用C++写一个算法可以实现单像空间后方交会

单像空间后方交会是一种将三维空间坐标与二维图像坐标之间的关系相互转换的方法，一般用于计算相机的位姿和三维点的坐标。下面是一个简单的C++代码实现：

#include <iostream>
#include <Eigen/Dense>

using namespace Eigen;

// 定义相机内参和图像坐标
Matrix3d K;
Vector2d p;

// 定义相机位姿和世界坐标
Matrix3d R;
Vector3d t;
Vector3d P;

// 定义误差函数
struct ErrorFunc {
  ErrorFunc(const Vector2d& p, const Vector3d& P, const Matrix3d& K, const Matrix3d& R, const Vector3d& t) :
    p_(p), P_(P), K_(K), R_(R), t_(t) {};

  template<typename T>
  bool operator()(const T* const X, T* residuals) const {
    // 提取相机参数
    T fx = T(K_(0,0)), fy = T(K_(1,1)), cx = T(K_(0,2)), cy = T(K_(1,2));
    T k1 = T(0), k2 = T(0), p1 = T(0), p2 = T(0);
    T x = T(p_(0)), y = T(p_(1));
    T Xw = T(P_(0)), Yw = T(P_(1)), Zw = T(P_(2));
    T rw = T(R_(0,0)), rx = T(R_(0,1)), ry = T(R_(0,2)), 
      rz = T(R_(1,0)), r1 = T(R_(1,1)), r2 = T(R_(1,2)),
      r3 = T(R_(2,0)), r4 = T(R_(2,1)), r5 = T(R_(2,2));
    T tx = T(t_(0)), ty = T(t_(1)), tz = T(t_(2));

    // 计算投影误差
    T u = fx * (rw * (Xw - tx) + rx * (Yw - ty) + ry * (Zw - tz)) / (rz * (Xw - tx) + r1 * (Yw - ty) + r2 * (Zw - tz)) + cx;
    T v = fy * (r3 * (Xw - tx) + r4 * (Yw - ty) + r5 * (Zw - tz)) / (rz * (Xw - tx) + r1 * (Yw - ty) + r2 * (Zw - tz)) + cy;
    residuals[0] = u - x;
    residuals[1] = v - y;
    return true;
  }

  const Vector2d p_;
  const Vector3d P_;
  const Matrix3d K_;
  const Matrix3d R_;
  const Vector3d t_;
};

// 定义优化函数
void optimize() {
  // 定义初始值
  Vector3d X(0, 0, 1);
  Vector3d t(0, 0, 0);
  Matrix3d R = Matrix3d::Identity();
  double fx = K(0,0), fy = K(1,1), cx = K(0,2), cy = K(1,2);

  // 定义优化器
  ceres::Problem problem;
  ceres::LossFunction* loss_function = new ceres::HuberLoss(1.0);
  ceres::LocalParameterization* quaternion_parameterization = new ceres::QuaternionParameterization;

  // 添加误差项
  ceres::CostFunction* cost_function = new ceres::AutoDiffCostFunction<ErrorFunc, 2, 4>(
    new ErrorFunc(p, P, K, R, t));
  problem.AddResidualBlock(cost_function, loss_function, X.data(), R.data(), t.data());

  // 优化器设置
  ceres::Solver::Options options;
  options.linear_solver_type = ceres::DENSE_QR;
  options.minimizer_progress_to_stdout = true;
  ceres::Solver::Summary summary;
  ceres::Solve(options, &problem, &summary);

  // 输出结果
  std::cout << summary.FullReport() << std::endl;
  std::cout << "X = " << X.transpose() << std::endl;
  std::cout << "R = " << std::endl << R << std::endl;
  std::cout << "t = " << t.transpose() << std::endl;
}

int main() {
  // 初始化相机内参和图像坐标
  K << 1000, 0, 500, 0, 1000, 500, 0, 0, 1;
  p << 400, 300;

  // 初始化相机位姿和世界坐标
  R << 1, 0, 0, 0, 1, 0, 0, 0, 1;
  t << 0, 0, 0;
  P << 1, 2, 3;

  // 进行优化
  optimize();

  return 0;
}

这个程序使用了Ceres Solver库来进行优化，通过定义误差函数和优化器设置，可以很方便地实现单像空间后方交会。