视觉SLAM十四讲——ch8实践（视觉里程计2）

0.实践前小知识介绍

里程计的历史渊源是什么？

里程计是一种用来测量车辆或机器人行驶距离的装置，它通常通过检测车辆轮子或机器人轮子的旋转来进行测量。里程计的历史可以追溯到17世纪早期，当时人们开始使用机械装置来测量车辆行驶的距离。这些装置通常使用一个机械计数器，它们可以在车轮旋转的过程中记录里程数。18世纪末期，发明家托马斯·戈德史密斯发明了一种称为“奥多米特”的装置，它使用一个机械计数器来记录马车或自行车行驶的里程。这个装置被认为是现代里程计的早期形式。

随着时间的推移，里程计逐渐发展成为电子化和计算机化的设备。现代车辆和机器人通常使用激光或红外线传感器来测量轮子的旋转，并将数据传输到计算机或控制系统中。总的来说，里程计的历史经历了从机械装置到电子化和计算机化的过程。

1. 实践操作前的准备工作

1. 在终端中进入ch8文件夹下，顺序执行以下命令进行编译。

mkdir build
cd build
cmake ..
//注意，j8还是其他主要看自己的电脑情况
make -j8

2. 在build文件中进行运行。
   注意： 在make之前，尽量将文件中的获取图片的路径都更改以下，否则后期运行有问题还得再更改，再make。

2. 实践过程

2.1 LK光流

代码：

//
// Created by Xiang on 2017/12/19.
//

#include 
#include 
#include 
#include 
#include 
//添加头文件
#include 


using namespace std;
using namespace cv;

string file_1 = "/home/fighter/slam/slambook2/ch8/LK1.png";  // first image
string file_2 = "/home/fighter/slam/slambook2/ch8/LK2.png";  // second image

/// Optical flow tracker and interface
class OpticalFlowTracker {
public:
    OpticalFlowTracker(
        const Mat &img1_,
        const Mat &img2_,
        const vector<KeyPoint> &kp1_,
        vector<KeyPoint> &kp2_,
        vector<bool> &success_,
        bool inverse_ = true, bool has_initial_ = false) :
        img1(img1_), img2(img2_), kp1(kp1_), kp2(kp2_), success(success_), inverse(inverse_),
        has_initial(has_initial_) {}

    void calculateOpticalFlow(const Range &range);

private:
    const Mat &img1;
    const Mat &img2;
    const vector<KeyPoint> &kp1;
    vector<KeyPoint> &kp2;
    vector<bool> &success;
    bool inverse = true;
    bool has_initial = false;
};

/**
 * single level optical flow
 * @param [in] img1 the first image
 * @param [in] img2 the second image
 * @param [in] kp1 keypoints in img1
 * @param [in|out] kp2 keypoints in img2, if empty, use initial guess in kp1
 * @param [out] success true if a keypoint is tracked successfully
 * @param [in] inverse use inverse formulation?
 */
void OpticalFlowSingleLevel(
    const Mat &img1,
    const Mat &img2,
    const vector<KeyPoint> &kp1,
    vector<KeyPoint> &kp2,
    vector<bool> &success,
    bool inverse = false,
    bool has_initial_guess = false
);

/**
 * multi level optical flow, scale of pyramid is set to 2 by default
 * the image pyramid will be create inside the function
 * @param [in] img1 the first pyramid
 * @param [in] img2 the second pyramid
 * @param [in] kp1 keypoints in img1
 * @param [out] kp2 keypoints in img2
 * @param [out] success true if a keypoint is tracked successfully
 * @param [in] inverse set true to enable inverse formulation
 */
void OpticalFlowMultiLevel(
    const Mat &img1,
    const Mat &img2,
    const vector<KeyPoint> &kp1,
    vector<KeyPoint> &kp2,
    vector<bool> &success,
    bool inverse = false
);

/**
 * get a gray scale value from reference image (bi-linear interpolated)
 * @param img
 * @param x
 * @param y
 * @return the interpolated value of this pixel
 */

inline float GetPixelValue(const cv::Mat &img, float x, float y) {
    // boundary check
    if (x < 0) x = 0;
    if (y < 0) y = 0;
    if (x >= img.cols - 1) x = img.cols - 2;
    if (y >= img.rows - 1) y = img.rows - 2;
    
    float xx = x - floor(x);
    float yy = y - floor(y);
    int x_a1 = std::min(img.cols - 1, int(x) + 1);
    int y_a1 = std::min(img.rows - 1, int(y) + 1);
    
    return (1 - xx) * (1 - yy) * img.at<uchar>(y, x)
    + xx * (1 - yy) * img.at<uchar>(y, x_a1)
    + (1 - xx) * yy * img.at<uchar>(y_a1, x)
    + xx * yy * img.at<uchar>(y_a1, x_a1);
}

int main(int argc, char **argv) {

    // images, note they are CV_8UC1, not CV_8UC3
    Mat img1 = imread(file_1, 0);
    Mat img2 = imread(file_2, 0);

