E-9)
            {
                e[2] = std::atan2(R[3], R[0]); //Yaw aroundZ
                e[1] = std::asin(-R[6]);//Pitch aroundY
                e[0] = std::atan2(R[7], R[8]); //Roll aroundX
            }
            else if (vs_1 <= 1E-9)
            {
                e[2] = 0; //Yaw aroundZ
                e[1] = 3.14159265358979323846 * 0.5;//Pitch aroundY
                e[0] = e[2] + atan2(R[1], R[2]); //Roll aroundX
            }
            else
            {
                e[2] = 0; //Yaw aroundZ
                e[1] = -3.14159265358979323846 * 0.5;//Pitch aroundY
                e[0] = -e[2] + atan2(-R[1], -R[2]); //Roll aroundX
            }
        }
    };
    static void quat2matrix(double q[4], double R[9], bool forward = true)
    {
        if (forward)//refer to qglviwer
        {
            double L1 = std::sqrt(q[0] * q[0] + q[1] * q[1] + q[2] * q[2] + q[3] * q[3]);
            if (std::abs(L1 - 1) > 1E-9) { std::printf("Not uint quaternion: NormQ=%.9f
", L1); abort(); }

            double xx = 2.0 * q[1] * q[1];
            double yy = 2.0 * q[2] * q[2];
            double zz = 2.0 * q[3] * q[3];

            double xy = 2.0 * q[1] * q[2];
            double xz = 2.0 * q[1] * q[3];
            double wx = 2.0 * q[1] * q[0];

            double yz = 2.0 * q[2] * q[3];
            double wy = 2.0 * q[2] * q[0];

            double wz = 2.0 * q[3] * q[0];

            R[0] = 1.0 - yy - zz;
            R[4] = 1.0 - xx - zz;
            R[8] = 1.0 - xx - yy;

            R[1] = xy - wz;
            R[3] = xy + wz;

            R[2] = xz + wy;
            R[6] = xz - wy;

            R[5] = yz - wx;
            R[7] = yz + wx;
        }
        else
        {
            double onePlusTrace = 1.0 + R[0] + R[4] + R[8];// Compute one plus the trace of the matrix
            if (onePlusTrace > 1E-9)
            {
                double s = sqrt(onePlusTrace) * 2.0;
                double is = 1 / s;
                q[0] = 0.25 * s;
                q[1] = (R[7] - R[5]) * is;
                q[2] = (R[2] - R[6]) * is;
                q[3] = (R[3] - R[1]) * is;
            }
            else
            {
                std::printf("1+trace(R)=%.9f is too small and (R11,R22,R33)=(%.9f,%.9f,%.9f)
", onePlusTrace, R[0], R[4], R[8]);
                if ((R[0] > R[4]) && (R[0] > R[8]))//max(R00, R11, R22)=R00
                {
                    double s = sqrt(1.0 + R[0] - R[4] - R[8]) * 2.0;
                    double is = 1 / s;
                    q[0] = (R[5] - R[7]) * is;
                    q[1] = 0.25 * s;
                    q[2] = (R[1] + R[3]) * is;
                    q[3] = (R[2] + R[6]) * is;
                }
                else if (R[4] > R[8])//max(R00, R11, R22)=R11
                {
                    double s = sqrt(1.0 - R[0] + R[4] - R[8]) * 2.0;
                    double is = 1 / s;
                    q[0] = (R[2] - R[6]) * is;
                    q[1] = (R[1] + R[3]) * is;
                    q[2] = 0.25 * s;
                    q[3] = (R[5] + R[7]) * is;
                }
                else//max(R00, R11, R22)=R22
                {
                    double s = sqrt(1.0 - R[0] - R[4] + R[8]) * 2.0;
                    double is = 1 / s;
                    q[0] = (R[1] - R[3]) * is;
                    q[1] = (R[2] + R[6]) * is;
                    q[2] = (R[5] + R[7]) * is;
                    q[3] = 0.25 * s;
                }
            }
            double L1 = std::sqrt(q[0] * q[0] + q[1] * q[1] + q[2] * q[2] + q[3] * q[3]);
            if (L1 < 1e-9) { std::printf("Wrong rotation matrix: NormQ=%.9f
", L1); abort(); }
            else { L1 = 1 / L1; q[0] *= L1; q[1] *= L1; q[2] *= L1; q[3] *= L1; }
        }
    }
    static void vec2quat(double r[3], double q[4], bool forward = true)
    {
        if (forward)//refer to qglviwer
        {
            double theta = std::sqrt(r[0] * r[0] + r[1] * r[1] + r[2] * r[2]);
            if (std::abs(theta) < 1E-9)
            {
                q[0] = 1; q[1] = q[2] = q[3] = 0;
                std::printf("Rotation approximates zero: Theta=%.9f
", theta);
            };

            q[0] = std::cos(theta * 0.5);
            double ss = std::sin(theta * 0.5) / theta;
            q[1] = r[0] * ss;
            q[2] = r[1] * ss;
            q[3] = r[2] * ss;
        }
        else
        {
            double L1 = std::sqrt(q[0] * q[0] + q[1] * q[1] + q[2] * q[2] + q[3] * q[3]);
            if (std::abs(L1 - 1) > 1E-9) { std::printf("Not uint quaternion: NormQ=%.9f
", L1); abort(); }

