自学SLAM（9）《第五讲：特征点法视觉里程计》作业

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1.ORB特征点

1.1 ORB提取

ORB(Oriented FAST and BRIEF) 特征是 SLAM 中⼀种很常⽤的特征，由于其⼆进制特性，使得它可以⾮常快速地提取与计算 [1]。下⾯，你将按照本题的指导，⾃⾏书写 ORB 的提取、描述⼦的计算以及匹配的代码。代码框架参照computeORB.cpp ⽂件，图像见 1.png ⽂件和 2.png。

1.2 ORB描述

1.3 暴力匹配

提⽰：

1. 你需要按位计算两个描述⼦之间的汉明距离。
2. OpenCV 的 DMatch 结构， queryIdx 为第⼀图的特征 ID， trainIdx 为第⼆个图的特征 ID。
3. 作为验证，匹配之后输出图像应如图 2 所⽰。

1.4 最后，请结合实验，回答下⾯⼏个问题

最后，请结合实验，回答下⾯⼏个问题：

1. 为什么说 ORB 是⼀种⼆进制特征？
2. 为什么在匹配时使⽤ 50 作为阈值，取更⼤或更⼩值会怎么样？
3. 暴⼒匹配在你的机器上表现如何？你能想到什么减少计算量的匹配⽅法吗？

我直接把1.1，1.2，1.3的代码和结果放到一起
computeORB.cpp：

#include 

#include 

using namespace std;

// global variables
string first_file = "../1.png";
string second_file = "../2.png";

const double pi = 3.1415926;    // pi


// TODO implement this function
/**
 * compute the angle for ORB descriptor
 * @param [in] image input image
 * @param [in|out] detected keypoints
 */
void computeAngle(const cv::Mat &image, vector<cv::KeyPoint> &keypoints);

// TODO implement this function
/**
 * compute ORB descriptor
 * @param [in] image the input image
 * @param [in] keypoints detected keypoints
 * @param [out] desc descriptor
 */
typedef vector<bool> DescType;  // type of descriptor, 256 bools
void computeORBDesc(const cv::Mat &image, vector<cv::KeyPoint> &keypoints, vector<DescType> &desc);

// TODO implement this function
/**
 * brute-force match two sets of descriptors
 * @param desc1 the first descriptor
 * @param desc2 the second descriptor
 * @param matches matches of two images
 */
void bfMatch(const vector<DescType> &desc1, const vector<DescType> &desc2, vector<cv::DMatch> &matches);

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

    // load image
    cv::Mat first_image = cv::imread(first_file, 0);    // load grayscale image
    cv::Mat second_image = cv::imread(second_file, 0);  // load grayscale image

    // plot the image
    cv::imshow("first image", first_image);
    cv::imshow("second image", second_image);
    cv::waitKey(0);

    // detect FAST keypoints using threshold=40
    vector<cv::KeyPoint> keypoints;
    cv::FAST(first_image, keypoints, 40);
    cout << "keypoints: " << keypoints.size() << endl;

    // compute angle for each keypoint
    computeAngle(first_image, keypoints);

    // compute ORB descriptors
    vector<DescType> descriptors;
    computeORBDesc(first_image, keypoints, descriptors);

    // plot the keypoints
    cv::Mat image_show;
    cv::drawKeypoints(first_image, keypoints, image_show, cv::Scalar::all(-1),
                      cv::DrawMatchesFlags::DRAW_RICH_KEYPOINTS);
    cv::imshow("features", image_show);
    cv::imwrite("feat1.png", image_show);
    cv::waitKey(0);

    // we can also match descriptors between images
    // same for the second
    vector<cv::KeyPoint> keypoints2;
    cv::FAST(second_image, keypoints2, 40);
    cout << "keypoints: " << keypoints2.size() << endl;

    // compute angle for each keypoint
    computeAngle(second_image, keypoints2);

    // compute ORB descriptors
    vector<DescType> descriptors2;
    computeORBDesc(second_image, keypoints2, descriptors2);

    // find matches
    vector<cv::DMatch> matches;
    bfMatch(descriptors, descriptors2, matches);
    cout << "matches: " << matches.size() << endl;

    // plot the matches
    cv::drawMatches(first_image, keypoints, second_image, keypoints2, matches, image_show);
    cv::imshow("matches", image_show);
    cv::imwrite("matches.png", image_show);
    cv::waitKey(0);

    cout << "done." << endl;
    return 0;
}

// -------------------------------------------------------------------------------------------------- //

// compute the angle
void computeAngle(const cv::Mat &image, vector<cv::KeyPoint> &keypoints) {
    int half_patch_size = 8;
    for (auto &kp : keypoints) {
	// START YOUR CODE HERE (~7 lines)
         //judge if keypoint is on edge
        int x=cvRound(kp.pt.x);
        int y=cvRound(kp.pt.y);
        if( x-half_patch_size<0||x+half_patch_size>image.cols||
            y-half_patch_size<0||y+half_patch_size>image.rows)
            continue;  //结束当前循环，进入到下一次循环
        double m01=0,m10=0;   //定义变量的时候，要初始化，不然这里第一张图片所有kp.angle=0
        for(int i=-half_patch_size;i<half_patch_size;i++){    //-8
            for(int j=-half_patch_size;j<half_patch_size;j++){
                m01 += j*image.at<uchar>(y+j,x+i);              //真实坐标(j,i)+(y,x)
                m10 += i*image.at<uchar>(y+j,x+i);              //获得单个像素值image.at(y,x)
            }
        }
        kp.angle = atan(m01/m10)*180/pi;
        cout<<"m10 = "<<m01<<"   "<<"m01 = "<<m10<<"  "<<"kp.angle = "<<kp.angle<<endl;
        // END YOUR CODE HERE
    }
    return;
}

