Ceres

Hello World!

12(10−x)2.

cost function

struct CostFunctor {   template <typename T>   bool operator()(const T* const x, T* residual) const {     residual[0] = T(10.0) - x[0];     return true;   }};

problem

  // auto-differentiation to obtain the derivative (jacobian).  CostFunction* cost_function =      new AutoDiffCostFunction<CostFunctor, 1, 1>(new CostFunctor);  problem.AddResidualBlock(cost_function, NULL, &x);

Numeric

cost function

struct NumericDiffCostFunctor {  bool operator()(const double* const x, double* residual) const {    residual[0] = 10.0 - x[0];    return true;  }};

Problem

CostFunction* cost_function =  new NumericDiffCostFunction<NumericDiffCostFunctor, ceres::CENTRAL, 1, 1>(      new NumericDiffCostFunctor);problem.AddResidualBlock(cost_function, NULL, &x);

Powell’s Function

f1(x)f2(x)f3(x)f4(x)F(x)=x1+10x2=5√(x3−x4)=(x2−2x3)2=10‾‾‾√(x1−x4)2=[f1(x), f2(x), f3(x), f4(x)]

cost function

struct F4 {  template <typename T>  bool operator()(const T* const x1, const T* const x4, T* residual) const {    residual[0] = T(sqrt(10.0)) * (x1[0] - x4[0]) * (x1[0] - x4[0]);    return true;  }};

用同样的方法构造F1、F2、F3的costfunction。

problem

problem.AddResidualBlock(new AutoDiffCostFunction<F1, 1, 1, 1>(new F1), NULL, &x1, &x2);problem.AddResidualBlock(new AutoDiffCostFunction<F2, 1, 1, 1>(new F2), NULL, &x3, &x4);problem.AddResidualBlock(new AutoDiffCostFunction<F3, 1, 1, 1>(new F3), NULL, &x2, &x3)problem.AddResidualBlock(new AutoDiffCostFunction<F4, 1, 1, 1>(new F4), NULL, &x1, &x4);

Curve Fitting(曲线拟合)

y=emx+c.

cost dunction

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] = T(y_) - exp(m[0] * T(x_) + c[0]);    return true;  } private:  // Observations for a sample.  const double x_;  const double y_;};

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]));  problem.AddResidualBlock(cost_function, NULL, &m, &c);}

拟合情况如下图

Robust Curve Fitting

​ 假设在曲线拟合是我们得到的数据有一些不符合噪音异常值，那么我们使用上面的方法来拟合时就会发现拟合曲线将会偏离真实值。（注意在图像最上方的异常值）

我们可以使用lostfuncion来解决这个问题。将代码

problem.AddResidualBlock(cost_function, NULL , &m, &c);

改为

problem.AddResidualBlock(cost_function, new CauchyLoss(0.5) , &m, &c);

Cauchyloss是ceres求解器的一种损失函数。我们可以得到以下结果。

Bundle Adjustment

​ 使用ceres主要就是为了解决BA问题，而BA问题通常是一个非线性优化问题。首先使用一个定义好的模块化函数来计算重投影，该函数的结构和取幂函数ExponentialResidual类似，都有一个实例来观察每个image。

​ 每一个BAL（Bundle Adjustment in Large）问题都需要三维点和有九个参数的相机模型，九个参数分别是三个旋转参数、三个平移参数、一个深度参数和两个畸变参数。

cost function

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] 是轴角参数。    T p[3];    ceres::AngleAxisRotatePoint(camera, point, p);    // camera[3,4,5] 是平移参数。    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 = T(1.0) + r2  * (l1 + l2  * r2);    // Compute final projected point position.    //计算最终投影点位置。    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.    //误差是预测和观察到的位置的差值。    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;};

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,                           NULL /* squared loss */,                           bal_problem.mutable_camera_for_observation(i),                           bal_problem.mutable_point_for_observation(i));}