Ceres-Solver学习日志：捆绑优化使用样例与单目视觉中的尺度问题

         前面通过几个篇幅阐述了Ceres的基本应用，本篇基于之前的样例展示一个稍加复杂的样例。在给出使用样例之前，我们先来谈谈捆绑优化(Bundle Adjustment)和单目视觉中的尺度问题。

1.捆绑优化

         捆绑优化的概念追溯久远，要从其原始出处说起，还真不是一两话能说清楚，涉及视觉领域相当专业的概念。也许大多数视觉领域的研究者都理解捆绑优化是什么，用得也如火纯青，但真要通俗易懂的说出来就不知道有几人啦，相关书籍和博文也很少有对捆绑优化的通俗阐述。这里，我根据自己的理解，避开视觉中的概念，从更广义的角度来说说自己对捆绑优化的理解。

         假设有优化模型A(a)=0，这里A代表线性或非线性或两者混合的方程组，a代表被优化的变量组成的向量。常规而言，只要方程的数量足够，迭代优化即可得到最优值。然而，有时候方程数量再多，得到的解也可能不正确或者说不够精确，至于其原由这里不展开，从哲学的角度来说，大意就是你一直站在角度A看问题，不够全面，自然得到结果a也不全面(即不精确)，如果能同时站在其它角度B、C、D来看问题，你看到的a是否会更正确了？答案很明显。但是当你站在B、C、D的角度来看问题时，包含的变量通常就不只是a了，我们假设站在B的角度会增加变量b、站在C的角度会增加变量c、站在D的角度会增加变量d。于是优化模型变为：

         A(a)=0;B(a,b)=0;C(a,c)=0;D(a,d)=0

以上是比较理想的情况，也有可能是如下情况：

         A(a)=0;B(a,b)=0;C(a, b,c)=0; D(a,b,c,d)=0。

         或

         A(a)=0;B(a,b)=0;C(b,c)=0;D(c,b)=0。

         具体形式，取决于实际模型。但至少需要存在一个中间角度将a和后续变量关联起来，才有意义。

         于是问题转化为求解最优abcd。这里将abcd捆绑优化，也正好呼应了捆绑优化这一概念。

         我们来分析之前和本篇的使用样例：

         OptimizeRt：这是之前的样例，基于一次观察(即一个角度)计算Rt，这还不能算捆绑优化。

         OptimizeKDRt：这是之前的样例，基于多次观察(即多个角度)计算KD和所有Rt，KD出现在所有观察，每个Rt对应一组观察，这就是典型的捆绑优化。

         OptimizeKDTP：这是本篇的样例，基于多次观察(即多个角度)计算KD、所有Rt及所有Point3D，KD出现在所有观察，每个Rt对应一组观察，每个Point3D对应至少N组(N大于等于2)观察，这也是典型的捆绑优化。

2.单目视觉中的尺度

         单目视觉中，当我们要基于多次观察计算每次观察的Rt和每次观察中某些(这里不说所有是因为一个三维点通常需要被多次观察才能很好地确定，对于没有被多次观察到的则将其排除)三维点时，必须首先得到一些确定的Rt或一些确定的三维点才能使模型收敛到正确的结果。通常有两种方案：

         (1)给定位姿：给定一次观察的Rt和另一次观察的t(或t模值)、或给定一次观察的Rt和两次观察的之间的相对t(或t模值)。

         (2)给定三维点：给定三个至少能出现在N次(N大于等于2)观察中的Point3D。

         以上要求的是最少数量，数量越多越好。关于原理分析，可参见视觉Slam那些古老的论文。

3.使用样例

         提供捆绑优化K、D、R、t、XYZ的使用样例，命名为OptimizeKDTP，这是在OptimizeKDRt的基础上增加了对世界坐标的优化。对此使用样例作如下说明：