            double theta = 2 * acos(q[0]);
            if (theta > 3.14159265358979323846) theta = 2 * 3.14159265358979323846 - theta;
            double thetaEx = theta / std::sin(theta * 0.5);
            r[0] = q[1] * thetaEx;
            r[1] = q[2] * thetaEx;
            r[2] = q[3] * thetaEx;
        }
    }
    static void vec2matrix(double r[3], double R[9], bool forward = true, int argc = 0, char** argv = 0)
    {
        if (argc > 0)
        {
            int N = 999;
            for (int k = 0; k < N; ++k) //refer to the subsequent article for more details
            {
                //1.GenerateData
                Matx31d r0 = r0.randu(-999, 999);
                Matx33d R0; cv::Rodrigues(r0, R0);

                //2.CalcByOpenCV
                Matx33d R1;
                Matx31d r1;
                cv::Rodrigues(r0, R1);
                cv::Rodrigues(R0, r1);

                //3.CalcByDIY
                Matx33d R2;
                Matx31d r2;
                vec2matrix(r0.val, R2.val, true);
                vec2matrix(r2.val, R0.val, false);

                //4.AnalyzeError
                double infR1R2 = norm(R1, R2, NORM_INF);
                double infr1r2 = norm(r1, r2, NORM_INF);

                //5.PrintError
                cout << endl << "LoopCount: " << k << endl;
                if (infR1R2 > 1E-12 || infr1r2 > 1E-12)
                {
                    cout << endl << "5.1PrintError" << endl;
                    cout << endl << "infR1R2: " << infR1R2 << endl;
                    cout << endl << "infr1r2: " << infr1r2 << endl;
                    if (0)
                    {
                        cout << endl << "5.2PrintDiff" << endl;
                        cout << endl << "R1: " << endl << R1 << endl;
                        cout << endl << "R2: " << endl << R2 << endl;
                        cout << endl;
                        cout << endl << "r1: " << endl << r1.t() << endl;
                        cout << endl << "r2: " << endl << r2.t() << endl;
                        cout << endl << "5.3PrintOthers" << endl;
                    }
                    cout << endl << "Press any key to continue" << endl; std::getchar();
                }
            }
            return;
        }

        if (forward)
        {
            double theta = std::sqrt(r[0] * r[0] + r[1] * r[1] + r[2] * r[2]);
            if (theta < 1E-9)
            {
                R[0] = R[4] = R[8] = 1.0;
                R[1] = R[2] = R[3] = R[5] = R[6] = R[7] = 0.0;
                std::printf("Rotation approximates zero: Theta=%.9f
", theta);
                return;
            }
            double cs = cos(theta);
            double sn = sin(theta);
            double itheta = 1. / theta;
            double cs1 = 1 - cs;
            double nx = r[0] * itheta;
            double ny = r[1] * itheta;
            double nz = r[2] * itheta;

            double nxnx = nx * nx, nyny = ny * ny, nznz = nz * nz;
            double nxny = nx * ny, nxnz = nx * nz, nynz = ny * nz;
            double nxsn = nx * sn, nysn = ny * sn, nzsn = nz * sn;

            R[0] = nxnx * cs1 + cs;
            R[3] = nxny * cs1 + nzsn;
            R[6] = nxnz * cs1 - nysn;

            R[1] = nxny * cs1 - nzsn;
            R[4] = nyny * cs1 + cs;
            R[7] = nynz * cs1 + nxsn;

            R[2] = nxnz * cs1 + nysn;
            R[5] = nynz * cs1 - nxsn;
            R[8] = nznz * cs1 + cs;

            if (0)
            {
                Mat_<double> dRdu({ 9, 4 }, {
                    2 * nx * cs1, 0, 0, (nxnx - 1) * sn,
                    ny * cs1, nx * cs1, -sn, nxny * sn - nz * cs,
                    nz * cs1, sn, nx * cs1, nxnz * sn + ny * cs,
                    ny * cs1, nx * cs1, sn, nxny * sn + nz * cs,
                    0, 2 * ny * cs1, 0, (nyny - 1) * sn,
                    -sn, nz * cs1, ny * cs1, nynz * sn - nx * cs,
                    nz * cs1, -sn, nx * cs1, nxnz * sn - ny * cs,
                    sn, nz * cs1, ny * cs1, nynz * sn + nx * cs,
                    0, 0, 2 * nz * cs1, (nznz - 1) * sn });

                Mat_<double> dudv({ 4, 4 }, {
                    itheta, 0, 0, -nx * itheta,
                    0, itheta, 0, -ny * itheta,
                    0, 0, itheta, -nz * itheta,
                    0, 0, 0, 1 });

                Mat_<double> dvdr({ 4, 3 }, {
                    1, 0, 0,
                    0, 1, 0,
                    0, 0, 1,
                    nx, ny, nz });