// -------------------------------------------------------------------------------------------------- //
// ORB pattern
int ORB_pattern[256 * 4] = {
        8, -3, 9, 5/*mean (0), correlation (0)*/,
        4, 2, 7, -12/*mean (1.12461e-05), correlation (0.0437584)*/,
        -11, 9, -8, 2/*mean (3.37382e-05), correlation (0.0617409)*/,
        7, -12, 12, -13/*mean (5.62303e-05), correlation (0.0636977)*/,
        2, -13, 2, 12/*mean (0.000134953), correlation (0.085099)*/,
        1, -7, 1, 6/*mean (0.000528565), correlation (0.0857175)*/,
        -2, -10, -2, -4/*mean (0.0188821), correlation (0.0985774)*/,
        -13, -13, -11, -8/*mean (0.0363135), correlation (0.0899616)*/,
        -13, -3, -12, -9/*mean (0.121806), correlation (0.099849)*/,
        10, 4, 11, 9/*mean (0.122065), correlation (0.093285)*/,
        -13, -8, -8, -9/*mean (0.162787), correlation (0.0942748)*/,
        -11, 7, -9, 12/*mean (0.21561), correlation (0.0974438)*/,
        7, 7, 12, 6/*mean (0.160583), correlation (0.130064)*/,
        -4, -5, -3, 0/*mean (0.228171), correlation (0.132998)*/,
        -13, 2, -12, -3/*mean (0.00997526), correlation (0.145926)*/,
        -9, 0, -7, 5/*mean (0.198234), correlation (0.143636)*/,
        12, -6, 12, -1/*mean (0.0676226), correlation (0.16689)*/,
        -3, 6, -2, 12/*mean (0.166847), correlation (0.171682)*/,
        -6, -13, -4, -8/*mean (0.101215), correlation (0.179716)*/,
        11, -13, 12, -8/*mean (0.200641), correlation (0.192279)*/,
        4, 7, 5, 1/*mean (0.205106), correlation (0.186848)*/,
        5, -3, 10, -3/*mean (0.234908), correlation (0.192319)*/,
        3, -7, 6, 12/*mean (0.0709964), correlation (0.210872)*/,
        -8, -7, -6, -2/*mean (0.0939834), correlation (0.212589)*/,
        -2, 11, -1, -10/*mean (0.127778), correlation (0.20866)*/,
        -13, 12, -8, 10/*mean (0.14783), correlation (0.206356)*/,
        -7, 3, -5, -3/*mean (0.182141), correlation (0.198942)*/,
        -4, 2, -3, 7/*mean (0.188237), correlation (0.21384)*/,
        -10, -12, -6, 11/*mean (0.14865), correlation (0.23571)*/,
        5, -12, 6, -7/*mean (0.222312), correlation (0.23324)*/,
        5, -6, 7, -1/*mean (0.229082), correlation (0.23389)*/,
        1, 0, 4, -5/*mean (0.241577), correlation (0.215286)*/,
        9, 11, 11, -13/*mean (0.00338507), correlation (0.251373)*/,
        4, 7, 4, 12/*mean (0.131005), correlation (0.257622)*/,
        2, -1, 4, 4/*mean (0.152755), correlation (0.255205)*/,
        -4, -12, -2, 7/*mean (0.182771), correlation (0.244867)*/,
        -8, -5, -7, -10/*mean (0.186898), correlation (0.23901)*/,
        4, 11, 9, 12/*mean (0.226226), correlation (0.258255)*/,
        0, -8, 1, -13/*mean (0.0897886), correlation (0.274827)*/,
        -13, -2, -8, 2/*mean (0.148774), correlation (0.28065)*/,
        -3, -2, -2, 3/*mean (0.153048), correlation (0.283063)*/,
        -6, 9, -4, -9/*mean (0.169523), correlation (0.278248)*/,
        8, 12, 10, 7/*mean (0.225337), correlation (0.282851)*/,
        0, 9, 1, 3/*mean (0.226687), correlation (0.278734)*/,
        7, -5, 11, -10/*mean (0.00693882), correlation (0.305161)*/,
        -13, -6, -11, 0/*mean (0.0227283), correlation (0.300181)*/,
        10, 7, 12, 1/*mean (0.125517), correlation (0.31089)*/,
        -6, -3, -6, 12/*mean (0.131748), correlation (0.312779)*/,
        10, -9, 12, -4/*mean (0.144827), correlation (0.292797)*/,
        -13, 8, -8, -12/*mean (0.149202), correlation (0.308918)*/,
        -13, 0, -8, -4/*mean (0.160909), correlation (0.310013)*/,
        3, 3, 7, 8/*mean (0.177755), correlation (0.309394)*/,
        5, 7, 10, -7/*mean (0.212337), correlation (0.310315)*/,
        -1, 7, 1, -12/*mean (0.214429), correlation (0.311933)*/,
        3, -10, 5, 6/*mean (0.235807), correlation (0.313104)*/,
        2, -4, 3, -10/*mean (0.00494827), correlation (0.344948)*/,
        -13, 0, -13, 5/*mean (0.0549145), correlation (0.344675)*/,
        -13, -7, -12, 12/*mean (0.103385), correlation (0.342715)*/,
        -13, 3, -11, 8/*mean (0.134222), correlation (0.322922)*/,
        -7, 12, -4, 7/*mean (0.153284), correlation (0.337061)*/,
        6, -10, 12, 8/*mean (0.154881), correlation (0.329257)*/,
        -9, -1, -7, -6/*mean (0.200967), correlation (0.33312)*/,
        -2, -5, 0, 12/*mean (0.201518), correlation (0.340635)*/,
        -12, 5, -7, 5/*mean (0.207805), correlation (0.335631)*/,
        3, -10, 8, -13/*mean (0.224438), correlation (0.34504)*/,
        -7, -7, -4, 5/*mean (0.239361), correlation (0.338053)*/,
        -3, -2, -1, -7/*mean (0.240744), correlation (0.344322)*/,
        2, 9, 5, -11/*mean (0.242949), correlation (0.34145)*/,
        -11, -13, -5, -13/*mean (0.244028), correlation (0.336861)*/,
        -1, 6, 0, -1/*mean (0.247571), correlation (0.343684)*/,
        5, -3, 5, 2/*mean (0.000697256), correlation (0.357265)*/,
        -4, -13, -4, 12/*mean (0.00213675), correlation (0.373827)*/,
        -9, -6, -9, 6/*mean (0.0126856), correlation (0.373938)*/,
        -12, -10, -8, -4/*mean (0.0152497), correlation (0.364237)*/,
        10, 2, 12, -3/*mean (0.0299933), correlation (0.345292)*/,
        7, 12, 12, 12/*mean (0.0307242), correlation (0.366299)*/,
        -7, -13, -6, 5/*mean (0.0534975), correlation (0.368357)*/,
        -4, 9, -3, 4/*mean (0.099865), correlation (0.372276)*/,
        7, -1, 12, 2/*mean (0.117083), correlation (0.364529)*/,
        -7, 6, -5, 1/*mean (0.126125), correlation (0.369606)*/,
        -13, 11, -12, 5/*mean (0.130364), correlation (0.358502)*/,
        -3, 7, -2, -6/*mean (0.131691), correlation (0.375531)*/,
        7, -8, 12, -7/*mean (0.160166), correlation (0.379508)*/,
        -13, -7, -11, -12/*mean (0.167848), correlation (0.353343)*/,
