ceres-solver / ceres-solver / 4c969a6c1c84d4dd3d4410dab326d75c90c00c9c / . / internal / ceres / corrector.cc

// Ceres Solver - A fast non-linear least squares minimizer | |

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// Author: sameeragarwal@google.com (Sameer Agarwal) | |

#include "ceres/corrector.h" | |

#include <cmath> | |

#include <cstddef> | |

#include "ceres/internal/eigen.h" | |

#include "glog/logging.h" | |

namespace ceres::internal { | |

Corrector::Corrector(const double sq_norm, const double rho[3]) { | |

CHECK_GE(sq_norm, 0.0); | |

sqrt_rho1_ = sqrt(rho[1]); | |

// If sq_norm = 0.0, the correction becomes trivial, the residual | |

// and the jacobian are scaled by the square root of the derivative | |

// of rho. Handling this case explicitly avoids the divide by zero | |

// error that would occur below. | |

// | |

// The case where rho'' < 0 also gets special handling. Technically | |

// it shouldn't, and the computation of the scaling should proceed | |

// as below, however we found in experiments that applying the | |

// curvature correction when rho'' < 0, which is the case when we | |

// are in the outlier region slows down the convergence of the | |

// algorithm significantly. | |

// | |

// Thus, we have divided the action of the robustifier into two | |

// parts. In the inliner region, we do the full second order | |

// correction which re-wights the gradient of the function by the | |

// square root of the derivative of rho, and the Gauss-Newton | |

// Hessian gets both the scaling and the rank-1 curvature | |

// correction. Normally, alpha is upper bounded by one, but with this | |

// change, alpha is bounded above by zero. | |

// | |

// Empirically we have observed that the full Triggs correction and | |

// the clamped correction both start out as very good approximations | |

// to the loss function when we are in the convex part of the | |

// function, but as the function starts transitioning from convex to | |

// concave, the Triggs approximation diverges more and more and | |

// ultimately becomes linear. The clamped Triggs model however | |

// remains quadratic. | |

// | |

// The reason why the Triggs approximation becomes so poor is | |

// because the curvature correction that it applies to the gauss | |

// newton hessian goes from being a full rank correction to a rank | |

// deficient correction making the inversion of the Hessian fraught | |

// with all sorts of misery and suffering. | |

// | |

// The clamped correction retains its quadratic nature and inverting it | |

// is always well formed. | |

if ((sq_norm == 0.0) || (rho[2] <= 0.0)) { | |

residual_scaling_ = sqrt_rho1_; | |

alpha_sq_norm_ = 0.0; | |

return; | |

} | |

// We now require that the first derivative of the loss function be | |

// positive only if the second derivative is positive. This is | |

// because when the second derivative is non-positive, we do not use | |

// the second order correction suggested by BAMS and instead use a | |

// simpler first order strategy which does not use a division by the | |

// gradient of the loss function. | |

CHECK_GT(rho[1], 0.0); | |

// Calculate the smaller of the two solutions to the equation | |

// | |

// 0.5 * alpha^2 - alpha - rho'' / rho' * z'z = 0. | |

// | |

// Start by calculating the discriminant D. | |

const double D = 1.0 + 2.0 * sq_norm * rho[2] / rho[1]; | |

// Since both rho[1] and rho[2] are guaranteed to be positive at | |

// this point, we know that D > 1.0. | |

const double alpha = 1.0 - sqrt(D); | |

// Calculate the constants needed by the correction routines. | |

residual_scaling_ = sqrt_rho1_ / (1 - alpha); | |

alpha_sq_norm_ = alpha / sq_norm; | |

} | |

void Corrector::CorrectResiduals(const int num_rows, double* residuals) { | |

DCHECK(residuals != nullptr); | |

// Equation 11 in BAMS. | |

VectorRef(residuals, num_rows) *= residual_scaling_; | |

} | |

void Corrector::CorrectJacobian(const int num_rows, | |

const int num_cols, | |

double* residuals, | |

double* jacobian) { | |

DCHECK(residuals != nullptr); | |

DCHECK(jacobian != nullptr); | |

// The common case (rho[2] <= 0). | |

if (alpha_sq_norm_ == 0.0) { | |

VectorRef(jacobian, num_rows * num_cols) *= sqrt_rho1_; | |

return; | |

} | |

// Equation 11 in BAMS. | |

// | |

// J = sqrt(rho) * (J - alpha^2 r * r' J) | |

// | |

// In days gone by this loop used to be a single Eigen expression of | |

// the form | |

// | |

// J = sqrt_rho1_ * (J - alpha_sq_norm_ * r* (r.transpose() * J)); | |

// | |

// Which turns out to about 17x slower on bal problems. The reason | |

// is that Eigen is unable to figure out that this expression can be | |

// evaluated columnwise and ends up creating a temporary. | |

for (int c = 0; c < num_cols; ++c) { | |

double r_transpose_j = 0.0; | |

for (int r = 0; r < num_rows; ++r) { | |

r_transpose_j += jacobian[r * num_cols + c] * residuals[r]; | |

} | |

for (int r = 0; r < num_rows; ++r) { | |

jacobian[r * num_cols + c] = | |

sqrt_rho1_ * (jacobian[r * num_cols + c] - | |

alpha_sq_norm_ * residuals[r] * r_transpose_j); | |

} | |

} | |

} | |

} // namespace ceres::internal |