                Mat_<double> Jacobian = dRdu * dudv * dvdr;//rows=9 cols=3
            }
        }
        else
        {
            double sx = R[7] - R[5];
            double sy = R[2] - R[6];
            double sz = R[3] - R[1];
            double sn = sqrt(sx * sx + sy * sy + sz * sz) * 0.5;
            double cs = (R[0] + R[4] + R[8] - 1) * 0.5;
            double theta = acos(cs);
            double ss = 2 * sn;
            double iss = 1. / ss;
            double tss = theta * iss;
            r[0] = tss * sx;
            r[1] = tss * sy;
            r[2] = tss * sz;

            if (0)
            {
                Mat_<double> drdu({ 3, 4 }, {
                    tss, 0, 0, (sn - theta * cs) * iss * iss * sx * 2,
                    0, tss, 0, (sn - theta * cs) * iss * iss * sy * 2,
                    0, 0, tss, (sn - theta * cs) * iss * iss * sz * 2 });

                Mat_<double> dudR({ 4, 9 }, {
                    0, 0, 0, 0, 0, -1, 0, 1, 0,
                    0, 0, 1, 0, 0, 0, -1, 0, 0,
                    0, -1, 0, 1, 0, 0, 0, 0, 0,
                    -iss, 0, 0, 0, -iss, 0, 0, 0, -iss });

                Mat_<double> Jacobian = drdu * dudR;//rows=3 cols=9
            }
        }
    }

private:
    const int nHorPoint3D = 100;
    const int nVerPoint3D = 100;
    const double varPoint3DXY = 10.;
    const double minPoint3DZ = 1.;
    const double maxPoint3DZ = 99.;
    const double minCamZ = 101.;
    const double maxCamZ = 150.;
    const double varCamDegree = 10.;
    Mat_<Vec3d> allPoint3D = Mat_<Vec3d>(nVerPoint3D * nHorPoint3D, 1);
    Mat_<double> allPoint3DZ = Mat_<double>(nVerPoint3D * nHorPoint3D, 1);
    Mat_<double> K;
    Mat_<double> D;
    const double deg2rad = 3.14159265358979323846 * 0.0055555555555555556;
    const double rad2deg = 180 * 0.3183098861837906715;

public:
    int camRows = 480;
    int camCols = 640;
    int camFovY = 90;
    int camFovX = 90;
    int camRand = 10;//append random[0,camRand] to camera intrinsics
    int nCamDist = 5;//refer to opencv for value domain
    int nMinMotion = 32; // no less than X motion views
    int nMaxMotion = INT_MAX; // no more than X motion views
    int nPoint2DThenExit = 32;//exit when less than X pixies
    int rotMode = 1 + 2 + 4;//0=noRot 1=xAxis 2=yAxis 4=zAxis
    bool noTrans = false;//translate or not while motion
    bool world2D = false;//planar world or not
    bool rndSeek = true;//use random seek or not
    bool enableVerbose = false;//check motions one by one or not
    vector<MotionView> motionViews;//World Information: RightX, FrontY, DownZ
    MotionSim(bool run = true, bool world2D0 = false, bool noTrans0 = false, int rotMode0 = 7) { if (run) runMotion(world2D0, noTrans0, rotMode0); }

public:
    void runMotion(bool world2D0 = false, bool noTrans0 = false, int rotMode0 = 7)
    {
        world2D = world2D0;
        noTrans = noTrans0;
        rotMode = rotMode0;
        motionViews.clear();
        if (rndSeek) cv::setRNGSeed(clock());
        while (motionViews.size() < nMinMotion)
        {
            //1.GetAllPoint3D
            if (world2D) allPoint3DZ = 0.;
            else cv::randu(allPoint3DZ, -maxPoint3DZ, -minPoint3DZ);//DownZ
            for (int i = 0, k = 0; i < nVerPoint3D; ++i)
                for (int j = 0; j < nHorPoint3D; ++j, ++k)
                    allPoint3D(k) = Vec3d((j + cv::randu<double>()) * varPoint3DXY, (i + cv::randu<double>()) * varPoint3DXY, allPoint3DZ(i, j));

            //2.GetCamParams
            double camFx = camCols / 2. / std::tan(camFovX / 2. * deg2rad) + cv::randu<double>() * camRand;
            double camFy = camRows / 2. / std::tan(camFovY / 2. * deg2rad) + cv::randu<double>() * camRand;
            double camCx = camCols / 2. + cv::randu<double>() * camRand;
            double camCy = camRows / 2. + cv::randu<double>() * camRand;
            K.create(3, 3); K << camFx, 0, camCx, 0, camFy, camCy, 0, 0, 1;
            D.create(nCamDist, 1); cv::randu(D, -1.0, 1.0);

            //3.GetAllMotionView
            motionViews.clear();
            for (int64 k = 0; ; ++k)
            {
                //3.1 JoinCamParams
                MotionView view;
                view.K = K.clone();
                view.D = D.clone();

                //3.2 GetCamTrans
                if (k == 0) view.t(0) = view.t(1) = 0;
                else
                {
                    view.t(0) = motionViews[k - 1].t(0) + cv::randu<double>() * varPoint3DXY;
                    view.t(1) = motionViews[k - 1].t(1) + cv::randu<double>() * varPoint3DXY;
                }
                view.t(2) = minCamZ + cv::randu<double>() * (maxCamZ - minCamZ);
                view.t(2) = -view.t(2);//DownZ
                if (noTrans && k != 0) { view.t(0) = motionViews[0].t(0); view.t(1) = motionViews[0].t(1); view.t(2) = motionViews[0].t(2); }

