单张像片空间后方交会C#

根据提供的引用内容，单张像片空间后方交会是一种计算摄影测量学中相机外部定位参数的方法。在C#中，可以使用Math.NET Numerics库来实现单张像片空间后方交会。具体步骤如下：

1. 安装Math.NET Numerics库。可以使用NuGet包管理器或手动下载安装。

2. 定义像片上四个像点的框标坐标及其对应的地面坐标。可以使用数组或列表来存储这些数据。

3. 定义相机主距和像片比例尺。

4. 导入Math.NET Numerics库中的Matrix和Vector类。

5. 定义一个误差函数，用于计算像点的重投影误差。可以使用以下代码：

private static double[] ComputeError(double[] x, double[] y, double[] X, double[] Y, double[] Z, double f, double k1, double k2, double p1, double p2)
{
    int n = x.Length;
    double[] error = new double[n];
    for (int i = 0; i < n; i++)
    {
        double r2 = x[i] * x[i] + y[i] * y[i];
        double r4 = r2 * r2;
        double r6 = r4 * r2;
        double radial = 1 + k1 * r2 + k2 * r4;
        double tangential_x = 2 * p1 * x[i] * y[i] + p2 * (r2 + 2 * x[i] * x[i]);
        double tangential_y = p1 * (r2 + 2 * y[i] * y[i]) + 2 * p2 * x[i] * y[i];
        double u = f * (x[i] * radial + tangential_x);
        double v = f * (y[i] * radial + tangential_y);
        double[] p = { X[i], Y[i], Z[i], 1 };
        Matrix<double> P = Matrix<double>.Build.DenseOfRowMajor(1, 4, p);
        Matrix<double> R = Matrix<double>.Build.DenseOfArray(new double[,] { { x[i] / u, y[i] / u, 1 } });
        Matrix<double> K = Matrix<double>.Build.DenseOfArray(new double[,] { { f, 0, 0 }, { 0, f, 0 }, { 0, 0, 1 } });
        Matrix<double> T = Matrix<double>.Build.DenseOfArray(new double[,] { { 1, 0, 0, 0 }, { 0, 1, 0, 0 }, { 0, 0, 1, 0 } });
        Matrix<double> M = K * T;
        Matrix<double> Xc = M.Inverse() * R.Transpose() * u;
        Matrix<double> xh = M * P.Transpose();
        xh = xh / xh[2, 0];
        error[i] = Math.Sqrt((xh[0, 0] - x[i]) * (xh[0, 0] - x[i]) + (xh[1, 0] - y[i]) * (xh[1, 0] - y[i]));
    }
    return error;
}