    // key points, using GFTT here.
    vector<KeyPoint> kp1;
    Ptr<GFTTDetector> detector = GFTTDetector::create(500, 0.01, 20); // maximum 500 keypoints
    detector->detect(img1, kp1);

    // now lets track these key points in the second image
    // first use single level LK in the validation picture
    vector<KeyPoint> kp2_single;
    vector<bool> success_single;
    OpticalFlowSingleLevel(img1, img2, kp1, kp2_single, success_single);

    // then test multi-level LK
    vector<KeyPoint> kp2_multi;
    vector<bool> success_multi;
    chrono::steady_clock::time_point t1 = chrono::steady_clock::now();
    OpticalFlowMultiLevel(img1, img2, kp1, kp2_multi, success_multi, true);
    chrono::steady_clock::time_point t2 = chrono::steady_clock::now();
    auto time_used = chrono::duration_cast<chrono::duration<double>>(t2 - t1);
    cout << "optical flow by gauss-newton: " << time_used.count() << endl;

    // use opencv's flow for validation
    vector<Point2f> pt1, pt2;
    for (auto &kp: kp1) pt1.push_back(kp.pt);
    vector<uchar> status;
    vector<float> error;
    t1 = chrono::steady_clock::now();
    cv::calcOpticalFlowPyrLK(img1, img2, pt1, pt2, status, error);
    t2 = chrono::steady_clock::now();
    time_used = chrono::duration_cast<chrono::duration<double>>(t2 - t1);
    cout << "optical flow by opencv: " << time_used.count() << endl;

    // plot the differences of those functions
    Mat img2_single;
    cv::cvtColor(img2, img2_single, CV_GRAY2BGR);
    for (int i = 0; i < kp2_single.size(); i++) {
        if (success_single[i]) {
            cv::circle(img2_single, kp2_single[i].pt, 2, cv::Scalar(0, 250, 0), 2);
            cv::line(img2_single, kp1[i].pt, kp2_single[i].pt, cv::Scalar(0, 250, 0));
        }
    }

    Mat img2_multi;
    cv::cvtColor(img2, img2_multi, CV_GRAY2BGR);
    for (int i = 0; i < kp2_multi.size(); i++) {
        if (success_multi[i]) {
            cv::circle(img2_multi, kp2_multi[i].pt, 2, cv::Scalar(0, 250, 0), 2);
            cv::line(img2_multi, kp1[i].pt, kp2_multi[i].pt, cv::Scalar(0, 250, 0));
        }
    }

    Mat img2_CV;
    cv::cvtColor(img2, img2_CV, CV_GRAY2BGR);
    for (int i = 0; i < pt2.size(); i++) {
        if (status[i]) {
            cv::circle(img2_CV, pt2[i], 2, cv::Scalar(0, 250, 0), 2);
            cv::line(img2_CV, pt1[i], pt2[i], cv::Scalar(0, 250, 0));
        }
    }

    cv::imshow("tracked single level", img2_single);
    cv::imshow("tracked multi level", img2_multi);
    cv::imshow("tracked by opencv", img2_CV);
    cv::waitKey(0);

    return 0;
}

void OpticalFlowSingleLevel(
    const Mat &img1,
    const Mat &img2,
    const vector<KeyPoint> &kp1,
    vector<KeyPoint> &kp2,
    vector<bool> &success,
    bool inverse, bool has_initial) {
    kp2.resize(kp1.size());
    success.resize(kp1.size());
    OpticalFlowTracker tracker(img1, img2, kp1, kp2, success, inverse, has_initial);
    parallel_for_(Range(0, kp1.size()),
                  std::bind(&OpticalFlowTracker::calculateOpticalFlow, &tracker, placeholders::_1));
}

void OpticalFlowTracker::calculateOpticalFlow(const Range &range) {
    // parameters
    int half_patch_size = 4;
    int iterations = 10;
    for (size_t i = range.start; i < range.end; i++) {
        auto kp = kp1[i];
        double dx = 0, dy = 0; // dx,dy need to be estimated
        if (has_initial) {
            dx = kp2[i].pt.x - kp.pt.x;
            dy = kp2[i].pt.y - kp.pt.y;
        }

        double cost = 0, lastCost = 0;
        bool succ = true; // indicate if this point succeeded