        1, -3, 12, 12/*mean (0.183378), correlation (0.371916)*/,
        2, -6, 3, 0/*mean (0.228711), correlation (0.371761)*/,
        -4, 3, -2, -13/*mean (0.247211), correlation (0.364063)*/,
        -1, -13, 1, 9/*mean (0.249325), correlation (0.378139)*/,
        7, 1, 8, -6/*mean (0.000652272), correlation (0.411682)*/,
        1, -1, 3, 12/*mean (0.00248538), correlation (0.392988)*/,
        9, 1, 12, 6/*mean (0.0206815), correlation (0.386106)*/,
        -1, -9, -1, 3/*mean (0.0364485), correlation (0.410752)*/,
        -13, -13, -10, 5/*mean (0.0376068), correlation (0.398374)*/,
        7, 7, 10, 12/*mean (0.0424202), correlation (0.405663)*/,
        12, -5, 12, 9/*mean (0.0942645), correlation (0.410422)*/,
        6, 3, 7, 11/*mean (0.1074), correlation (0.413224)*/,
        5, -13, 6, 10/*mean (0.109256), correlation (0.408646)*/,
        2, -12, 2, 3/*mean (0.131691), correlation (0.416076)*/,
        3, 8, 4, -6/*mean (0.165081), correlation (0.417569)*/,
        2, 6, 12, -13/*mean (0.171874), correlation (0.408471)*/,
        9, -12, 10, 3/*mean (0.175146), correlation (0.41296)*/,
        -8, 4, -7, 9/*mean (0.183682), correlation (0.402956)*/,
        -11, 12, -4, -6/*mean (0.184672), correlation (0.416125)*/,
        1, 12, 2, -8/*mean (0.191487), correlation (0.386696)*/,
        6, -9, 7, -4/*mean (0.192668), correlation (0.394771)*/,
        2, 3, 3, -2/*mean (0.200157), correlation (0.408303)*/,
        6, 3, 11, 0/*mean (0.204588), correlation (0.411762)*/,
        3, -3, 8, -8/*mean (0.205904), correlation (0.416294)*/,
        7, 8, 9, 3/*mean (0.213237), correlation (0.409306)*/,
        -11, -5, -6, -4/*mean (0.243444), correlation (0.395069)*/,
        -10, 11, -5, 10/*mean (0.247672), correlation (0.413392)*/,
        -5, -8, -3, 12/*mean (0.24774), correlation (0.411416)*/,
        -10, 5, -9, 0/*mean (0.00213675), correlation (0.454003)*/,
        8, -1, 12, -6/*mean (0.0293635), correlation (0.455368)*/,
        4, -6, 6, -11/*mean (0.0404971), correlation (0.457393)*/,
        -10, 12, -8, 7/*mean (0.0481107), correlation (0.448364)*/,
        4, -2, 6, 7/*mean (0.050641), correlation (0.455019)*/,
        -2, 0, -2, 12/*mean (0.0525978), correlation (0.44338)*/,
        -5, -8, -5, 2/*mean (0.0629667), correlation (0.457096)*/,
        7, -6, 10, 12/*mean (0.0653846), correlation (0.445623)*/,
        -9, -13, -8, -8/*mean (0.0858749), correlation (0.449789)*/,
        -5, -13, -5, -2/*mean (0.122402), correlation (0.450201)*/,
        8, -8, 9, -13/*mean (0.125416), correlation (0.453224)*/,
        -9, -11, -9, 0/*mean (0.130128), correlation (0.458724)*/,
        1, -8, 1, -2/*mean (0.132467), correlation (0.440133)*/,
        7, -4, 9, 1/*mean (0.132692), correlation (0.454)*/,
        -2, 1, -1, -4/*mean (0.135695), correlation (0.455739)*/,
        11, -6, 12, -11/*mean (0.142904), correlation (0.446114)*/,
        -12, -9, -6, 4/*mean (0.146165), correlation (0.451473)*/,
        3, 7, 7, 12/*mean (0.147627), correlation (0.456643)*/,
        5, 5, 10, 8/*mean (0.152901), correlation (0.455036)*/,
        0, -4, 2, 8/*mean (0.167083), correlation (0.459315)*/,
        -9, 12, -5, -13/*mean (0.173234), correlation (0.454706)*/,
        0, 7, 2, 12/*mean (0.18312), correlation (0.433855)*/,
        -1, 2, 1, 7/*mean (0.185504), correlation (0.443838)*/,
        5, 11, 7, -9/*mean (0.185706), correlation (0.451123)*/,
        3, 5, 6, -8/*mean (0.188968), correlation (0.455808)*/,
        -13, -4, -8, 9/*mean (0.191667), correlation (0.459128)*/,
        -5, 9, -3, -3/*mean (0.193196), correlation (0.458364)*/,
        -4, -7, -3, -12/*mean (0.196536), correlation (0.455782)*/,
        6, 5, 8, 0/*mean (0.1972), correlation (0.450481)*/,
        -7, 6, -6, 12/*mean (0.199438), correlation (0.458156)*/,
        -13, 6, -5, -2/*mean (0.211224), correlation (0.449548)*/,
        1, -10, 3, 10/*mean (0.211718), correlation (0.440606)*/,
        4, 1, 8, -4/*mean (0.213034), correlation (0.443177)*/,
        -2, -2, 2, -13/*mean (0.234334), correlation (0.455304)*/,
        2, -12, 12, 12/*mean (0.235684), correlation (0.443436)*/,
        -2, -13, 0, -6/*mean (0.237674), correlation (0.452525)*/,
        4, 1, 9, 3/*mean (0.23962), correlation (0.444824)*/,
        -6, -10, -3, -5/*mean (0.248459), correlation (0.439621)*/,
        -3, -13, -1, 1/*mean (0.249505), correlation (0.456666)*/,
        7, 5, 12, -11/*mean (0.00119208), correlation (0.495466)*/,
        4, -2, 5, -7/*mean (0.00372245), correlation (0.484214)*/,
        -13, 9, -9, -5/*mean (0.00741116), correlation (0.499854)*/,
        7, 1, 8, 6/*mean (0.0208952), correlation (0.499773)*/,
        7, -8, 7, 6/*mean (0.0220085), correlation (0.501609)*/,
        -7, -4, -7, 1/*mean (0.0233806), correlation (0.496568)*/,
        -8, 11, -7, -8/*mean (0.0236505), correlation (0.489719)*/,
        -13, 6, -12, -8/*mean (0.0268781), correlation (0.503487)*/,
        2, 4, 3, 9/*mean (0.0323324), correlation (0.501938)*/,
        10, -5, 12, 3/*mean (0.0399235), correlation (0.494029)*/,
        -6, -5, -6, 7/*mean (0.0420153), correlation (0.486579)*/,
        8, -3, 9, -8/*mean (0.0548021), correlation (0.484237)*/,
        2, -12, 2, 8/*mean (0.0616622), correlation (0.496642)*/,
        -11, -2, -10, 3/*mean (0.0627755), correlation (0.498563)*/,
        -12, -13, -7, -9/*mean (0.0829622), correlation (0.495491)*/,
        -11, 0, -10, -5/*mean (0.0843342), correlation (0.487146)*/,
        5, -3, 11, 8/*mean (0.0929937), correlation (0.502315)*/,
        -2, -13, -1, 12/*mean (0.113327), correlation (0.48941)*/,
        -1, -8, 0, 9/*mean (0.132119), correlation (0.467268)*/,