                //3.3 GetCamRot: degree-->radian-->matrix-->vector&quaternion
                view.degree = 0.;
                if (rotMode & 1) view.degree(0) = cv::randu<double>() * varCamDegree;
                if (rotMode & 2) view.degree(1) = cv::randu<double>() * varCamDegree;
                if (rotMode & 4) view.degree(2) = cv::randu<double>() * varCamDegree;
                view.radian = view.degree * deg2rad;
                euler2matrix(view.radian.ptr<double>(), view.R.ptr<double>());
                cv::Rodrigues(view.R, view.r);
                quat2matrix(view.q.ptr<double>(), view.R.ptr<double>(), false);
                cv::hconcat(view.R, view.t, view.T);
                cv::vconcat(view.r, view.t, view.rt);

                //3.4 GetPoint3DAndPoint2D
                Mat_<Vec2d> allPoint2D;
                cv::projectPoints(allPoint3D, -view.r, -view.R.t() * view.t, view.K, view.D, allPoint2D);
                for (int k = 0; k < allPoint2D.total(); ++k)
                    if (allPoint2D(k)[0] > 0 && allPoint2D(k)[0] < camCols && allPoint2D(k)[1] > 0 && allPoint2D(k)[1] < camRows)
                    {
                        view.point2D.push_back(allPoint2D(k));
                        view.point3D.push_back(allPoint3D(k));
                        view.point3DIds.push_back(k);
                    }

                //3.5 PrintDetails
                motionViews.push_back(view);
                if (enableVerbose)
                {
                    cout << endl << view.print();
                    cout << fmt::format("view={}   features={}
", k, view.point2D.rows);
                    double minV = 0, maxV = 0;//Distortion makes some minV next to maxV
                    int minId = 0, maxId = 0;
                    cv::minMaxIdx(allPoint2D.reshape(1, int(allPoint2D.total()) * allPoint2D.channels()), &minV, &maxV, &minId, &maxId);
                    cout << fmt::format("minInfo:({}, {})", minId, minV) << allPoint3D(minId / 2) << allPoint2D(minId / 2) << endl;
                    cout << fmt::format("maxInfo:({}, {})", maxId, maxV) << allPoint3D(maxId / 2) << allPoint2D(maxId / 2) << endl;
                    cout << "Press any key to continue" << endl; std::getchar();
                }
                if (view.point2D.rows < nPoint2DThenExit || motionViews.size() > nMaxMotion) break;
            }
        }
    }
    void visMotion()
    {
        //1.CreateWidgets
        Size2d validSize(nHorPoint3D * varPoint3DXY, nVerPoint3D * varPoint3DXY);
        Mat_<cv::Affine3d> camPoses(int(motionViews.size()), 1); for (int k = 0; k < camPoses.rows; ++k) camPoses(k) = cv::Affine3d(motionViews[k].T);
        viz::WText worldInfo(fmt::format("nMotionView: {}
K: {}
D: {}", motionViews.size(), cvarr2str(K), cvarr2str(D)), Point(10, 240), 10);
        viz::WCoordinateSystem worldCSys(1000);
        viz::WPlane worldGround(Point3d(validSize.width / 2, validSize.height / 2, 0), Vec3d(0, 0, 1), Vec3d(0, 1, 0), validSize);
        viz::WCloud worldPoints(allPoint3D, Mat_<Vec3b>(allPoint3D.size(), Vec3b(0, 255, 0)));
        viz::WTrajectory camTraj1(camPoses, viz::WTrajectory::FRAMES, 8);
        viz::WTrajectorySpheres camTraj2(camPoses, 100, 2);
        viz::WTrajectoryFrustums camTraj3(camPoses, Matx33d(K), 4., viz::Color::yellow());
        worldCSys.setRenderingProperty(viz::OPACITY, 0.1);
        worldGround.setRenderingProperty(viz::OPACITY, 0.1);
        camTraj2.setRenderingProperty(viz::OPACITY, 0.6);

        //2.ShowWidgets
        static viz::Viz3d viz3d(__FUNCTION__);
        viz3d.showWidget("worldInfo", worldInfo);
        viz3d.showWidget("worldCSys", worldCSys);
        viz3d.showWidget("worldGround", worldGround);
        viz3d.showWidget("worldPoints", worldPoints);
        viz3d.showWidget("camTraj1", camTraj1);
        viz3d.showWidget("camTraj2", camTraj2);
        viz3d.showWidget("camTraj3", camTraj3);