         (1)提供参数设置以决定是否固定K或D或R或t或P，以模拟视觉开发中的各种应用场景(如定位、标定、重建等)。

         (2)当要优化世界坐标时，提供参数设置以决定选用位姿真值还是三维点真值来估计尺度。

         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 OptimizeKDTP
{
public:
    using MotionView = MotionSim::MotionView;
    static void TestMe(int argc = 0, char** argv = 0)
    {
        int N = 99;
        for (int k = 0; k < N; ++k)
        {
            //0.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;
            const int keyViewId = 0;
            const int nMinMatch = 8;
            Mat_<Vec3d> point3D0;
            vector<vector<std::pair<int, int>>> idsPosePoint2DForAll3D;
            vector<bool> rtsHitted(motionSim.motionViews.size(), false);
            {
                //1.Get approapriate world points and its all observations             
                for (int k = 0; k < motionSim.motionViews[keyViewId].point3DIds.rows; ++k)
                {
                    int which3D = motionSim.motionViews[keyViewId].point3DIds(k);
                    vector<std::pair<int, int>> idsPosePoint2DForCur3D;
                    for (int i = 0; i < motionSim.motionViews.size(); ++i)//WhichPose==WhichView
                        for (int j = 0; j < motionSim.motionViews[i].point3DIds.rows; ++j)//WhichPoint3D=WhichPoint2D
                            if (which3D == motionSim.motionViews[i].point3DIds(j)) idsPosePoint2DForCur3D.push_back(std::pair<int, int>(i, j));
                    if (idsPosePoint2DForCur3D.size() > nMinMatch)
                    {
                        point3D0.push_back(motionSim.motionViews[keyViewId].point3D(k));
                        idsPosePoint2DForAll3D.push_back(idsPosePoint2DForCur3D);
                    }
                }
                //2.Check which views are used and check above world points whether echo its all observations
                Scalar sumDBG = 0;
                for (int k = 0; k < point3D0.rows; ++k)
                    for (int i = 0; i < idsPosePoint2DForAll3D[k].size(); ++i)
                    {
                        int whichView = idsPosePoint2DForAll3D[k][i].first;
                        int whichPoint2D = idsPosePoint2DForAll3D[k][i].second;
                        sumDBG += cv::sum(motionSim.motionViews[whichView].point3D(whichPoint2D) - point3D0(k));
                        Mat_<Vec2d> pt; cv::projectPoints(point3D0(k), rs0.rowRange(whichView * 3, whichView * 3 + 3), ts0.rowRange(whichView * 3, whichView * 3 + 3), Matx33d(K0(0), 0, K0(2), 0, K0(1), K0(3), 0, 0, 1), D0, pt);
                        sumDBG += cv::sum(motionSim.motionViews[whichView].point2D(whichPoint2D) - pt(0));
                        rtsHitted[whichView] = true;
                    }
                if (sumDBG[0] + sumDBG[1] + sumDBG[2] + sumDBG[3] != 0) spdlog::error("Error input: {}", sumDBG[0] + sumDBG[1] + sumDBG[2] + sumDBG[3]);
            }

            //1.CalcByCeresWithAutoDeriv
            bool fixK = false;
            bool fixD = false;
            bool fixRot = false;
            bool fixTrans = false;
            bool fixPoint3D = false;
            int scaleFrom = 1;//1=fromPose 2=fromPoint3D
            Mat_<double> Rs1, ts1;
            for (int k = 0; k < rs0.rows; k += 3)
            {
                double errRot = errRatio;
                double errTrans = errRatioTrans;
                if (fixRot || rtsHitted[k / 3] == false) errRot = 1.;
                if (fixTrans || rtsHitted[k / 3] == false) errTrans = 1.;
                if (fixPoint3D == false && scaleFrom == 1)
                {
                    if (k / 3 == keyViewId) errRot = 1.;//Fix at least one rotation frame for scale
                    if (k / 3 == keyViewId || k / 3 == keyViewId + 1) errTrans = 1.;//Fix at least two translation frames for scale
                }
                Mat_<double> _R; cv::Rodrigues(rs0.rowRange(k, k + 3) * errRot, _R);
                Rs1.push_back(_R);
                ts1.push_back(ts0.rowRange(k, k + 3) * errTrans);
            }
            Mat_<double> K1 = K0 * (fixK ? 1. : errRatio);
            Mat_<double> D1 = D0 * (fixD ? 1. : errRatio);
            Mat_<Vec3d> point3D1 = point3D0 * (fixPoint3D ? 1. : errRatio);
            if (fixPoint3D == false && scaleFrom == 2) for (int k = 0; k < 3; ++k) point3D1(k) = point3D0(k);//Fix at least three world points for scale
            ceres::Problem problem1;
            for (int k = 0; k < point3D1.rows; ++k)
                for (int i = 0; i < idsPosePoint2DForAll3D[k].size(); ++i)
                {
                    int whichView = idsPosePoint2DForAll3D[k][i].first;
                    int whichPoint2D = idsPosePoint2DForAll3D[k][i].second;
                    problem1.AddResidualBlock(new ceres::AutoDiffCostFunction<ProjectionModelKDTP, 2, 3, 9, 3, 4, nDist>(
                        new ProjectionModelKDTP(motionSim.motionViews[whichView].point2D.row(whichPoint2D), motionSim.nCamDist)),
                        NULL, point3D1(k).val, Rs1.ptr<double>(whichView * 3), ts1.ptr<double>(whichView * 3), K1.ptr<double>(), D1.ptr<double>());

