Curve fitting

This example is selected from the tutorial, which is a simple optimization problem and can clearly present the frame.

To start with, we need to define a templated object to evaluate the residual.

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:
  // Observations for a sample.
  const double x_;
  const double y_;
};

The observations are a $2n$ data array. Then, we can construct the problem and use AutoDiffCostFunction to differentiate automatically.

// Initiate the parameters.
double m = 0.0;
double c = 0.0;
//ceres::Problem
Problem problem;

for (int i = 0; i < kNumObservations; ++i) {
  CostFunction* cost_function =
       new AutoDiffCostFunction<ExponentialResidual, 1, 1, 1>(
           new ExponentialResidual(data[2 * i], data[2 * i + 1]));
  // Add every observation's residual to problem.
  problem.AddResidualBlock(cost_function, nullptr, &m, &c);
}

Solve the problem:

  //Set the solver's options.
  Solver::Options options;
  options.max_num_iterations = 25;
  options.linear_solver_type = ceres::DENSE_QR;
  options.minimizer_progress_to_stdout = true;

  Solver::Summary summary;
  Solve(options, &problem, &summary);

Bundle Adjustment

The structure :

struct SnavelyReprojectionError {
  SnavelyReprojectionError(double observed_x, double observed_y)
      : observed_x(observed_x), observed_y(observed_y) {}

  template <typename T>
  bool operator()(const T* const camera,
                  const T* const point,
                  T* residuals) const {
    // camera[0,1,2] are the angle-axis rotation.
    T p[3];
    ceres::AngleAxisRotatePoint(camera, point, p);
    // camera[3,4,5] are the translation.
    p[0] += camera[3]; p[1] += camera[4]; p[2] += camera[5];

    // Compute the center of distortion. The sign change comes from
    // the camera model that Noah Snavely's Bundler assumes, whereby
    // the camera coordinate system has a negative z axis.
    T xp = - p[0] / p[2];
    T yp = - p[1] / p[2];

    // Apply second and fourth order radial distortion.
    const T& l1 = camera[7];
    const T& l2 = camera[8];
    T r2 = xp*xp + yp*yp;
    T distortion = 1.0 + r2  * (l1 + l2  * r2);

    // Compute final projected point position.
    const T& focal = camera[6];
    T predicted_x = focal * distortion * xp;
    T predicted_y = focal * distortion * yp;

    // The error is the difference between the predicted and observed position.
    residuals[0] = predicted_x - T(observed_x);
    residuals[1] = predicted_y - T(observed_y);
    return true;
  }

   // Factory to hide the construction of the CostFunction object from
   // the client code.
   static ceres::CostFunction* Create(const double observed_x,
                                      const double observed_y) {
     return (new ceres::AutoDiffCostFunction<SnavelyReprojectionError, 2, 9, 3>(
                 new SnavelyReprojectionError(observed_x, observed_y)));
   }

  double observed_x;
  double observed_y;
};

Construct bundle adjustment problem:

ceres::Problem problem;
for (int i = 0; i < bal_problem.num_observations(); ++i) {
  ceres::CostFunction* cost_function =
      SnavelyReprojectionError::Create(
           bal_problem.observations()[2 * i + 0],
           bal_problem.observations()[2 * i + 1]);
  problem.AddResidualBlock(cost_function,
                           nullptr /* squared loss */,
                           bal_problem.mutable_camera_for_observation(i),
                           bal_problem.mutable_point_for_observation(i));
}

Optimization:

ceres::Solver::Options options;
options.linear_solver_type = ceres::DENSE_SCHUR;
options.minimizer_progress_to_stdout = true;
ceres::Solver::Summary summary;

Reference

http://ceres-solver.org/nnls_tutorial.html