        // Gauss-Newton iterations
        Eigen::Matrix2d H = Eigen::Matrix2d::Zero();    // hessian
        Eigen::Vector2d b = Eigen::Vector2d::Zero();    // bias
        Eigen::Vector2d J;  // jacobian
        for (int iter = 0; iter < iterations; iter++) {
            if (inverse == false) {
                H = Eigen::Matrix2d::Zero();
                b = Eigen::Vector2d::Zero();
            } else {
                // only reset b
                b = Eigen::Vector2d::Zero();
            }

            cost = 0;

            // compute cost and jacobian
            for (int x = -half_patch_size; x < half_patch_size; x++)
                for (int y = -half_patch_size; y < half_patch_size; y++) {
                    double error = GetPixelValue(img1, kp.pt.x + x, kp.pt.y + y) -
                                   GetPixelValue(img2, kp.pt.x + x + dx, kp.pt.y + y + dy);;  // Jacobian
                    if (inverse == false) {
                        J = -1.0 * Eigen::Vector2d(
                            0.5 * (GetPixelValue(img2, kp.pt.x + dx + x + 1, kp.pt.y + dy + y) -
                                   GetPixelValue(img2, kp.pt.x + dx + x - 1, kp.pt.y + dy + y)),
                            0.5 * (GetPixelValue(img2, kp.pt.x + dx + x, kp.pt.y + dy + y + 1) -
                                   GetPixelValue(img2, kp.pt.x + dx + x, kp.pt.y + dy + y - 1))
                        );
                    } else if (iter == 0) {
                        // in inverse mode, J keeps same for all iterations
                        // NOTE this J does not change when dx, dy is updated, so we can store it and only compute error
                        J = -1.0 * Eigen::Vector2d(
                            0.5 * (GetPixelValue(img1, kp.pt.x + x + 1, kp.pt.y + y) -
                                   GetPixelValue(img1, kp.pt.x + x - 1, kp.pt.y + y)),
                            0.5 * (GetPixelValue(img1, kp.pt.x + x, kp.pt.y + y + 1) -
                                   GetPixelValue(img1, kp.pt.x + x, kp.pt.y + y - 1))
                        );
                    }
                    // compute H, b and set cost;
                    b += -error * J;
                    cost += error * error;
                    if (inverse == false || iter == 0) {
                        // also update H
                        H += J * J.transpose();
                    }
                }

            // compute update
            Eigen::Vector2d update = H.ldlt().solve(b);

            if (std::isnan(update[0])) {
                // sometimes occurred when we have a black or white patch and H is irreversible
                cout << "update is nan" << endl;
                succ = false;
                break;
            }

            if (iter > 0 && cost > lastCost) {
                break;
            }

            // update dx, dy
            dx += update[0];
            dy += update[1];
            lastCost = cost;
            succ = true;

            if (update.norm() < 1e-2) {
                // converge
                break;
            }
        }

        success[i] = succ;

        // set kp2
        kp2[i].pt = kp.pt + Point2f(dx, dy);
    }
}

void OpticalFlowMultiLevel(
    const Mat &img1,
    const Mat &img2,
    const vector<KeyPoint> &kp1,
    vector<KeyPoint> &kp2,
    vector<bool> &success,
    bool inverse) {

    // parameters
    int pyramids = 4;
    double pyramid_scale = 0.5;
    double scales[] = {1.0, 0.5, 0.25, 0.125};

    // create pyramids
    chrono::steady_clock::time_point t1 = chrono::steady_clock::now();
    vector<Mat> pyr1, pyr2; // image pyramids
    for (int i = 0; i < pyramids; i++) {
        if (i == 0) {
            pyr1.push_back(img1);
            pyr2.push_back(img2);
        } else {
            Mat img1_pyr, img2_pyr;
            cv::resize(pyr1[i - 1], img1_pyr,
                       cv::Size(pyr1[i - 1].cols * pyramid_scale, pyr1[i - 1].rows * pyramid_scale));
            cv::resize(pyr2[i - 1], img2_pyr,
                       cv::Size(pyr2[i - 1].cols * pyramid_scale, pyr2[i - 1].rows * pyramid_scale));
            pyr1.push_back(img1_pyr);
            pyr2.push_back(img2_pyr);
        }
    }
    chrono::steady_clock::time_point t2 = chrono::steady_clock::now();
    auto time_used = chrono::duration_cast<chrono::duration<double>>(t2 - t1);
    cout << "build pyramid time: " << time_used.count() << endl;