        -13, -11, -12, -5/*mean (0.136269), correlation (0.498771)*/,
        -10, -2, -10, 11/*mean (0.142173), correlation (0.498714)*/,
        -3, 9, -2, -13/*mean (0.144141), correlation (0.491973)*/,
        2, -3, 3, 2/*mean (0.14892), correlation (0.500782)*/,
        -9, -13, -4, 0/*mean (0.150371), correlation (0.498211)*/,
        -4, 6, -3, -10/*mean (0.152159), correlation (0.495547)*/,
        -4, 12, -2, -7/*mean (0.156152), correlation (0.496925)*/,
        -6, -11, -4, 9/*mean (0.15749), correlation (0.499222)*/,
        6, -3, 6, 11/*mean (0.159211), correlation (0.503821)*/,
        -13, 11, -5, 5/*mean (0.162427), correlation (0.501907)*/,
        11, 11, 12, 6/*mean (0.16652), correlation (0.497632)*/,
        7, -5, 12, -2/*mean (0.169141), correlation (0.484474)*/,
        -1, 12, 0, 7/*mean (0.169456), correlation (0.495339)*/,
        -4, -8, -3, -2/*mean (0.171457), correlation (0.487251)*/,
        -7, 1, -6, 7/*mean (0.175), correlation (0.500024)*/,
        -13, -12, -8, -13/*mean (0.175866), correlation (0.497523)*/,
        -7, -2, -6, -8/*mean (0.178273), correlation (0.501854)*/,
        -8, 5, -6, -9/*mean (0.181107), correlation (0.494888)*/,
        -5, -1, -4, 5/*mean (0.190227), correlation (0.482557)*/,
        -13, 7, -8, 10/*mean (0.196739), correlation (0.496503)*/,
        1, 5, 5, -13/*mean (0.19973), correlation (0.499759)*/,
        1, 0, 10, -13/*mean (0.204465), correlation (0.49873)*/,
        9, 12, 10, -1/*mean (0.209334), correlation (0.49063)*/,
        5, -8, 10, -9/*mean (0.211134), correlation (0.503011)*/,
        -1, 11, 1, -13/*mean (0.212), correlation (0.499414)*/,
        -9, -3, -6, 2/*mean (0.212168), correlation (0.480739)*/,
        -1, -10, 1, 12/*mean (0.212731), correlation (0.502523)*/,
        -13, 1, -8, -10/*mean (0.21327), correlation (0.489786)*/,
        8, -11, 10, -6/*mean (0.214159), correlation (0.488246)*/,
        2, -13, 3, -6/*mean (0.216993), correlation (0.50287)*/,
        7, -13, 12, -9/*mean (0.223639), correlation (0.470502)*/,
        -10, -10, -5, -7/*mean (0.224089), correlation (0.500852)*/,
        -10, -8, -8, -13/*mean (0.228666), correlation (0.502629)*/,
        4, -6, 8, 5/*mean (0.22906), correlation (0.498305)*/,
        3, 12, 8, -13/*mean (0.233378), correlation (0.503825)*/,
        -4, 2, -3, -3/*mean (0.234323), correlation (0.476692)*/,
        5, -13, 10, -12/*mean (0.236392), correlation (0.475462)*/,
        4, -13, 5, -1/*mean (0.236842), correlation (0.504132)*/,
        -9, 9, -4, 3/*mean (0.236977), correlation (0.497739)*/,
        0, 3, 3, -9/*mean (0.24314), correlation (0.499398)*/,
        -12, 1, -6, 1/*mean (0.243297), correlation (0.489447)*/,
        3, 2, 4, -8/*mean (0.00155196), correlation (0.553496)*/,
        -10, -10, -10, 9/*mean (0.00239541), correlation (0.54297)*/,
        8, -13, 12, 12/*mean (0.0034413), correlation (0.544361)*/,
        -8, -12, -6, -5/*mean (0.003565), correlation (0.551225)*/,
        2, 2, 3, 7/*mean (0.00835583), correlation (0.55285)*/,
        10, 6, 11, -8/*mean (0.00885065), correlation (0.540913)*/,
        6, 8, 8, -12/*mean (0.0101552), correlation (0.551085)*/,
        -7, 10, -6, 5/*mean (0.0102227), correlation (0.533635)*/,
        -3, -9, -3, 9/*mean (0.0110211), correlation (0.543121)*/,
        -1, -13, -1, 5/*mean (0.0113473), correlation (0.550173)*/,
        -3, -7, -3, 4/*mean (0.0140913), correlation (0.554774)*/,
        -8, -2, -8, 3/*mean (0.017049), correlation (0.55461)*/,
        4, 2, 12, 12/*mean (0.01778), correlation (0.546921)*/,
        2, -5, 3, 11/*mean (0.0224022), correlation (0.549667)*/,
        6, -9, 11, -13/*mean (0.029161), correlation (0.546295)*/,
        3, -1, 7, 12/*mean (0.0303081), correlation (0.548599)*/,
        11, -1, 12, 4/*mean (0.0355151), correlation (0.523943)*/,
        -3, 0, -3, 6/*mean (0.0417904), correlation (0.543395)*/,
        4, -11, 4, 12/*mean (0.0487292), correlation (0.542818)*/,
        2, -4, 2, 1/*mean (0.0575124), correlation (0.554888)*/,
        -10, -6, -8, 1/*mean (0.0594242), correlation (0.544026)*/,
        -13, 7, -11, 1/*mean (0.0597391), correlation (0.550524)*/,
        -13, 12, -11, -13/*mean (0.0608974), correlation (0.55383)*/,
        6, 0, 11, -13/*mean (0.065126), correlation (0.552006)*/,
        0, -1, 1, 4/*mean (0.074224), correlation (0.546372)*/,
        -13, 3, -9, -2/*mean (0.0808592), correlation (0.554875)*/,
        -9, 8, -6, -3/*mean (0.0883378), correlation (0.551178)*/,
        -13, -6, -8, -2/*mean (0.0901035), correlation (0.548446)*/,
        5, -9, 8, 10/*mean (0.0949843), correlation (0.554694)*/,
        2, 7, 3, -9/*mean (0.0994152), correlation (0.550979)*/,
        -1, -6, -1, -1/*mean (0.10045), correlation (0.552714)*/,
        9, 5, 11, -2/*mean (0.100686), correlation (0.552594)*/,
        11, -3, 12, -8/*mean (0.101091), correlation (0.532394)*/,
        3, 0, 3, 5/*mean (0.101147), correlation (0.525576)*/,
        -1, 4, 0, 10/*mean (0.105263), correlation (0.531498)*/,
        3, -6, 4, 5/*mean (0.110785), correlation (0.540491)*/,
        -13, 0, -10, 5/*mean (0.112798), correlation (0.536582)*/,
        5, 8, 12, 11/*mean (0.114181), correlation (0.555793)*/,
        8, 9, 9, -6/*mean (0.117431), correlation (0.553763)*/,
        7, -4, 8, -12/*mean (0.118522), correlation (0.553452)*/,
        -10, 4, -10, 9/*mean (0.12094), correlation (0.554785)*/,
        7, 3, 12, 4/*mean (0.122582), correlation (0.555825)*/,
        9, -7, 10, -2/*mean (0.124978), correlation (0.549846)*/,
        7, 0, 12, -2/*mean (0.127002), correlation (0.537452)*/,
        -1, -6, 0, -11/*mean (0.127148), correlation (0.547401)*/
};