        //3.UpdateWidghts
        static const vector<MotionView>& views = motionViews;
        viz3d.registerKeyboardCallback([](const viz::KeyboardEvent& keyboarEvent, void* pVizBorad)->void
            {
                if (keyboarEvent.action != viz::KeyboardEvent::KEY_DOWN) return;
                static int pos = 0;
                if (keyboarEvent.code == ' ')
                {
                    size_t num = views.size();
                    size_t ind = pos % num;
                    double xmin3D = DBL_MAX, ymin3D = DBL_MAX, xmin2D = DBL_MAX, ymin2D = DBL_MAX;
                    double xmax3D = -DBL_MAX, ymax3D = -DBL_MAX, xmax2D = -DBL_MAX, ymax2D = -DBL_MAX;
                    for (size_t k = 0; k < views[ind].point3D.rows; ++k)
                    {
                        Vec3d pt3 = views[ind].point3D(int(k));
                        Vec2d pt2 = views[ind].point2D(int(k));
                        if (pt3[0] < xmin3D) xmin3D = pt3[0];
                        if (pt3[0] > xmax3D) xmax3D = pt3[0];
                        if (pt3[1] < ymin3D) ymin3D = pt3[1];
                        if (pt3[1] > ymax3D) ymax3D = pt3[1];
                        if (pt2[0] < xmin2D) xmin2D = pt2[0];
                        if (pt2[0] > xmax2D) xmax2D = pt2[0];
                        if (pt2[1] < ymin2D) ymin2D = pt2[1];
                        if (pt2[1] > ymax2D) ymax2D = pt2[1];
                    }
                    if (pos != 0)
                    {
                        for (int k = 0; k < views[ind == 0 ? num - 1 : ind - 1].point3D.rows; ++k) viz3d.removeWidget("active" + std::to_string(k));
                        viz3d.removeWidget("viewInfo");
                        viz3d.removeWidget("camSolid");
                    }
                    for (int k = 0; k < views[ind].point3D.rows; ++k) viz3d.showWidget("active" + std::to_string(k), viz::WSphere(views[ind].point3D(k), 5, 10));
                    viz3d.showWidget("viewInfo", viz::WText(fmt::format("CurrentMotion: {}
ValidPoints: {}
Min3DXY_Min2DXY: {}, {}, {}, {}
Max3DXY_Max2DXY: {}, {}, {}, {}
Rot_Trans_Euler: {}
",
                        ind, views[ind].point3D.rows, xmin3D, ymin3D, xmin2D, ymin2D, xmax3D, ymax3D, xmax2D, ymax2D,
                        cvarr2str(views[ind].r.t()) + cvarr2str(views[ind].t.t()) + cvarr2str(views[ind].degree.t())), Point(10, 10), 10));
                    viz3d.showWidget("camSolid", viz::WCameraPosition(Matx33d(views[ind].K), 10, viz::Color::yellow()), cv::Affine3d(views[ind].T));
                    ++pos;
                }
            }, 0);
        viz3d.spin();
    }
};

class OptimizeRt
{
public:
    using MotionView = MotionSim::MotionView;
    static void TestMe(int argc = 0, char** argv = 0)
    {
        int N = 99;
        for (int k = 0; k < N; ++k)
        {
            //1.GenerateData
            bool world2D = k % 2;
            int rotMode = k % 7 + 1;
            MotionSim motionSim(false);
            motionSim.camFovX = 90;
            motionSim.camFovY = 90;
            motionSim.camRand = 10;
            motionSim.nMinMotion = 16;//2
            motionSim.nMaxMotion = 32;//4
            motionSim.rndSeek = false;
            motionSim.nCamDist = 5;
            motionSim.runMotion(world2D, false, rotMode);
            //motionSim.visMotion();
            int rndInd = int(motionSim.motionViews.size() * cv::randu<double>());
            Mat_<double> r0 = -motionSim.motionViews[rndInd].r;
            Mat_<double> t0 = -motionSim.motionViews[rndInd].R.t() * motionSim.motionViews[rndInd].t;
            const MotionView& motionView = motionSim.motionViews[rndInd];
            double errRatio = 0.9;

            //2.CalcByCeres
            Mat_<double> r1 = r0 * errRatio;
            Mat_<double> t1 = t0 * errRatio;
            ceres::Problem problem;
            problem.AddResidualBlock(new ceres::AutoDiffCostFunction<ProjectionModelRt, ceres::DYNAMIC, 3, 3>(
                new ProjectionModelRt(motionView, motionSim.nCamDist), motionView.point3D.rows * 2), NULL, r1.ptr<double>(), t1.ptr<double>());
            ceres::Solver::Options options;
            ceres::Solver::Summary summary;
            ceres::Solve(options, &problem, &summary);
            int nIter1 = (int)summary.iterations.size();

            //3.AnalyzeError
            double infr0r0 = norm(r0, r0 * errRatio, NORM_INF);
            double infr0r1 = norm(r0, r1, NORM_INF);
            double inft0t0 = norm(t0, t0 * errRatio, NORM_INF);
            double inft0t1 = norm(t0, t1, NORM_INF);