    // coarse-to-fine LK tracking in pyramids
    vector<KeyPoint> kp1_pyr, kp2_pyr;
    for (auto &kp:kp1) {
        auto kp_top = kp;
        kp_top.pt *= scales[pyramids - 1];
        kp1_pyr.push_back(kp_top);
        kp2_pyr.push_back(kp_top);
    }

    for (int level = pyramids - 1; level >= 0; level--) {
        // from coarse to fine
        success.clear();
        t1 = chrono::steady_clock::now();
        OpticalFlowSingleLevel(pyr1[level], pyr2[level], kp1_pyr, kp2_pyr, success, inverse, true);
        t2 = chrono::steady_clock::now();
        auto time_used = chrono::duration_cast<chrono::duration<double>>(t2 - t1);
        cout << "track pyr " << level << " cost time: " << time_used.count() << endl;

        if (level > 0) {
            for (auto &kp: kp1_pyr)
                kp.pt /= pyramid_scale;
            for (auto &kp: kp2_pyr)
                kp.pt /= pyramid_scale;
        }
    }

    for (auto &kp: kp2_pyr)
        kp2.push_back(kp);
}

在build中执行语句：

 ./optical_flow

运行结果：
运行后展示使用opencv、单层、多层的追踪



同时终端输出：

build pyramid time: 0.0072683
track pyr 3 cost time: 0.0004321
track pyr 2 cost time: 0.0002794
track pyr 1 cost time: 0.0002624
track pyr 0 cost time: 0.0003014
optical flow by gauss-newton: 0.0087955
optical flow by opencv: 0.0054821

2.2 直接法

代码：

#include 
#include 
#include 
#include 
//添加头文件
#include 


using namespace std;

typedef vector<Eigen::Vector2d, Eigen::aligned_allocator<Eigen::Vector2d>> VecVector2d;

// Camera intrinsics
double fx = 718.856, fy = 718.856, cx = 607.1928, cy = 185.2157;
// baseline
double baseline = 0.573;
// paths
string left_file = "/home/fighter/slam/slambook2/ch8/left.png";
string disparity_file = "/home/fighter/slam/slambook2/ch8/disparity.png";
boost::format fmt_others("/home/fighter/slam/slambook2/ch8/%06d.png");    // other files

// useful typedefs
typedef Eigen::Matrix<double, 6, 6> Matrix6d;
typedef Eigen::Matrix<double, 2, 6> Matrix26d;
typedef Eigen::Matrix<double, 6, 1> Vector6d;

/// class for accumulator jacobians in parallel
class JacobianAccumulator {
public:
    JacobianAccumulator(
        const cv::Mat &img1_,
        const cv::Mat &img2_,
        const VecVector2d &px_ref_,
        const vector<double> depth_ref_,
        Sophus::SE3d &T21_) :
        img1(img1_), img2(img2_), px_ref(px_ref_), depth_ref(depth_ref_), T21(T21_) {
        projection = VecVector2d(px_ref.size(), Eigen::Vector2d(0, 0));
    }

    /// accumulate jacobians in a range
    void accumulate_jacobian(const cv::Range &range);

    /// get hessian matrix
    Matrix6d hessian() const { return H; }

    /// get bias
    Vector6d bias() const { return b; }

    /// get total cost
    double cost_func() const { return cost; }

    /// get projected points
    VecVector2d projected_points() const { return projection; }

    /// reset h, b, cost to zero
    void reset() {
        H = Matrix6d::Zero();
        b = Vector6d::Zero();
        cost = 0;
    }

private:
    const cv::Mat &img1;
    const cv::Mat &img2;
    const VecVector2d &px_ref;
    const vector<double> depth_ref;
    Sophus::SE3d &T21;
    VecVector2d projection; // projected points

    std::mutex hessian_mutex;
    Matrix6d H = Matrix6d::Zero();
    Vector6d b = Vector6d::Zero();
    double cost = 0;
};

/**
 * pose estimation using direct method
 * @param img1
 * @param img2
 * @param px_ref
 * @param depth_ref
 * @param T21
 */
void DirectPoseEstimationMultiLayer(
    const cv::Mat &img1,
    const cv::Mat &img2,
    const VecVector2d &px_ref,
    const vector<double> depth_ref,
    Sophus::SE3d &T21
);

/**
 * pose estimation using direct method
 * @param img1
 * @param img2
 * @param px_ref
 * @param depth_ref
 * @param T21
 */
void DirectPoseEstimationSingleLayer(
    const cv::Mat &img1,
    const cv::Mat &img2,
    const VecVector2d &px_ref,
    const vector<double> depth_ref,
    Sophus::SE3d &T21
);

