1、最小二乘题目：

        假设有三个WIFI热点，位置分别在(x1,y1), (x2,y2), (x3,y3), 移动端测量到每一个热点的距离L1,L2和L3，要求解移动端的位置. 

#include <iostream>
#include <vector>
#include <cmath> 

class Point {
public:
    double x, y;
};

Point test(const std::vector<Point>& location, const std::vector<double>& distances) {

    double x1 = location[0].x, y1 = location[0].y, L1 = distances[0];
    double x2 = location[1].x, y2 = location[1].y, L2 = distances[1];
    double x3 = location[2].x, y3 = location[2].y, L3 = distances[2];

    double a1 = 2 * (x2 - x1);
    double b1 = 2 * (y2 - y1);
    double c1 = L1 * L1 - L2 * L2 + x2 * x2 - x1 * x1 + y2 * y2 - y1 * y1;

    double a2 = 2 * (x3 - x1);
    double b2 = 2 * (y3 - y1);
    double c2 = L1 * L1 - L3 * L3 + x3 * x3 - x1 * x1 + y3 * y3 - y1 * y1;

    Point result;
    result.x = (c1 * b2 - c2 * b1) / (a1 * b2 - a2 * b1);
    result.y = (a1 * c2 - a2 * c1) / (a1 * b2 - a2 * b1);

    return result;
}

int main() {
    std::vector<Point> location = {{0, 0}, {1, 0}, {0, 1}};
    double sqrt_2 = std::sqrt(2.0);
    std::vector<double> distances = {sqrt_2, 1.0, 1.0};
    Point result = test(location, distances);
    std::cout << "移动端的位置是 (" << result.x << ", " << result.y << ")" << std::endl;
    return 0;
}

2、图像题目

已知相机内参：520.9, 0, 325.1, 0, 521.0, 249.7, 0, 0, 1。求解两帧图像之间的运动。

    a.部署OpenCV等环境，使用熟悉的特征点匹配后计算。

    b.运动求解过程需要亲自编写程序，不允许调用第三方库。

CMakeLists.txt

cmake_minimum_required(VERSION 2.8)
project(vo1)

set(CMAKE_BUILD_TYPE "Release")
add_definitions("-DENABLE_SSE")
set(CMAKE_CXX_FLAGS "-std=c++14 -O2 ${SSE_FLAGS} -msse4")
#list(APPEND CMAKE_MODULE_PATH ${PROJECT_SOURCE_DIR}/cmake)

find_package(OpenCV 3.4.15 REQUIRED)
#find_package(G2O REQUIRED)
#find_package(Sophus REQUIRED)

include_directories(
        ${OpenCV_INCLUDE_DIRS}
       # ${G2O_INCLUDE_DIRS}
       # ${Sophus_INCLUDE_DIRS}
        "/usr/include/eigen3/"
)

# add_executable( pose_estimation_2d2d pose_estimation_2d2d.cpp extra.cpp ) # use this if in OpenCV2 
add_executable(pose_estimation_2d2d pose_estimation_2d2d.cpp)
target_link_libraries(pose_estimation_2d2d ${OpenCV_LIBS})

主函数

#include <iostream>
#include <opencv2/core/core.hpp>
#include <opencv2/features2d/features2d.hpp>
#include <opencv2/highgui/highgui.hpp>
#include <opencv2/calib3d/calib3d.hpp>
#include <vector>
#include <random>
#include <ctime>
#include <Eigen/Dense>
using namespace std;
using namespace cv;
using namespace Eigen;

void find_feature_matches(
  const Mat &img_1, const Mat &img_2,
  std::vector<KeyPoint> &keypoints_1,
  std::vector<KeyPoint> &keypoints_2,
  std::vector<DMatch> &matches);

void pose_estimation_2d2d(
  std::vector<KeyPoint> keypoints_1,
  std::vector<KeyPoint> keypoints_2,
  std::vector<DMatch> matches,
  Mat &R, Mat &t);

const int MAX_ITER = 100000;  // 最大迭代次数
const float THRESHOLD = 0.001;  // 内点距离阈值


float calculateError(const Matrix3f& F, const Point2f& pt1, const Point2f& pt2) {
    Vector3f p1(pt1.x, pt1.y, 1.0);
    Vector3f p2(pt2.x, pt2.y, 1.0);
    Vector3f line = F * p1;
    return fabs(p2.dot(line)) / sqrt(line[0] * line[0] + line[1] * line[1]);
}