// compute the descriptor
void computeORBDesc(const cv::Mat &image, vector<cv::KeyPoint> &keypoints, vector<DescType> &desc) {
    for (auto &kp: keypoints) {
        DescType d(256, false);
        for (int i = 0; i < 256; i++) {
            // START YOUR CODE HERE (~7 lines)
            auto cos_ = float(cos(kp.angle*pi/180)); //将角度转换成弧度再进行cos、sin的计算
            auto sin_ = float(sin(kp.angle*pi/180));
            //注意pattern中的数如何取
            cv::Point2f p_r(cos_*ORB_pattern[4*i]-sin_*ORB_pattern[4*i+1],
                    sin_*ORB_pattern[4*i]+cos_*ORB_pattern[4*i+1]);
            cv::Point2f q_r(cos_*ORB_pattern[4*i+2]-sin_*ORB_pattern[4*i+3],
                    sin_*ORB_pattern[4*i+2]+cos_*ORB_pattern[4*i+3]);
 
            cv::Point2f p(kp.pt+p_r); //获取p'与q'的真实坐标，才能获得其像素值
            cv::Point2f q(kp.pt+q_r);
 
            // if kp goes outside, set d.clear()
            if(p.x<0||p.y<0||p.x>image.cols||p.y>image.rows||
            q.x<0||q.y<0||q.x>image.cols||q.y>image.rows){
                d.clear();
                break;
            }
            //像素值比较
             d[i]=image.at<uchar>(p)>image.at<uchar>(q)?0:1; 
	    // END YOUR CODE HERE
        }
        desc.push_back(d);
    }

    int bad = 0;
    for (auto &d: desc) {
        if (d.empty()) bad++;
    }
    cout << "bad/total: " << bad << "/" << desc.size() << endl;
    return;
}

// brute-force matching
void bfMatch(const vector<DescType> &desc1, const vector<DescType> &desc2, vector<cv::DMatch> &matches) {
    int d_max = 50;

    // START YOUR CODE HERE (~12 lines)
    // find matches between desc1 and desc2. 
    for(int i=0;i<desc1.size();i++){
        if(desc1[i].empty())
            continue;
        int d_min=256 ,index=-1; //必须定义在这里，每次循环重新初始化
        for(int j=0;j<desc2.size();j++){ //这个for循环，取出最小的d_min
            if(desc2[j].empty())
                continue;
            int d=0; //必须定义在这里，每次循环重新初始化
            for(int k=0;k<256;k++){
                d += desc1[i][k]^desc2[j][k]; //异或：不同为1;
            }
            if(d<d_min){
                d_min=d;
                index=j;
            }
        }
        if(d_min<=d_max){
            cv::DMatch match(i,index,d_min);
            matches.push_back(match);
        }
    }
    // END YOUR CODE HERE

    for (auto &m: matches) {
        cout << m.queryIdx << ", " << m.trainIdx << ", " << m.distance << endl;
    }
    return;
}

CMakeLists.txt：

cmake_minimum_required( VERSION 2.8 )
project(stereoVision)
set( CMAKE_CXX_FLAGS "-std=c++11 -O3")
 
include_directories("/usr/include/eigen3")
find_package(Pangolin REQUIRED)
include_directories( ${Pangolin_INCLUDE_DIRS} )
 
find_package(OpenCV 3.0 QUIET) #find_package(<Name>)命令首先会在模块路径中寻找 Find<name>.cmake
if(NOT OpenCV_FOUND)
    find_package(OpenCV 2.4.3 QUIET)
    if(NOT OpenCV_FOUND)
        message(FATAL_ERROR "OpenCV > 2.4.3 not found.")
    endif()
endif()

include_directories(${OpenCV_INCLUDE_DIRS})

add_executable(computeORB computeORB.cpp)
target_link_libraries(computeORB ${OpenCV_LIBS})