            //4.PrintError
            cout << fmt::format("LoopCount: {}      CeresSolver.iters: {}
", k, nIter1);
            if (infr0r1 > 1e-8 || inft0t1 > 1e-8)
            {
                cout << fmt::format("infr0r1: {:<15.9}		{:<15.9}
", infr0r1, infr0r0);
                cout << fmt::format("inft0t1: {:<15.9}		{:<15.9}
", inft0t1, inft0t0);
                cout << "Press any key to continue" << endl; std::getchar();
            }
        }
    }

public:
    struct ProjectionModelRt
    {
        const int nDist;
        double K[4];
        const double* D;
        Mat_<Vec2d> point2D;
        Mat_<Vec3d> point3D;
        ProjectionModelRt(const MotionView& motionView0, const int nDist0) : nDist(nDist0)
        {
            K[0] = motionView0.K(0, 0);
            K[1] = motionView0.K(1, 1);
            K[2] = motionView0.K(0, 2);
            K[3] = motionView0.K(1, 2);
            D = motionView0.D.ptr<double>();
            point2D = motionView0.point2D;
            point3D = motionView0.point3D;
        }
        template <typename tp> bool operator()(const tp* const rot, const tp* const t, tp* errPoint2D) const
        {
            //1.Projection params
            double fx = K[0];
            double fy = K[1];
            double cx = K[2];
            double cy = K[3];

            //2.Distortion params
            double k1 = D[0];
            double k2 = D[1];
            double p1 = D[2];
            double p2 = D[3];
            double k3, k4, k5, k6;
            double s1, s2, s3, s4;
            if (nDist > 4) k3 = D[4];
            if (nDist > 5) { k4 = D[5]; k5 = D[6]; k6 = D[7]; }
            if (nDist > 8) { s1 = D[8]; s2 = D[9]; s3 = D[10]; s4 = D[11]; }

            //3.Translation params
            tp tx = t[0];
            tp ty = t[1];
            tp tz = t[2];

            //4.Rotation params
            tp R11, R12, R13, R21, R22, R23, R31, R32, R33;
            {
                tp theta = sqrt(rot[0] * rot[0] + rot[1] * rot[1] + rot[2] * rot[2]);
                tp cs = cos(theta);
                tp sn = sin(theta);
                tp itheta = 1. / theta;//if denominator==0
                tp cs1 = 1. - cs;
                tp nx = rot[0] * itheta;
                tp ny = rot[1] * itheta;
                tp nz = rot[2] * itheta;

                tp nxnx = nx * nx, nyny = ny * ny, nznz = nz * nz;
                tp nxny = nx * ny, nxnz = nx * nz, nynz = ny * nz;
                tp nxsn = nx * sn, nysn = ny * sn, nzsn = nz * sn;

                R11 = nxnx * cs1 + cs;
                R21 = nxny * cs1 + nzsn;
                R31 = nxnz * cs1 - nysn;

                R12 = nxny * cs1 - nzsn;
                R22 = nyny * cs1 + cs;
                R32 = nynz * cs1 + nxsn;

                R13 = nxnz * cs1 + nysn;
                R23 = nynz * cs1 - nxsn;
                R33 = nznz * cs1 + cs;
            }

            //5.ReProjection
            const Vec2d* data2D = point2D.ptr<Vec2d>();
            const Vec3d* data3D = point3D.ptr<Vec3d>();
            Vec<tp, 2>* err2D = (Vec<tp, 2>*)errPoint2D;
            for (int k = 0; k < point3D.rows; ++k)
            {
                //5.1 WorldCoordinate
                double X = data3D[k][0];
                double Y = data3D[k][1];
                double Z = data3D[k][2];

                //5.2 CameraCoordinate
                tp x = R11 * X + R12 * Y + R13 * Z + tx;
                tp y = R21 * X + R22 * Y + R23 * Z + ty;
                tp z = R31 * X + R32 * Y + R33 * Z + tz;

                //5.3 StandardPhysicsCoordinate
                tp iz = 1. / z; //if denominator==0
                tp xc = x * iz;
                tp yc = y * iz;

                //5.4 DistortionPhysicsCoordinate
                tp xc2 = xc * xc;
                tp yc2 = yc * yc;
                tp d2 = xc2 + yc2;
                tp xcyc = 2. * xc * yc;
                tp d4 = d2 * d2;
                tp d6 = d2 * d4;
                tp d2xc2 = d2 + 2. * xc2;
                tp d2yc2 = d2 + 2. * yc2;
                tp nu, de, xd, yd;
                if (nDist < 5)
                {
                    nu = 1. + k1 * d2 + k2 * d4;
                    xd = xc * nu + p1 * xcyc + p2 * d2xc2;
                    yd = yc * nu + p2 * xcyc + p1 * d2yc2;
                }
                else if (nDist < 8)
                {
                    nu = 1. + k1 * d2 + k2 * d4 + k3 * d6;
                    xd = xc * nu + p1 * xcyc + p2 * d2xc2;
                    yd = yc * nu + p2 * xcyc + p1 * d2yc2;
                }
                else if (nDist < 12)
                {
                    nu = 1. + k1 * d2 + k2 * d4 + k3 * d6;
                    de = 1. / (1. + k4 * d2 + k5 * d4 + k6 * d6);//if denominator==0
                    xd = xc * nu * de + p1 * xcyc + p2 * d2xc2;
                    yd = yc * nu * de + p2 * xcyc + p1 * d2yc2;
                }
                else if (nDist < 14)
                {
                    nu = 1. + k1 * d2 + k2 * d4 + k3 * d6;
                    de = 1. / (1. + k4 * d2 + k5 * d4 + k6 * d6);//if denominator==0
                    xd = xc * nu * de + p1 * xcyc + p2 * d2xc2 + s1 * d2 + s2 * d4;
                    yd = yc * nu * de + p2 * xcyc + p1 * d2yc2 + s3 * d2 + s4 * d4;
                }
                err2D[k][0] = xd * fx + cx - data2D[k][0];
                err2D[k][1] = yd * fy + cy - data2D[k][1];
            }
            return true;
        }
    };
};