Matrix3f ransacFundamentalMat(const vector<Point2f>& points1, const vector<Point2f>& points2) {
    int bestInliersCount = 0;
    Matrix3f bestF;
    std::default_random_engine rng(time(0));
    std::uniform_int_distribution<int> uniDist(0, points1.size() - 1);

    for (int iter = 0; iter < MAX_ITER; ++iter) {
        vector<Point2f> samplePoints1, samplePoints2;
        for (int i = 0; i < 8; ++i) {
            int idx = uniDist(rng);
            samplePoints1.push_back(points1[idx]);
            samplePoints2.push_back(points2[idx]);
        }

        int N = samplePoints1.size();
        MatrixXf A(N, 9);

        for (int i = 0; i < N; ++i) {
            float x1 = samplePoints1[i].x;
            float y1 = samplePoints1[i].y;
            float x2 = samplePoints2[i].x;
            float y2 = samplePoints2[i].y;
            A.row(i) << x2 * x1, x2 * y1, x2, y2 * x1, y2 * y1, y2, x1, y1, 1;
        }

        JacobiSVD<MatrixXf> svd(A, ComputeFullV);
        VectorXf f = svd.matrixV().col(8);

        Matrix3f F;
        F << f(0), f(1), f(2),
                f(3), f(4), f(5),
                f(6), f(7), f(8);

        JacobiSVD<Matrix3f> svdF(F, ComputeFullU | ComputeFullV);
        Vector3f singularValues = svdF.singularValues();
        singularValues[2] = 0;
        F = svdF.matrixU() * singularValues.asDiagonal() * svdF.matrixV().transpose();

        int inliersCount = 0;
        for (int i = 0; i < points1.size(); ++i) {
            if (calculateError(F, points1[i], points2[i]) < THRESHOLD) {
                ++inliersCount;
            }
        }

        if (inliersCount > bestInliersCount) {
            bestInliersCount = inliersCount;
            cout<<bestInliersCount<<endl;
            bestF = F;
        }
    }

    return bestF;
}
void recoverPose_my(const Matrix3f& essential_matrix, const vector<Point2f>& points1, const vector<Point2f>& points2, Matrix3f& R, Vector3f& t, double focal_length, const Point2d& principal_point) {
    // SVD分解本质矩阵
    JacobiSVD<Matrix3f> svd(essential_matrix, ComputeFullU | ComputeFullV);
    Matrix3f U = svd.matrixU();
    Matrix3f Vt = svd.matrixV().transpose();

    // 构造W矩阵和Z矩阵
    Matrix3f W;
    W << 0, -1, 0,
            1, 0, 0,
            0, 0, 1;

    Matrix3f Z;
    Z << 0, 1, 0,
            -1, 0, 0,
            0, 0, 0;

    // 可能的两种旋转矩阵
    Matrix3f R1 = U * W * Vt;
    if (R1.determinant() < 0) {
        R1 = -R1;
    }
    Matrix3f R2 = U * W.transpose() * Vt;
    if (R2.determinant() < 0) {
        R2 = -R2;
    }

    // 平移向量
    Vector3f t1 = U.col(2);
    Vector3f t2 = -U.col(2);

    // 对两个可能的解进行验证，选择最合适的那个
    int valid_count1 = 0;
    int valid_count2 = 0;

    for (int i = 0; i < points1.size(); ++i) {
        Vector3f p1(points1[i].x, points1[i].y, 1.0);
        Vector3f p2(points2[i].x, points2[i].y, 1.0);

        p1[0] = (p1[0] - principal_point.x) / focal_length;
        p1[1] = (p1[1] - principal_point.y) / focal_length;
        p2[0] = (p2[0] - principal_point.x) / focal_length;
        p2[1] = (p2[1] - principal_point.y) / focal_length;

        // 计算重投影误差来判断是否在相机前方
        Vector3f p1_proj = R1 * p2 + t1;
        if (p1_proj[2] > 0) {
            ++valid_count1;
        }

        p1_proj = R2 * p2 + t2;
        if (p1_proj[2] > 0) {
            ++valid_count2;
        }
    }

    if (valid_count1 > valid_count2) {
        R = R1;
        t = t1;
    } else {
        R = R2;
        t = t2;
    }
}

int main() {

  //-- 读取图像
  Mat img_1 = imread("../1.png", CV_LOAD_IMAGE_COLOR);
  Mat img_2 = imread("../2.png", CV_LOAD_IMAGE_COLOR);
    if (img_1.empty() || img_2.empty()) {
        cerr << "无法加载图片" << endl;
        return -1;
    }

  vector<KeyPoint> keypoints_1, keypoints_2;
  vector<DMatch> matches;
  find_feature_matches(img_1, img_2, keypoints_1, keypoints_2, matches);
  cout << "一共找到了" << matches.size() << "组匹配点" << endl;