然后就是五件套
mkdir build
cd build
cmake …
make
./computeORB

然后是简答题：

1.为什么说 ORB 是⼀种⼆进制特征？
ORB使用改进的BRIEF特征描述，而BRIEF是一种二进制的描述子，其描述向量由许多个0和1组成。也就是说ORB采用二进制的描述子用来描述每个特征点的特征信息。

2.为什么在匹配时使⽤ 50 作为阈值，取更⼤或更⼩值会怎么样？ 当阈值为50的时候，可以检测出的特征对有95个匹配的特征对。但存在一些误匹配的点对。

当阈值为30的时候，可以检测到的特征点对很少，当然我还检测了20的时候，到20就一对特征点对也检测不出来了。

当阈值设置为90时,可以检测到非常多的个点对,误匹配很多

3.暴⼒匹配在你的机器上表现如何？你能想到什么减少计算量的匹配⽅法吗？
运行时间如图所示，使用快速近似最近邻的方法（FLANN）。

2.从 E 恢复 R，t

首先说一下运行中碰到的问题吧：
第一个问题：

如果碰到这个情况，那就是我们的E2Rt文件中找不到#include ，只需要改为#include 就可以了。
第二个问题：

如果是这种情况，我们就需要把E2Rt文件中62、63行的（文件中所有的）so3d改为so3即可。

E2Rt.cpp

#include 
#include 
#include 

using namespace Eigen;

#include 

#include 

using namespace std;

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

    // 给定Essential矩阵
    Matrix3d E;
    E << -0.0203618550523477, -0.4007110038118445, -0.03324074249824097,
            0.3939270778216369, -0.03506401846698079, 0.5857110303721015,
            -0.006788487241438284, -0.5815434272915686, -0.01438258684486258;

    // 待计算的R,t
    Matrix3d R;
    Vector3d t;

    // SVD and fix sigular values
    // START YOUR CODE HERE
      JacobiSVD<MatrixXd> svd(E,ComputeThinU | ComputeThinV);
    Matrix3d U=svd.matrixU();
    Matrix3d V=svd.matrixV();
    VectorXd sigma_value=svd.singularValues();
    Matrix3d SIGMA=U.inverse()*E*V.transpose().inverse();
    Vector3d sigma_value2={(sigma_value[0]+sigma_value[1])/2,(sigma_value[0]+sigma_value[1])/2,0};
    Matrix3d SIGMA2=sigma_value2.asDiagonal();
    cout<<"SIGMA=\n"<<SIGMA<<endl;
    cout<<"sigma_value=\n"<<sigma_value<<endl;
    cout<<"SIGMA2=\n"<<SIGMA<<endl;
    cout<<"sigma_value2=\n"<<sigma_value<<endl;
    // END YOUR CODE HERE

    // set t1, t2, R1, R2 
    // START YOUR CODE HERE
    Matrix3d t_wedge1;
    Matrix3d t_wedge2;

    Matrix3d R1;
    Matrix3d R2;
     Matrix3d RZ1=AngleAxisd(M_PI/2,Vector3d(0,0,1)).toRotationMatrix();
    Matrix3d RZ2=AngleAxisd(-M_PI/2,Vector3d(0,0,1)).toRotationMatrix();
    t_wedge1=U*RZ1*SIGMA2*U.transpose();
    t_wedge2=U*RZ2*SIGMA2*U.transpose();
    R1=U*RZ1.transpose()*V.transpose();
    R2=U*RZ2.transpose()*V.transpose();
    // END YOUR CODE HERE

    cout << "R1 = " << R1 << endl;
    cout << "R2 = " << R2 << endl;
    cout << "t1 = " << Sophus::SO3::vee(t_wedge1) << endl;
    cout << "t2 = " << Sophus::SO3::vee(t_wedge2) << endl;

    // check t^R=E up to scale
    Matrix3d tR = t_wedge1 * R1;
    cout << "t^R = " << tR << endl;

    return 0;
}

CMakeLists.txt：

cmake_minimum_required(VERSION 3.0)
project(E2RT)
set(CMAKE_CXX_STANDARD 11)
set(CMAKE_BUILD_TYPE "Release")

#添加头文件
include_directories( "/usr/include/eigen3")
find_package(Sophus REQUIRED)
include_directories(${Sophus_INCLUDE_DIRS})

add_executable(E2Rt E2Rt.cpp)
#链接OpenCV库
target_link_libraries(E2Rt ${Sophus_LIBRARIES})

运行结果如下：

3.用 G-N 实现 Bundle Adjustment

这里如果cmake …
make有问题的话，和上一题的解决方法是一样的。

GN-BA.cpp

#include 
#include 

using namespace Eigen;

#include 
#include 
#include 
#include 

#include "sophus/se3.h"

using namespace std;

typedef vector<Vector3d, Eigen::aligned_allocator<Vector3d>> VecVector3d;
typedef vector<Vector2d, Eigen::aligned_allocator<Vector3d>> VecVector2d;
typedef Matrix<double, 6, 1> Vector6d;

string p3d_file = "../p3d.txt";
string p2d_file = "../p2d.txt";

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

    VecVector2d p2d;
    VecVector3d p3d;
    Matrix3d K;
    double fx = 520.9, fy = 521.0, cx = 325.1, cy = 249.7;
    K << fx, 0, cx, 0, fy, cy, 0, 0, 1;

    // load points in to p3d and p2d 
    // START YOUR CODE HERE
    ifstream p3d_fin(p3d_file);
    ifstream p2d_fin(p2d_file);
    Vector3d p3d_input;
    Vector2d p2d_input;
    if (!p3d_fin) {
        cerr << "p3d_fin " << p3d_file << " not found." << endl;
    }
    while (!p3d_fin.eof()) {
        p3d_fin >> p3d_input(0) >> p3d_input(1) >> p3d_input(2);
        p3d.push_back(p3d_input);
    }
    p3d_fin.close();

    if (!p2d_fin) {
        cerr << "p2d_fin " << p2d_file << " not found." << endl;
    }
    while (!p2d_fin.eof()) {
        p2d_fin >> p2d_input(0) >> p2d_input(1);
        p2d.push_back(p2d_input);
    }
    p2d_fin.close();
    // END YOUR CODE HERE
    assert(p3d.size() == p2d.size());

    int iterations = 100;
    double cost = 0, lastCost = 0;
    int nPoints = p3d.size();
    cout << "points: " << nPoints << endl;