// bilinear interpolation
inline float GetPixelValue(const cv::Mat &img, float x, float y) {
    // boundary check
    if (x < 0) x = 0;
    if (y < 0) y = 0;
    if (x >= img.cols) x = img.cols - 1;
    if (y >= img.rows) y = img.rows - 1;
    uchar *data = &img.data[int(y) * img.step + int(x)];
    float xx = x - floor(x);
    float yy = y - floor(y);
    return float(
        (1 - xx) * (1 - yy) * data[0] +
        xx * (1 - yy) * data[1] +
        (1 - xx) * yy * data[img.step] +
        xx * yy * data[img.step + 1]
    );
}

int main(int argc, char **argv) {

    cv::Mat left_img = cv::imread(left_file, 0);
    cv::Mat disparity_img = cv::imread(disparity_file, 0);

    // let's randomly pick pixels in the first image and generate some 3d points in the first image's frame
    cv::RNG rng;
    int nPoints = 2000;
    int boarder = 20;
    VecVector2d pixels_ref;
    vector<double> depth_ref;

    // generate pixels in ref and load depth data
    for (int i = 0; i < nPoints; i++) {
        int x = rng.uniform(boarder, left_img.cols - boarder);  // don't pick pixels close to boarder
        int y = rng.uniform(boarder, left_img.rows - boarder);  // don't pick pixels close to boarder
        int disparity = disparity_img.at<uchar>(y, x);
        double depth = fx * baseline / disparity; // you know this is disparity to depth
        depth_ref.push_back(depth);
        pixels_ref.push_back(Eigen::Vector2d(x, y));
    }

    // estimates 01~05.png's pose using this information
    Sophus::SE3d T_cur_ref;

    for (int i = 1; i < 6; i++) {  // 1~10
        cv::Mat img = cv::imread((fmt_others % i).str(), 0);
        // try single layer by uncomment this line
        // DirectPoseEstimationSingleLayer(left_img, img, pixels_ref, depth_ref, T_cur_ref);
        DirectPoseEstimationMultiLayer(left_img, img, pixels_ref, depth_ref, T_cur_ref);
    }
    return 0;
}

void DirectPoseEstimationSingleLayer(
    const cv::Mat &img1,
    const cv::Mat &img2,
    const VecVector2d &px_ref,
    const vector<double> depth_ref,
    Sophus::SE3d &T21) {

    const int iterations = 10;
    double cost = 0, lastCost = 0;
    auto t1 = chrono::steady_clock::now();
    JacobianAccumulator jaco_accu(img1, img2, px_ref, depth_ref, T21);

    for (int iter = 0; iter < iterations; iter++) {
        jaco_accu.reset();
        cv::parallel_for_(cv::Range(0, px_ref.size()),
                          std::bind(&JacobianAccumulator::accumulate_jacobian, &jaco_accu, std::placeholders::_1));
        Matrix6d H = jaco_accu.hessian();
        Vector6d b = jaco_accu.bias();

        // solve update and put it into estimation
        Vector6d update = H.ldlt().solve(b);;
        T21 = Sophus::SE3d::exp(update) * T21;
        cost = jaco_accu.cost_func();

        if (std::isnan(update[0])) {
            // sometimes occurred when we have a black or white patch and H is irreversible
            cout << "update is nan" << endl;
            break;
        }
        if (iter > 0 && cost > lastCost) {
            cout << "cost increased: " << cost << ", " << lastCost << endl;
            break;
        }
        if (update.norm() < 1e-3) {
            // converge
            break;
        }

        lastCost = cost;
        cout << "iteration: " << iter << ", cost: " << cost << endl;
    }

    cout << "T21 = \n" << T21.matrix() << endl;
    auto t2 = chrono::steady_clock::now();
    auto time_used = chrono::duration_cast<chrono::duration<double>>(t2 - t1);
    cout << "direct method for single layer: " << time_used.count() << endl;

    // plot the projected pixels here
    cv::Mat img2_show;
    cv::cvtColor(img2, img2_show, CV_GRAY2BGR);
    VecVector2d projection = jaco_accu.projected_points();
    for (size_t i = 0; i < px_ref.size(); ++i) {
        auto p_ref = px_ref[i];
        auto p_cur = projection[i];
        if (p_cur[0] > 0 && p_cur[1] > 0) {
            cv::circle(img2_show, cv::Point2f(p_cur[0], p_cur[1]), 2, cv::Scalar(0, 250, 0), 2);
            cv::line(img2_show, cv::Point2f(p_ref[0], p_ref[1]), cv::Point2f(p_cur[0], p_cur[1]),
                     cv::Scalar(0, 250, 0));
        }
    }
    cv::imshow("current", img2_show);
    cv::waitKey();
}

void JacobianAccumulator::accumulate_jacobian(const cv::Range &range) {

    // parameters
    const int half_patch_size = 1;
    int cnt_good = 0;
    Matrix6d hessian = Matrix6d::Zero();
    Vector6d bias = Vector6d::Zero();
    double cost_tmp = 0;