6. 定义一个函数，用于计算相机的外方位元素。可以使用以下代码：

public static void ComputeExteriorOrientation(double[] x, double[] y, double[] X, double[] Y, double[] Z, double f, double k1, double k2, double p1, double p2, out double[] omega, out double[] phi, out double[] kappa, out double[] X0, out double[] Y0, out double[] Z0)
{
    int n = x.Length;
    double[] omega0 = { 0, 0, 0 };
    double[] phi0 = { 0, 0, 0 };
    double[] kappa0 = { 0, 0, 0 };
    double[] X00 = { 0, 0, 0 };
    double[] Y00 = { 0, 0, 0 };
    double[] Z00 = { 0, 0, 0 };
    double[] error = ComputeError(x, y, X, Y, Z, f, k1, k2, p1, p2);
    double error0 = error.Sum();
    double delta = 1e-10;
    double[] delta_omega = { delta, 0, 0 };
    double[] delta_phi = { 0, delta, 0 };
    double[] delta_kappa = { 0, 0, delta };
    for (int i = 0; i < n; i++)
    {
        double[] X1 = { X[i], Y[i], Z[i], 1 };
        Matrix<double> P = Matrix<double>.Build.DenseOfRowMajor(1, 4, X1);
        Matrix<double> R = Matrix<double>.Build.DenseOfArray(new double[,] { { x[i] / f, y[i] / f, 1 } });
        Matrix<double> K = Matrix<double>.Build.DenseOfArray(new double[,] { { f, 0, 0 }, { 0, f, 0 }, { 0, 0, 1 } });
        Matrix<double> T = Matrix<double>.Build.DenseOfArray(new double[,] { { 1, 0, 0, 0 }, { 0, 1, 0, 0 }, { 0, 0, 1, 0 } });
        Matrix<double> M = K * T;
        Matrix<double> Xc = M.Inverse() * R.Transpose() * f;
        Matrix<double> xh = M * P.Transpose();
        xh = xh / xh[2, 0];
        double[] X2 = { X[i] + delta, Y[i], Z[i], 1 };
        P = Matrix<double>.Build.DenseOfRowMajor(1, 4, X2);
        Matrix<double> Xc2 = M.Inverse() * R.Transpose() * f;
        Matrix<double> xh2 = M * P.Transpose();
        xh2 = xh2 / xh2[2, 0];
        double[] X3 = { X[i], Y[i] + delta, Z[i], 1 };
        P = Matrix<double>.Build.DenseOfRowMajor(1, 4, X3);
        Matrix<double> Xc3 = M.Inverse() * R.Transpose() * f;
        Matrix<double> xh3 = M * P.Transpose();
        xh3 = xh3 / xh3[2, 0];
        double[] X4 = { X[i], Y[i], Z[i] + delta, 1 };
        P = Matrix<double>.Build.DenseOfRowMajor(1, 4, X4);
        Matrix<double> Xc4 = M.Inverse() * R.Transpose() * f;
        Matrix<double> xh4 = M * P.Transpose();
        xh4 = xh4 / xh4[2, 0];
        double[] J = { (xh2[0, 0] - xh[0, 0]) / delta, (xh3[0, 0] - xh[0, 0]) / delta, (xh4[0, 0] - xh[0, 0]) / delta, (xh2[1, 0] - xh[1, 0]) / delta, (xh3[1, 0] - xh[1, 0]) / delta, (xh4[1, 0] - xh[1, 0]) / delta };
        Matrix<double> J1 = Matrix<double>.Build.DenseOfRowMajor(2, 3, J);
        Matrix<double> J2 = Matrix<double>.Build.DenseOfArray(new double[,] { { -Xc[0, 0], -Xc[0, 1], -Xc[0, 2] }, { -Xc[1, 0], -Xc[1, 1], -Xc[1, 2] } });
        Matrix<double> J3 = Matrix<double>.Build.DenseOfArray(new double[,] { { f * (1 + k1 * x[i] * x[i] + k2 * x[i] * x[i] * x[i] * x[i]) + 2 * p1 * x[i] * y[i] + p2 * (3 * x[i] * x[i] + y[i] * y[i]), f * (2 * p1 * x[i] * y[i] + p2 * (x[i] * x[i] + 3 * y[i] * y[i]) + k1 * 2 * x[i] * y[i] + k2 * 2 * x[i] * x[i] * y[i] * y[i]), 0 } });
        Matrix<double> J4 = Matrix<double>.Build.DenseOfArray(new double[,] { { f * (2 * p1 * x[i] * y[i] + p2 * (x[i] * x[i] + 3 * y[i] * y[i]) + k1 * 2 * x[i] * y[i] + k2 * 2 * x[i] * x[i] * y[i] * y[i]), f * (p1 * (x[i] * x[i] + 3 * y[i] * y[i]) + 2 * p2 * x[i] * y[i] + k1 * (x[i] * x[i] + y[i] * y[i] * 2) + k2 * (x[i] * x[i] * 2 + y[i] * y[i] * 3)), 0 } });
        Matrix<double> J5 = Matrix<double>.Build.DenseOfArray(new double[,] { { 0, 0, f } });
        Matrix<double> J6 = Matrix<double>.Build.DenseOfArray(new double[,] { { -1, 0, xh[0, 0] / xh[2, 0] }, { 0, -1, xh[1, 0] / xh[2, 0] } });
        Matrix<double> J7 = Matrix<double>.Build.DenseOfArray(new double[,] { { 1, 0, xh[0, 0] / xh[2, 0] }, { 0, 1, xh[1, 0] / xh[2, 0] } });
        Matrix<double> J8 = J1 * J2 * J3;
        Matrix<double> J9 = J4 * J5;
        Matrix<double> J10 = J6 * J7;
        Matrix<double> J11 = J8 * J9 * J10;
        Matrix<double> J12 = Matrix<double>.Build.DenseOfArray(new double[,] { { J11[0, 0], J11[0, 1], J11[0, 2], J11[0, 3], J11[0, 4], J11[0, 5] }, { J11[1, 0], J11[1, 1], J11[1, 2], J11[1, 3], J11[1, 4], J11[1, 5] } });
        Matrix<double> J13 = J12.Transpose() * J12;
        Matrix<double> J14 = J12.Transpose() * error[i];
        Matrix<double> J15 = J13.Inverse() * J14;
        omega0[0] += J15[0, 0];
        omega0[1] += J15[1, 0];
        omega0[2] += J15[2, 0];
        phi0[0] += J15[3, 0];
        phi0[1] += J15[4, 0];
        phi0[2] += J15[5, 0];
    }
    double error1 = error.Sum();
    while (error1 < error0)
    {
        error0 = error1;
        double[] omega1 = { omega0[0], omega0[1], omega0[2] };
        double[] phi1 = { phi0[0], phi0[1], phi0[2] };
        double[] kappa1 = { kappa0[0], kappa0[1], kappa0[2] };
        double[] X01 = { X00[0], X00[1], X00[2] };
        double[] Y01 = { Y00[0], Y00[1], Y00[2] };
        double[] Z01 = { Z00[0], Z00[1], Z00[2] };
        for (int i = 0; i < n; i++)
        {
            double[] X1 = { X[i], Y[i], Z[i], 1 };
            Matrix<double> P = Matrix<double>.Build.DenseOfRowMajor(1, 4, X1);
            Matrix<double> R = Matrix<double>.Build.DenseOfArray(new double[,] { { x[i] / f, y[i] / f, 1 } });
            Matrix<double> K = Matrix<double>.Build.DenseOfArray(new double[,] { { f, 0, 0 }, { 0, f, 0 }, { 0, 0, 1 } });
            Matrix<double> T = Matrix<double>.Build.DenseOfArray(new double[,] { { 1, 0, 0, 0 }, { 0, 1, 0, 0 }, { 0, 0, 1, 0 } });
            Matrix<double> M = K * T;
            Matrix<double> Xc = M.Inverse() * R.Transpose() * f;
            Matrix<double> xh = M * P.Transpose();
            xh = xh / xh[2, 0];
            double[] X2 = { X[i] + delta, Y[i], Z[i], 1 };
            P = Matrix<double>.Build.DenseOfRowMajor(1,