  //-- 估计两张图像间运动
  Mat R, t;
  pose_estimation_2d2d(keypoints_1, keypoints_2, matches, R, t);

  return 0;
}

void find_feature_matches(const Mat &img_1, const Mat &img_2,
                          std::vector<KeyPoint> &keypoints_1,
                          std::vector<KeyPoint> &keypoints_2,
                          std::vector<DMatch> &matches) {
  //-- 初始化
  Mat descriptors_1, descriptors_2;
  // used in OpenCV3
  Ptr<FeatureDetector> detector = ORB::create();
  Ptr<DescriptorExtractor> descriptor = ORB::create();
  Ptr<DescriptorMatcher> matcher = DescriptorMatcher::create("BruteForce-Hamming");
  //-- 第一步:检测 Oriented FAST 角点位置
  detector->detect(img_1, keypoints_1);
  detector->detect(img_2, keypoints_2);

  //-- 第二步:根据角点位置计算 BRIEF 描述子
  descriptor->compute(img_1, keypoints_1, descriptors_1);
  descriptor->compute(img_2, keypoints_2, descriptors_2);

  //-- 第三步:对两幅图像中的BRIEF描述子进行匹配，使用 Hamming 距离
  vector<DMatch> match;
  //BFMatcher matcher ( NORM_HAMMING );
  matcher->match(descriptors_1, descriptors_2, match);

  //-- 第四步:匹配点对筛选
  double min_dist = 10000, max_dist = 0;

  //找出所有匹配之间的最小距离和最大距离, 即是最相似的和最不相似的两组点之间的距离
  for (int i = 0; i < descriptors_1.rows; i++) {
    double dist = match[i].distance;
    if (dist < min_dist) min_dist = dist;
    if (dist > max_dist) max_dist = dist;
  }

  printf("-- Max dist : %f \n", max_dist);
  printf("-- Min dist : %f \n", min_dist);

  //当描述子之间的距离大于两倍的最小距离时,即认为匹配有误.但有时候最小距离会非常小,设置一个经验值30作为下限.
  for (int i = 0; i < descriptors_1.rows; i++) {
    if (match[i].distance <= max(2 * min_dist, 30.0)) {
      matches.push_back(match[i]);
    }
  }
    Mat img_matches;
    drawMatches(img_1, keypoints_1, img_2, keypoints_2, matches, img_matches);

    // 显示匹配结果
    imshow("按任意键继续！", img_matches);
    waitKey();
}


void pose_estimation_2d2d(std::vector<KeyPoint> keypoints_1,
                          std::vector<KeyPoint> keypoints_2,
                          std::vector<DMatch> matches,
                          Mat &R, Mat &t) {
  // 相机内参,TUM Freiburg2
  Mat K = (Mat_<double>(3, 3) << 520.9, 0, 325.1, 0, 521.0, 249.7, 0, 0, 1);

  //-- 把匹配点转换为vector<Point2f>的形式
  vector<Point2f> points1;
  vector<Point2f> points2;

  for (int i = 0; i < (int) matches.size(); i++) {
    points1.push_back(keypoints_1[matches[i].queryIdx].pt);
    points2.push_back(keypoints_2[matches[i].trainIdx].pt);
  }

  //-- 计算本质矩阵
  Point2d principal_point(325.1, 249.7);  //相机光心
  double focal_length = 521;      //相机焦距
  Mat essential_matrix;
  essential_matrix = findEssentialMat(points1, points2, focal_length, principal_point);

  // 内参矩阵K
  Matrix3f k;
  k << focal_length, 0, principal_point.x,
   0, focal_length, principal_point.y,
   0, 0, 1;

  // 计算本质矩阵
  Matrix3f fundamental_matrix_ransac;
  fundamental_matrix_ransac =  ransacFundamentalMat(points1, points2);
  Matrix3f E = k.transpose() * fundamental_matrix_ransac * k;
  JacobiSVD<Matrix3f> svdE(E, ComputeFullU | ComputeFullV);
  Vector3f singularValues = svdE.singularValues();
  singularValues[2] = 0;
  E = svdE.matrixU() * singularValues.asDiagonal() * svdE.matrixV().transpose();
  //-- 从本质矩阵中恢复旋转和平移信息.
  recoverPose(essential_matrix, points1, points2, R, t, focal_length, principal_point);
  Matrix3f R1;
  Vector3f t1;
  recoverPose_my(  E , points1, points2, R1, t1, focal_length, principal_point);
  cout << "调用opencv计算的结果： " << endl;
  cout << "R is " << endl << R << endl;
  cout << "t is " << endl << t << endl;
  cout <<  "使用ransac自行计算的结果： " << endl;
  cout << "R is " << endl << R1 << endl;
  cout << "t is " << endl << t1 << endl;
}