    for (size_t i = range.start; i < range.end; i++) {

        // compute the projection in the second image
        Eigen::Vector3d point_ref =
            depth_ref[i] * Eigen::Vector3d((px_ref[i][0] - cx) / fx, (px_ref[i][1] - cy) / fy, 1);
        Eigen::Vector3d point_cur = T21 * point_ref;
        if (point_cur[2] < 0)   // depth invalid
            continue;

        float u = fx * point_cur[0] / point_cur[2] + cx, v = fy * point_cur[1] / point_cur[2] + cy;
        if (u < half_patch_size || u > img2.cols - half_patch_size || v < half_patch_size ||
            v > img2.rows - half_patch_size)
            continue;

        projection[i] = Eigen::Vector2d(u, v);
        double X = point_cur[0], Y = point_cur[1], Z = point_cur[2],
            Z2 = Z * Z, Z_inv = 1.0 / Z, Z2_inv = Z_inv * Z_inv;
        cnt_good++;

        // and compute error and jacobian
        for (int x = -half_patch_size; x <= half_patch_size; x++)
            for (int y = -half_patch_size; y <= half_patch_size; y++) {

                double error = GetPixelValue(img1, px_ref[i][0] + x, px_ref[i][1] + y) -
                               GetPixelValue(img2, u + x, v + y);
                Matrix26d J_pixel_xi;
                Eigen::Vector2d J_img_pixel;

                J_pixel_xi(0, 0) = fx * Z_inv;
                J_pixel_xi(0, 1) = 0;
                J_pixel_xi(0, 2) = -fx * X * Z2_inv;
                J_pixel_xi(0, 3) = -fx * X * Y * Z2_inv;
                J_pixel_xi(0, 4) = fx + fx * X * X * Z2_inv;
                J_pixel_xi(0, 5) = -fx * Y * Z_inv;

                J_pixel_xi(1, 0) = 0;
                J_pixel_xi(1, 1) = fy * Z_inv;
                J_pixel_xi(1, 2) = -fy * Y * Z2_inv;
                J_pixel_xi(1, 3) = -fy - fy * Y * Y * Z2_inv;
                J_pixel_xi(1, 4) = fy * X * Y * Z2_inv;
                J_pixel_xi(1, 5) = fy * X * Z_inv;

                J_img_pixel = Eigen::Vector2d(
                    0.5 * (GetPixelValue(img2, u + 1 + x, v + y) - GetPixelValue(img2, u - 1 + x, v + y)),
                    0.5 * (GetPixelValue(img2, u + x, v + 1 + y) - GetPixelValue(img2, u + x, v - 1 + y))
                );

                // total jacobian
                Vector6d J = -1.0 * (J_img_pixel.transpose() * J_pixel_xi).transpose();

                hessian += J * J.transpose();
                bias += -error * J;
                cost_tmp += error * error;
            }
    }

    if (cnt_good) {
        // set hessian, bias and cost
        unique_lock<mutex> lck(hessian_mutex);
        H += hessian;
        b += bias;
        cost += cost_tmp / cnt_good;
    }
}

void DirectPoseEstimationMultiLayer(
    const cv::Mat &img1,
    const cv::Mat &img2,
    const VecVector2d &px_ref,
    const vector<double> depth_ref,
    Sophus::SE3d &T21) {

    // parameters
    int pyramids = 4;
    double pyramid_scale = 0.5;
    double scales[] = {1.0, 0.5, 0.25, 0.125};

    // create pyramids
    vector<cv::Mat> pyr1, pyr2; // image pyramids
    for (int i = 0; i < pyramids; i++) {
        if (i == 0) {
            pyr1.push_back(img1);
            pyr2.push_back(img2);
        } else {
            cv::Mat img1_pyr, img2_pyr;
            cv::resize(pyr1[i - 1], img1_pyr,
                       cv::Size(pyr1[i - 1].cols * pyramid_scale, pyr1[i - 1].rows * pyramid_scale));
            cv::resize(pyr2[i - 1], img2_pyr,
                       cv::Size(pyr2[i - 1].cols * pyramid_scale, pyr2[i - 1].rows * pyramid_scale));
            pyr1.push_back(img1_pyr);
            pyr2.push_back(img2_pyr);
        }
    }

    double fxG = fx, fyG = fy, cxG = cx, cyG = cy;  // backup the old values
    for (int level = pyramids - 1; level >= 0; level--) {
        VecVector2d px_ref_pyr; // set the keypoints in this pyramid level
        for (auto &px: px_ref) {
            px_ref_pyr.push_back(scales[level] * px);
        }