                    problem1.SetParameterization(Rs1.ptr<double>(whichView * 3), new LocalParamRWithGeneralJ);//it is to be set more than one time
                    if (fixK) problem1.SetParameterBlockConstant(K1.ptr<double>());
                    if (fixD) problem1.SetParameterBlockConstant(D1.ptr<double>());
                    if (fixRot) problem1.SetParameterBlockConstant(Rs1.ptr<double>(whichView * 3));
                    if (fixTrans) problem1.SetParameterBlockConstant(ts1.ptr<double>(whichView * 3));
                    if (fixPoint3D) problem1.SetParameterBlockConstant(point3D1(k).val);
                    if (fixPoint3D == false && scaleFrom == 1)
                    {
                        if (whichView == keyViewId) problem1.SetParameterBlockConstant(Rs1.ptr<double>(whichView * 3));//Fix at least one rotation and two translations for scale
                        if (whichView == keyViewId || whichView == keyViewId + 1) problem1.SetParameterBlockConstant(ts1.ptr<double>(whichView * 3));
                    }
                    if (fixPoint3D == false && scaleFrom == 2 && k < 3) problem1.SetParameterBlockConstant(point3D1(k).val);//Fix at least three world points for scale
                }
            ceres::Solver::Options options;
            ceres::Solver::Summary summary;
            ceres::Solve(options, &problem1, &summary);
            Mat_<double> rs1; for (int k = 0; k < Rs1.rows; k += 3) { Mat_<double> _r; cv::Rodrigues(Rs1.rowRange(k, k + 3), _r); rs1.push_back(_r); }
            int nIter1 = (int)summary.iterations.size();

            //2.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);
            double infPTS0PTS0 = norm(point3D0, point3D0 * errRatio, NORM_INF);
            double infPTS0PTS1 = norm(point3D0, point3D1, NORM_INF);

            //3.PrintError
            cout << fmt::format("LoopCount: {}      AutoDeriv.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 << fmt::format("infPTS0PTS1  : {:<15.9}		{:<15.9}
", infPTS0PTS1, infPTS0PTS0);
                if (1)
                {
                    //cout << endl << "" << << endl;
                }
                cout << "Press any key to continue" << endl; std::getchar();
            }
        }
    }

public:
    struct ProjectionModelKDTP
    {
        const int nDist;
        const Mat_<Vec2d> point2D;
        ProjectionModelKDTP(const Mat_<Vec2d> point2D0, const int nDist0) : point2D(point2D0), nDist(nDist0) {}
        template <typename tp> bool operator()(const tp* const point3D, 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.Rotation params
            tp R11, R12, R13, R21, R22, R23, R31, R32, R33;
            {
                R11 = rot[0]; R12 = rot[1]; R13 = rot[2];
                R21 = rot[3]; R22 = rot[4]; R23 = rot[5];
                R31 = rot[6]; R32 = rot[7]; R33 = rot[8];
            }

            //5.ReProjection
            const Vec2d* data2D = point2D.ptr<Vec2d>();
            Vec<tp, 3>* data3D = (Vec<tp, 3>*)point3D;
            Vec<tp, 2>* err2D = (Vec<tp, 2>*)errPoint2D;
            for (int k = 0; k < point2D.rows; ++k)
            {
                //5.1 WorldCoordinate
                tp X = data3D[k][0];
                tp Y = data3D[k][1];
                tp 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;
        }
    };
    struct LocalParamRWithGeneralJ : public ceres::LocalParameterization
    {
        bool Plus(const double* preParam, const double* deltaParam, double* newParam) const
        {
            Matx33d ambientPre(preParam);
            Matx33d ambientDelta; cv::Rodrigues(Matx31d(deltaParam), ambientDelta);
            Matx33d ambientNew = ambientDelta * ambientPre;
            memcpy(newParam, ambientNew.val, sizeof(double) * 9);
            return true;
        }

        bool ComputeJacobian(const double* param, double* jacobian) const
        {
            Mat_<double> dRdr(9, 3, jacobian);
            dRdr << 0, param[6], -param[3], 0, param[7], -param[4], 0, param[8], -param[5],
                -param[6], 0, param[0], -param[7], 0, param[1], -param[8], 0, param[2],
                param[3], -param[0], 0, param[4], -param[1], 0, param[5], -param[2], 0;
            return true;

            Matx33d R(param);
            Mat_<double> dRdrT({ 3, 9 }, {
                0, 0, 0, -R(2, 0), -R(2, 1), -R(2, 2), R(1, 0), R(1, 1), R(1, 2),
                R(2, 0), R(2, 1), R(2, 2), 0, 0, 0, -R(0, 0), -R(0, 1), -R(0, 2),
                -R(1, 0), -R(1, 1), -R(1, 2), R(0, 0), R(0, 1), R(0, 2), 0, 0, 0 });
            cout << endl << cv::norm(dRdr, dRdrT.t(), NORM_INF) << endl;
        }

        int GlobalSize() const { return 9; }
        int LocalSize() const { return 3; }
    };
};

int main(int argc, char** argv) { OptimizeKDTP::TestMe(argc, argv); return 0; }