稳健的曲线拟合¶
现在假设我们得到的数据有一些异常值，即我们有一些不服从噪声模型的点。如果我们使用上面的代码来拟合这些数据，我们将得到如下所示的拟合。注意拟合曲线如何偏离基本事实。
要处理异常值，标准技术是使用 LossFunction. 损失函数减少了具有高残差的残差块的影响，通常是那些对应于异常值的块。
为了将损失函数与残差块相关联，我们更改
problem.AddResidualBlock(cost_function, nullptr , &m, &c);
到
problem.AddResidualBlock(cost_function, new CauchyLoss(0.5) , &m, &c);
// Data generated using the following octave code.
// randn('seed', 23497);
// m = 0.3;
// c = 0.1;
// x=[0:0.075:5];
// y = exp(m * x + c);
// noise = randn(size(x)) * 0.2;
// outlier_noise = rand(size(x)) < 0.05;
// y_observed = y + noise + outlier_noise;
// data = [x', y_observed'];
const int kNumObservations = 67;
// clang-format off
const double data[] = {
0.000000e+00, 1.133898e+00,
7.500000e-02, 1.334902e+00,
1.500000e-01, 1.213546e+00,
2.250000e-01, 1.252016e+00,
3.000000e-01, 1.392265e+00,
3.750000e-01, 1.314458e+00,
4.500000e-01, 1.472541e+00,
5.250000e-01, 1.536218e+00,
6.000000e-01, 1.355679e+00,
6.750000e-01, 1.463566e+00,
7.500000e-01, 1.490201e+00,
8.250000e-01, 1.658699e+00,
9.000000e-01, 1.067574e+00,
9.750000e-01, 1.464629e+00,
1.050000e+00, 1.402653e+00,
1.125000e+00, 1.713141e+00,
1.200000e+00, 1.527021e+00,
1.275000e+00, 1.702632e+00,
1.350000e+00, 1.423899e+00,
1.425000e+00, 5.543078e+00, // Outlier point
1.500000e+00, 5.664015e+00, // Outlier point
1.575000e+00, 1.732484e+00,
1.650000e+00, 1.543296e+00,
1.725000e+00, 1.959523e+00,
1.800000e+00, 1.685132e+00,
1.875000e+00, 1.951791e+00,
1.950000e+00, 2.095346e+00,
2.025000e+00, 2.361460e+00,
2.100000e+00, 2.169119e+00,
2.175000e+00, 2.061745e+00,
2.250000e+00, 2.178641e+00,
2.325000e+00, 2.104346e+00,
2.400000e+00, 2.584470e+00,
2.475000e+00, 1.914158e+00,
2.550000e+00, 2.368375e+00,
2.625000e+00, 2.686125e+00,
2.700000e+00, 2.712395e+00,
2.775000e+00, 2.499511e+00,
2.850000e+00, 2.558897e+00,
2.925000e+00, 2.309154e+00,
3.000000e+00, 2.869503e+00,
3.075000e+00, 3.116645e+00,
3.150000e+00, 3.094907e+00,
3.225000e+00, 2.471759e+00,
3.300000e+00, 3.017131e+00,
3.375000e+00, 3.232381e+00,
3.450000e+00, 2.944596e+00,
3.525000e+00, 3.385343e+00,
3.600000e+00, 3.199826e+00,
3.675000e+00, 3.423039e+00,
3.750000e+00, 3.621552e+00,
3.825000e+00, 3.559255e+00,
3.900000e+00, 3.530713e+00,
3.975000e+00, 3.561766e+00,
4.050000e+00, 3.544574e+00,
4.125000e+00, 3.867945e+00,
4.200000e+00, 4.049776e+00,
4.275000e+00, 3.885601e+00,
4.350000e+00, 4.110505e+00,
4.425000e+00, 4.345320e+00,
4.500000e+00, 4.161241e+00,
4.575000e+00, 4.363407e+00,
4.650000e+00, 4.161576e+00,
4.725000e+00, 4.619728e+00,
4.800000e+00, 4.737410e+00,
4.875000e+00, 4.727863e+00,
4.950000e+00, 4.669206e+00
};
// clang-format on
using ceres::AutoDiffCostFunction;
using ceres::CauchyLoss;
using ceres::CostFunction;
using ceres::Problem;
using ceres::Solve;
using ceres::Solver;
struct ExponentialResidual {
ExponentialResidual(double x, double y) : x_(x), y_(y) {}
template <typename T>
bool operator()(const T* const m, const T* const c, T* residual) const {
residual[0] = y_ - exp(m[0] * x_ + c[0]);
return true;
}
private:
const double x_;
const double y_;
};
int main(int argc, char** argv) {
google::InitGoogleLogging(argv[0]);
double m = 0.0;
double c = 0.0;
Problem problem;
for (int i = 0; i < kNumObservations; ++i) {// 循环遍历所有观测数据 x y 67组
//自动导数（AutoDiffCostFunction）：由ceres自行决定导数的计算方式，最常用的求导方式。
//ExponentialResidual, 1, 1, 1
//模板参数依次为仿函数（functor）类型CostFunctor，残差维数residualDim 1 参数维数paramDim 参数m 1维 参数c 1维 接受参数类型为仿函数指针CostFunctor*。
CostFunction* cost_function =
new AutoDiffCostFunction<ExponentialResidual, 1, 1, 1>( //所有的观测数据 逐条加入yi=data[2 * i], xi=data[2 * i + 1]
new ExponentialResidual(data[2 * i], data[2 * i + 1]));
/*
标准技术是使用 LossFunction 损失函数减少了具有高残差的残差块的影响，通常是那些对应于异常值的块
problem.AddResidualBlock(cost_function, nullptr , &m, &c);
改为
problem.AddResidualBlock(cost_function, new CauchyLoss(0.5) , &m, &c);
*/
problem.AddResidualBlock(cost_function, new CauchyLoss(0.5), &m, &c);
}
Solver::Options options;
options.linear_solver_type = ceres::DENSE_QR;
options.minimizer_progress_to_stdout = true;
Solver::Summary summary;
Solve(options, &problem, &summary);
std::cout << summary.BriefReport() << "\n";
std::cout << "Initial m: " << 0.0 << " c: " << 0.0 << "\n";
std::cout << "Final m: " << m << " c: " << c << "\n";
return 0;
}
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标签：曲线拟合,00,01,const,0.1,cereas,ceres,double,using From： https://www.cnblogs.com/gooutlook/p/17020940.html