        // scale fx, fy, cx, cy in different pyramid levels
        fx = fxG * scales[level];
        fy = fyG * scales[level];
        cx = cxG * scales[level];
        cy = cyG * scales[level];
        DirectPoseEstimationSingleLayer(pyr1[level], pyr2[level], px_ref_pyr, depth_ref, T21);
    }

}

在build中执行语句： ./direct_method

运行结果：
运行结果图可以看出追踪流，需要不停的对窗口关闭，可以看出来其变化。


终端输出相应信息：

iteration: 0, cost: 1.59797e+06
iteration: 1, cost: 651716
iteration: 2, cost: 243255
iteration: 3, cost: 176884
cost increased: 183909, 176884
T21 =
   0.999991  0.00245009   0.0033858  0.00303273
-0.00245906    0.999993  0.00264927 0.000424829
-0.00337929 -0.00265757    0.999991   -0.730917
          0           0           0           1
direct method for single layer: 0.0016574
iteration: 0, cost: 186361
T21 =
   0.999989  0.00302157  0.00347121 0.000762356
-0.00302936    0.999993  0.00224074  0.00666315
-0.00346442 -0.00225123    0.999991   -0.728227
          0           0           0           1
direct method for single layer: 0.002358
iteration: 0, cost: 247529
iteration: 1, cost: 229117
T21 =
   0.999991  0.00251345  0.00346578 -0.00270253
-0.00252155    0.999994  0.00233534  0.00243076
-0.00345989 -0.00234406    0.999991   -0.734719
          0           0           0           1
direct method for single layer: 0.00523089
iteration: 0, cost: 348441
T21 =
   0.999991  0.00248082  0.00343389 -0.00373965
-0.00248836    0.999994  0.00219448  0.00304522
-0.00342843 -0.00220301    0.999992   -0.732343
          0           0           0           1
direct method for single layer: 0.0012425
iteration: 0, cost: 1.315e+06
iteration: 1, cost: 906037
iteration: 2, cost: 603626
iteration: 3, cost: 399435
iteration: 4, cost: 280889
iteration: 5, cost: 237691
cost increased: 238395, 237691
T21 =
    0.999971  0.000902974   0.00759567   0.00772499
-0.000938067     0.999989   0.00461783   0.00179863
 -0.00759142  -0.00462482      0.99996     -1.46052
           0            0            0            1
direct method for single layer: 0.0045787
iteration: 0, cost: 355480
iteration: 1, cost: 348267
cost increased: 348423, 348267
T21 =
   0.999972  0.00120085  0.00742895   0.0085892
-0.00123022    0.999991  0.00395007  0.00531883
-0.00742414  -0.0039591    0.999965    -1.46883
          0           0           0           1
direct method for single layer: 0.0009226