class OptimizeKDRt
{
public:
    using MotionView = MotionSim::MotionView;
    static void TestMe(int argc = 0, char** argv = 0)
    {
        int N = 99;
        for (int k = 0; k < N; ++k)
        {
            //1.GenerateData
            bool world2D = k % 2;
            int rotMode = k % 7 + 1;
            MotionSim motionSim(false);
            motionSim.camFovX = 90;
            motionSim.camFovY = 90;
            motionSim.camRand = 10;
            motionSim.nMinMotion = 16;//2
            motionSim.nMaxMotion = 32;//4
            motionSim.rndSeek = false;
            static const int nDist = 5;
            motionSim.nCamDist = nDist;
            motionSim.runMotion(world2D, false, rotMode);
            //motionSim.visMotion();
            Mat_<double> rs0; for (size_t k = 0; k < motionSim.motionViews.size(); ++k) rs0.push_back(-motionSim.motionViews[k].r);
            Mat_<double> ts0; for (size_t k = 0; k < motionSim.motionViews.size(); ++k) ts0.push_back(-motionSim.motionViews[k].R.t() * motionSim.motionViews[k].t);
            Mat_<double> K0({ 4, 1 }, { motionSim.motionViews[0].K(0, 0), motionSim.motionViews[0].K(1, 1), motionSim.motionViews[0].K(0, 2), motionSim.motionViews[0].K(1, 2) });
            Mat_<double> D0 = motionSim.motionViews[0].D.clone();
            double errRatio = 0.9;
            double errRatioTrans = 0.99;

            //2.CalcByCeres
            Mat_<double> rs1 = rs0 * errRatio;
            Mat_<double> ts1 = ts0 * errRatioTrans;
            Mat_<double> K1 = K0 * errRatio;
            Mat_<double> D1 = D0 * errRatio;
            ceres::Problem problem;
            for (int k = 0; k < motionSim.motionViews.size(); ++k)
                problem.AddResidualBlock(new ceres::AutoDiffCostFunction<ProjectionModelKDRt, ceres::DYNAMIC, 3, 3, 4, nDist>(
                    new ProjectionModelKDRt(motionSim.motionViews[k], motionSim.nCamDist), motionSim.motionViews[k].point3D.rows * 2), NULL,
                    rs1.ptr<double>(k * 3), ts1.ptr<double>(k * 3),
                    K1.ptr<double>(), D1.ptr<double>());
            ceres::Solver::Options options;
            ceres::Solver::Summary summary;
            ceres::Solve(options, &problem, &summary);
            int nIter1 = (int)summary.iterations.size();

            //3.AnalyzeError
            double infrs0rs0 = norm(rs0, rs0 * errRatio, NORM_INF);
            double infrs0rs1 = norm(rs0, rs1, NORM_INF);
            double infts0ts0 = norm(ts0, ts0 * errRatioTrans, NORM_INF);
            double infts0ts1 = norm(ts0, ts1, NORM_INF);
            double infK0K0 = norm(K0, K0 * errRatio, NORM_INF);
            double infK0K1 = norm(K0, K1, NORM_INF);
            double infD0D0 = norm(D0, D0 * errRatio, NORM_INF);
            double infD0D1 = norm(D0, D1, NORM_INF);

            //4.PrintError
            cout << fmt::format("LoopCount: {}      CeresSolver.iters: {}
", k, nIter1);
            if (infrs0rs1 > 1e-8 || infts0ts1 > 1e-8 || infK0K1 > 1e-8 || infD0D1 > 1e-8)
            {
                cout << fmt::format("infrs0rs1: {:<15.9}		{:<15.9}
", infrs0rs1, infrs0rs0);
                cout << fmt::format("infts0ts1: {:<15.9}		{:<15.9}
", infts0ts1, infts0ts0);
                cout << fmt::format("infK0K1  : {:<15.9}		{:<15.9}
", infK0K1, infK0K0);
                cout << fmt::format("infD0D1  : {:<15.9}		{:<15.9}
", infD0D1, infD0D0);
                cout << "Press any key to continue" << endl; std::getchar();
            }
        }
    }

public:
    struct ProjectionModelKDRt
    {
        const int nDist;
        Mat_<Vec2d> point2D;
        Mat_<Vec3d> point3D;
        ProjectionModelKDRt(const MotionView& motionView0, const int nDist0) : nDist(nDist0)
        {
            point2D = motionView0.point2D;
            point3D = motionView0.point3D;
        }
        template <typename tp> bool operator()(const tp* const rot, const tp* const t, const tp* const K, const tp* const D, tp* errPoint2D) const
        {
            //1.Projection params
            tp fx = K[0];
            tp fy = K[1];
            tp cx = K[2];
            tp cy = K[3];