    Sophus::SE3 T_esti; // estimated pose

    for (int iter = 0; iter < iterations; iter++) {

        Matrix<double, 6, 6> H = Matrix<double, 6, 6>::Zero();
        Vector6d b = Vector6d::Zero();

        cost = 0;
        // compute cost
        for (int i = 0; i < nPoints; i++) {
            // compute cost for p3d[I] and p2d[I]
            // START YOUR CODE HERE 
            Eigen::Vector3d pc = T_esti * p3d[i];
            Eigen::Vector2d proj(fx * pc[0] / pc[2] + cx, fy * pc[1] / pc[2] + cy);
            Eigen::Vector2d e = p2d[i] - proj;

            cost += e.squaredNorm()/2;
	    // END YOUR CODE HERE

	    // compute jacobian
            Matrix<double, 2, 6> J;
            // START YOUR CODE HERE
            double inv_z = 1.0 / pc[2];
            double inv_z2 = inv_z * inv_z;
            J << -fx * inv_z,
                    0,
                    fx * pc[0] * inv_z2,
                    fx * pc[0] * pc[1] * inv_z2,
                    -fx - fx * pc[0] * pc[0] * inv_z2,
                    fx * pc[1] * inv_z,
                    0,
                    -fy * inv_z,
                    fy * pc[1] * inv_z2,
                    fy + fy * pc[1] * pc[1] * inv_z2,
                    -fy * pc[0] * pc[1] * inv_z2,
                    -fy * pc[0] * inv_z;
	    // END YOUR CODE HERE

            H += J.transpose() * J;
            b += -J.transpose() * e;
        }

	// solve dx 
        Vector6d dx;

        // START YOUR CODE HERE 
        dx = H.ldlt().solve(b);
        // END YOUR CODE HERE

        if (isnan(dx[0])) {
            cout << "result is nan!" << endl;
            break;
        }

        if (iter > 0 && cost >= lastCost) {
            // cost increase, update is not good
            cout << "cost: " << cost << ", last cost: " << lastCost << endl;
            break;
        }

        // update your estimation
        // START YOUR CODE HERE 
        T_esti = Sophus::SE3::exp(dx) * T_esti;
        // END YOUR CODE HERE
        
        lastCost = cost;

        cout << "iteration " << iter << " cost=" << cout.precision(12) << cost << endl;
    }

    cout << "estimated pose: \n" << T_esti.matrix() << endl;
    return 0;
}

CMakeLists.txt:

cmake_minimum_required(VERSION 3.0)

project(E2RT)
set(CMAKE_CXX_STANDARD 11)
set(CMAKE_BUILD_TYPE "Release")

#添加头文件
include_directories( "/usr/include/eigen3")
find_package(Sophus REQUIRED)
include_directories(${Sophus_INCLUDE_DIRS})


add_executable(gn_ba GN-BA.cpp)

#链接OpenCV库
target_link_libraries(gn_ba ${Sophus_LIBRARIES})

运行结果：

1.如何定义重投影误差？

像素位置与空间点的位置关系如下：

写成矩阵形式为S_iu_i=KTP_i

由于相机位姿未知及观测点的噪声，该等式存在一个误差。把误差求和，构建最小二乘问题，然后寻找最好的相机位姿，使它最小化：

将该问题的误差项，是将3D点的投影与观测位置做差，称之为重投影误差。

2.该误差关于⾃变量的雅可⽐矩阵是什么？

3.解出更新量之后，如何更新⾄之前的估计上？
左乘或右乘微小扰动exp(dx)
代码中为左乘

4.* 用 ICP 实现轨迹对齐

icp.cpp

#include 
#include 
#include 
#include 
#include 
#include 
#include 
#include 

using namespace std;
using namespace Eigen;
using namespace cv;

string trajectory_file = "../compare.txt";

void pose_estimation_3d3d(const vector<Point3f> &pts1,const vector<Point3f> &pts2, Eigen::Matrix3d &R_, Eigen::Vector3d &t_);
void DrawTrajectory(vector<Sophus::SE3, Eigen::aligned_allocator<Sophus::SE3>> poses_e,
        vector<Sophus::SE3, Eigen::aligned_allocator<Sophus::SE3>> poses_g,
        const string& ID);

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

    vector<Sophus::SE3, Eigen::aligned_allocator<Sophus::SE3>> poses_e;
    vector<Sophus::SE3, Eigen::aligned_allocator<Sophus::SE3>> poses_g;
    vector<Sophus::SE3, Eigen::aligned_allocator<Sophus::SE3>> poses_gt;
    vector<Point3f> pts_e,pts_g;
    ifstream fin(trajectory_file);
    if(!fin){
        cerr<<"can't find file at "<<trajectory_file<<endl;
        return 1;
    }
    while(!fin.eof()){
        double t1,tx1,ty1,tz1,qx1,qy1,qz1,qw1;
        double t2,tx2,ty2,tz2,qx2,qy2,qz2,qw2;
        fin>>t1>>tx1>>ty1>>tz1>>qx1>>qy1>>qz1>>qw1>>t2>>tx2>>ty2>>tz2>>qx2>>qy2>>qz2>>qw2;
        pts_e.push_back(Point3f(tx1,ty1,tz1));
        pts_g.push_back(Point3f(tx2,ty2,tz2));
        poses_e.push_back(Sophus::SE3(Quaterniond(qw1,qx1,qy1,qz1),Vector3d(tx1,ty1,tz1)));
        poses_g.push_back(Sophus::SE3(Quaterniond(qw2,qx2,qy2,qz2),Vector3d(tx2,ty2,tz2)));
    }