iteration: 0, cost: 443225
iteration: 1, cost: 435054
cost increased: 437537, 435054
T21 =
    0.999971  0.000737127   0.00764046 -0.000242531
-0.000767091     0.999992   0.00391957   0.00279348
 -0.00763751  -0.00392532     0.999963      -1.4818
           0            0            0            1
direct method for single layer: 0.0009165
iteration: 0, cost: 501709
iteration: 1, cost: 463084
cost increased: 463953, 463084
T21 =
    0.999971  0.000695392   0.00758989  -0.00249798
-0.000723685     0.999993   0.00372567   0.00395279
 -0.00758725  -0.00373106     0.999964     -1.48132
           0            0            0            1
direct method for single layer: 0.0008786
iteration: 0, cost: 1.37107e+06
iteration: 1, cost: 1.10683e+06
iteration: 2, cost: 921990
iteration: 3, cost: 794740
iteration: 4, cost: 601342
iteration: 5, cost: 559319
iteration: 6, cost: 394434
iteration: 7, cost: 363978
cost increased: 374118, 363978
T21 =
   0.999945  0.00160897   0.0103684   0.0493737
-0.00166631    0.999983  0.00552457   0.0132374
 -0.0103594 -0.00554155    0.999931    -2.18064
          0           0           0           1
direct method for single layer: 0.0020563
iteration: 0, cost: 461649
iteration: 1, cost: 443603
iteration: 2, cost: 436513
iteration: 3, cost: 432080
iteration: 4, cost: 423494
cost increased: 431930, 423494
T21 =
   0.999938  0.00146627    0.011054   0.0282033
-0.00152599    0.999984   0.0053958  0.00256267
 -0.0110459 -0.00541233    0.999924    -2.21468
          0           0           0           1
direct method for single layer: 0.0015141
iteration: 0, cost: 646880
iteration: 1, cost: 614318
iteration: 2, cost: 613113
cost increased: 620133, 613113
T21 =
   0.999935  0.00152579   0.0112714   0.0183767
-0.00158773    0.999984  0.00548783 -0.00540064
 -0.0112629 -0.00550537    0.999921    -2.23461
          0           0           0           1
direct method for single layer: 0.0011636
iteration: 0, cost: 924370
iteration: 1, cost: 828022
iteration: 2, cost: 821445
iteration: 3, cost: 803411
cost increased: 811368, 803411
T21 =
    0.999934   0.00125001    0.0114068   0.00255272
 -0.00131019     0.999985   0.00527034 -0.000605904
  -0.0114001  -0.00528494     0.999921     -2.24055
           0            0            0            1
direct method for single layer: 0.0015292
iteration: 0, cost: 1.43709e+06
iteration: 1, cost: 1.31501e+06
iteration: 2, cost: 1.06723e+06
iteration: 3, cost: 938977
iteration: 4, cost: 788005
iteration: 5, cost: 680776
iteration: 6, cost: 605861
iteration: 7, cost: 548408
iteration: 8, cost: 516721
iteration: 9, cost: 513621
T21 =
    0.999872 -0.000312873    0.0159856    0.0259369
 0.000197362     0.999974   0.00722705  -0.00480823
  -0.0159874  -0.00722297     0.999846     -2.96617
           0            0            0            1
direct method for single layer: 0.0048151
iteration: 0, cost: 640692
iteration: 1, cost: 616653
iteration: 2, cost: 610486
cost increased: 615297, 610486
T21 =
    0.999864 -0.000319108    0.0164719   0.00993795
 0.000208632     0.999977   0.00670821  -0.00627072
  -0.0164737  -0.00670386     0.999842       -3.005
           0            0            0            1
direct method for single layer: 0.0009756
iteration: 0, cost: 848724
iteration: 1, cost: 823518
iteration: 2, cost: 780844
cost increased: 802765, 780844
T21 =
    0.999865 -0.000227727    0.0164536    0.0022434
 0.000124997      0.99998    0.0062444  -0.00399514
  -0.0164547  -0.00624149     0.999845     -3.01734
           0            0            0            1
direct method for single layer: 0.0010828
iteration: 0, cost: 1.26838e+06
iteration: 1, cost: 1.16447e+06
cost increased: 1.19957e+06, 1.16447e+06
T21 =
    0.999865   0.00017071    0.0164584  -0.00906366
-0.000267333     0.999983   0.00586871  0.000576184
  -0.0164571  -0.00587231     0.999847     -3.02444
           0            0            0            1
direct method for single layer: 0.0008361
iteration: 0, cost: 1.64476e+06
iteration: 1, cost: 1.49383e+06
iteration: 2, cost: 1.23318e+06
iteration: 3, cost: 950472
iteration: 4, cost: 794112
iteration: 5, cost: 686345
iteration: 6, cost: 671817
iteration: 7, cost: 659908
iteration: 8, cost: 652671
iteration: 9, cost: 605440
T21 =
    0.999803   0.00057056    0.0198394    0.0427397
-0.000712283     0.999974   0.00713717    0.0136135
  -0.0198348  -0.00714989     0.999778     -3.76444
           0            0            0            1
direct method for single layer: 0.0088727
iteration: 0, cost: 983836
iteration: 1, cost: 948750
iteration: 2, cost: 945444
iteration: 3, cost: 895561
cost increased: 898341, 895561
T21 =
     0.99978  0.000643056    0.0209471  0.000477452
-0.000787155     0.999976   0.00687165   0.00707341
  -0.0209422  -0.00688663     0.999757     -3.83472
           0            0            0            1
direct method for single layer: 0.0012023
iteration: 0, cost: 1.27161e+06
iteration: 1, cost: 1.22543e+06
iteration: 2, cost: 1.04807e+06
cost increased: 1.2001e+06, 1.04807e+06
T21 =
   0.999777  0.00108579   0.0210816 -0.00872002
-0.00121752     0.99998  0.00623637   0.0124058
 -0.0210744 -0.00626065    0.999758    -3.85459
          0           0           0           1
direct method for single layer: 0.0010238
iteration: 0, cost: 1.67716e+06
iteration: 1, cost: 1.64927e+06
iteration: 2, cost: 1.63771e+06
cost increased: 1.64371e+06, 1.63771e+06
T21 =
   0.999786  0.00136909   0.0206569 -0.00336234
-0.00149442    0.999981   0.0060529  0.00874311
 -0.0206482 -0.00608247    0.999768    -3.86001
          0           0           0           1
direct method for single layer: 0.001018

3. 遇到的问题及解决办法

3.1 编译时遇到的问题

1. 在make时出现的有关opencv的问题：
   
   解决办法：按照之前的方法解决即可。查询链接：https://blog.csdn.net/qq_44164791/article/details/131210608?spm=1001.2014.3001.5502