            //2.Distortion params
            tp k1 = D[0];
            tp k2 = D[1];
            tp p1 = D[2];
            tp p2 = D[3];
            tp k3, k4, k5, k6;
            tp s1, s2, s3, s4;
            if (nDist > 4) k3 = D[4];
            if (nDist > 5) { k4 = D[5]; k5 = D[6]; k6 = D[7]; }
            if (nDist > 8) { s1 = D[8]; s2 = D[9]; s3 = D[10]; s4 = D[11]; }

            //3.Translation params
            tp tx = t[0];
            tp ty = t[1];
            tp tz = t[2];

            //4.
            tp R11, R12, R13, R21, R22, R23, R31, R32, R33;
            {
                tp theta = sqrt(rot[0] * rot[0] + rot[1] * rot[1] + rot[2] * rot[2]);
                tp cs = cos(theta);
                tp sn = sin(theta);
                tp itheta = 1. / theta;//if denominator==0
                tp cs1 = 1. - cs;
                tp nx = rot[0] * itheta;
                tp ny = rot[1] * itheta;
                tp nz = rot[2] * itheta;

                tp nxnx = nx * nx, nyny = ny * ny, nznz = nz * nz;
                tp nxny = nx * ny, nxnz = nx * nz, nynz = ny * nz;
                tp nxsn = nx * sn, nysn = ny * sn, nzsn = nz * sn;

                R11 = nxnx * cs1 + cs;
                R21 = nxny * cs1 + nzsn;
                R31 = nxnz * cs1 - nysn;

                R12 = nxny * cs1 - nzsn;
                R22 = nyny * cs1 + cs;
                R32 = nynz * cs1 + nxsn;

                R13 = nxnz * cs1 + nysn;
                R23 = nynz * cs1 - nxsn;
                R33 = nznz * cs1 + cs;
            }

            //5.ReProjection
            const Vec2d* data2D = point2D.ptr<Vec2d>();
            const Vec3d* data3D = point3D.ptr<Vec3d>();
            Vec<tp, 2>* err2D = (Vec<tp, 2>*)errPoint2D;
            for (int k = 0; k < point3D.rows; ++k)
            {
                //5.1 WorldCoordinate
                double X = data3D[k][0];
                double Y = data3D[k][1];
                double Z = data3D[k][2];

                //5.2 CameraCoordinate
                tp x = R11 * X + R12 * Y + R13 * Z + tx;
                tp y = R21 * X + R22 * Y + R23 * Z + ty;
                tp z = R31 * X + R32 * Y + R33 * Z + tz;

                //5.3 StandardPhysicsCoordinate
                tp iz = 1. / z; //if denominator==0
                tp xc = x * iz;
                tp yc = y * iz;

                //5.4 DistortionPhysicsCoordinate
                tp xc2 = xc * xc;
                tp yc2 = yc * yc;
                tp d2 = xc2 + yc2;
                tp xcyc = 2. * xc * yc;
                tp d4 = d2 * d2;
                tp d6 = d2 * d4;
                tp d2xc2 = d2 + 2. * xc2;
                tp d2yc2 = d2 + 2. * yc2;
                tp nu, de, xd, yd;
                if (nDist < 5)
                {
                    nu = 1. + k1 * d2 + k2 * d4;
                    xd = xc * nu + p1 * xcyc + p2 * d2xc2;
                    yd = yc * nu + p2 * xcyc + p1 * d2yc2;
                }
                else if (nDist < 8)
                {
                    nu = 1. + k1 * d2 + k2 * d4 + k3 * d6;
                    xd = xc * nu + p1 * xcyc + p2 * d2xc2;
                    yd = yc * nu + p2 * xcyc + p1 * d2yc2;
                }
                else if (nDist < 12)
                {
                    nu = 1. + k1 * d2 + k2 * d4 + k3 * d6;
                    de = 1. / (1. + k4 * d2 + k5 * d4 + k6 * d6);//if denominator==0
                    xd = xc * nu * de + p1 * xcyc + p2 * d2xc2;
                    yd = yc * nu * de + p2 * xcyc + p1 * d2yc2;
                }
                else if (nDist < 14)
                {
                    nu = 1. + k1 * d2 + k2 * d4 + k3 * d6;
                    de = 1. / (1. + k4 * d2 + k5 * d4 + k6 * d6);//if denominator==0
                    xd = xc * nu * de + p1 * xcyc + p2 * d2xc2 + s1 * d2 + s2 * d4;
                    yd = yc * nu * de + p2 * xcyc + p1 * d2yc2 + s3 * d2 + s4 * d4;
                }
                err2D[k][0] = xd * fx + cx - data2D[k][0];
                err2D[k][1] = yd * fy + cy - data2D[k][1];
            }
            return true;
        }
    };
};

int main(int argc, char** argv) { OptimizeKDRt::TestMe(argc, argv); return 0; }
int main1(int argc, char** argv) { OptimizeRt::TestMe(argc, argv); return 0; }
int main2(int argc, char** argv) { OptimizeKDRt::TestMe(argc, argv); return 0; }