    Matrix3d R;
    Vector3d t;
    pose_estimation_3d3d(pts_e,pts_g,R,t);
    Sophus::SE3 T_eg(R,t);
    for(auto SE_g:poses_g)    {
        Sophus::SE3 T_e=T_eg*SE_g;
        poses_gt.push_back(T_e);
    }
    DrawTrajectory(poses_e,poses_g," Before Align");
    DrawTrajectory(poses_e,poses_gt," After Align");
    return 0;
}

void pose_estimation_3d3d(const vector<Point3f> &pts1,
                          const vector<Point3f> &pts2,
                          Eigen::Matrix3d &R_, Eigen::Vector3d &t_) {
    Point3f p1, p2;     // center of mass
    int N = pts1.size();
    for (int i = 0; i < N; i++) {
        p1 += pts1[i];
        p2 += pts2[i];
    }
    p1 = Point3f(Vec3f(p1) / N);
    p2 = Point3f(Vec3f(p2) / N);
    vector<Point3f> q1(N), q2(N); // remove the center
    for (int i = 0; i < N; i++) {
        q1[i] = pts1[i] - p1;
        q2[i] = pts2[i] - p2;
    }

    // compute q1*q2^T
    Eigen::Matrix3d W = Eigen::Matrix3d::Zero();
    for (int i = 0; i < N; i++) {
        W += Eigen::Vector3d(q1[i].x, q1[i].y, q1[i].z) * Eigen::Vector3d(q2[i].x, q2[i].y, q2[i].z).transpose();
    }
    cout << "W=" << W << endl;

    // SVD on W
    Eigen::JacobiSVD<Eigen::Matrix3d> svd(W, Eigen::ComputeFullU | Eigen::ComputeFullV);
    Eigen::Matrix3d U = svd.matrixU();
    Eigen::Matrix3d V = svd.matrixV();

    cout << "U=" << U << endl;
    cout << "V=" << V << endl;

    R_ = U * (V.transpose());
    if (R_.determinant() < 0) {
        R_ = -R_;
    }
    t_ = Eigen::Vector3d(p1.x, p1.y, p1.z) - R_ * Eigen::Vector3d(p2.x, p2.y, p2.z);
}

void DrawTrajectory(vector<Sophus::SE3, Eigen::aligned_allocator<Sophus::SE3>> poses_e,
        vector<Sophus::SE3, Eigen::aligned_allocator<Sophus::SE3>> poses_g,
        const string& ID) {
    if (poses_e.empty() || poses_g.empty()) {
        cerr << "Trajectory is empty!" << endl;
        return;
    }

    string windowtitle = "Trajectory Viewer" + ID;
    // create pangolin window and plot the trajectory
    pangolin::CreateWindowAndBind(windowtitle, 1024, 768);
    glEnable(GL_DEPTH_TEST);
    glEnable(GL_BLEND);
    glBlendFunc(GL_SRC_ALPHA, GL_ONE_MINUS_SRC_ALPHA);

    pangolin::OpenGlRenderState s_cam(
            pangolin::ProjectionMatrix(1024, 768, 500, 500, 512, 389, 0.1, 1000),
            pangolin::ModelViewLookAt(0, -0.1, -1.8, 0, 0, 0, 0.0, -1.0, 0.0)
    );

    pangolin::View &d_cam = pangolin::CreateDisplay()
            .SetBounds(0.0, 1.0, pangolin::Attach::Pix(175), 1.0, -1024.0f / 768.0f)
            .SetHandler(new pangolin::Handler3D(s_cam));


    while (pangolin::ShouldQuit() == false) {
        glClear(GL_COLOR_BUFFER_BIT | GL_DEPTH_BUFFER_BIT);

        d_cam.Activate(s_cam);
        glClearColor(1.0f, 1.0f, 1.0f, 1.0f);

        glLineWidth(2);
        for (size_t i = 0; i < poses_e.size() - 1; i++) {
            glColor3f(1.0f, 0.0f, 0.0f);
            glBegin(GL_LINES);
            auto p1 = poses_e[i], p2 = poses_e[i + 1];
            glVertex3d(p1.translation()[0], p1.translation()[1], p1.translation()[2]);
            glVertex3d(p2.translation()[0], p2.translation()[1], p2.translation()[2]);
            glEnd();
        }
        for (size_t i = 0; i < poses_g.size() - 1; i++) {
            glColor3f(0.0f, 0.0f, 1.0f);
            glBegin(GL_LINES);
            auto p1 = poses_g[i], p2 = poses_g[i + 1];
            glVertex3d(p1.translation()[0], p1.translation()[1], p1.translation()[2]);
            glVertex3d(p2.translation()[0], p2.translation()[1], p2.translation()[2]);
            glEnd();
        }
        pangolin::FinishFrame();
        usleep(5000);   // sleep 5 ms
    }
}

compare.txt:

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1305031547.812317 -0.964506626 0.684349477 -0.429410160 -0.117394648 -0.022055248 -0.071717598 0.990246713 1305031547.812700 0.8942 -0.1371 1.2441 0.9011 0.2310 -0.0996 -0.3532
1305031547.844564 -0.962979019 0.686277032 -0.429381162 -0.114744045 -0.021526016 -0.070772044 0.990637004 1305031547.842800 0.8946 -0.1365 1.2442 0.9008 0.2305 -0.0994 -0.3543
1305031547.876362 -0.963006616 0.688939631 -0.429479182 -0.113952808 -0.021306587 -0.070929334 0.990721881 1305031547.872800 0.8950 -0.1360 1.2442 0.9016 0.2311 -0.0993 -0.3520
1305031547.912744 -0.960898936 0.685296535 -0.428291798 -0.116005875 -0.021983700 -0.069795489 0.990549266 1305031547.912800 0.8956 -0.1352 1.2445 0.9016 0.2297 -0.0992 -0.3529
1305031547.944304 -0.960632503 0.685537279 -0.428972363 -0.115375742 -0.021099383 -0.069957919 0.990630627 1305031547.943100 0.8959 -0.1347 1.2447 0.9019 0.2297 -0.0988 -0.3522

CMakeLists.txt:

cmake_minimum_required(VERSION 3.0)

project(E2RT)
set(CMAKE_CXX_STANDARD 11)
set(CMAKE_BUILD_TYPE "Release")

#添加头文件
include_directories( "/usr/include/eigen3")
find_package(Sophus REQUIRED)
find_package(Pangolin REQUIRED)
find_package(OpenCV REQUIRED)
#添加头文件
include_directories( ${OpenCV_INCLUDE_DIRS})
include_directories(${Pangolin_INCLUDE_DIRS})
include_directories(${Sophus_INCLUDE_DIRS})


add_executable(useICP icp.cpp)

#链接OpenCV库
target_link_libraries(useICP ${Sophus_LIBRARIES} ${Pangolin_LIBRARIES} ${OpenCV_LIBS})

运行结果如下：