internal/ceres/corrector.cc

// Ceres Solver - A fast non-linear least squares minimizer
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// http://code.google.com/p/ceres-solver/
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// Author: sameeragarwal@google.com (Sameer Agarwal)

#include "ceres/corrector.h"

#include <cstddef>
#include <cmath>
#include "ceres/internal/eigen.h"
#include "glog/logging.h"

namespace ceres {
namespace internal {

Corrector::Corrector(double sq_norm, const double rho[3]) {
  CHECK_GE(sq_norm, 0.0);
  CHECK_GT(rho[1], 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 squareroot 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. Normaly, 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;
  }

  // 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(int nrow, double* residuals) {
  DCHECK(residuals != NULL);
  VectorRef r_ref(residuals, nrow);
  // Equation 11 in BANS.
  r_ref *= residual_scaling_;
}

void Corrector::CorrectJacobian(int nrow, int ncol,
                                double* residuals, double* jacobian) {
  DCHECK(residuals != NULL);
  DCHECK(jacobian != NULL);
  ConstVectorRef r_ref(residuals, nrow);
  MatrixRef j_ref(jacobian, nrow, ncol);

  // Equation 11 in BANS.
  j_ref = sqrt_rho1_ * (j_ref - alpha_sq_norm_ *
                        r_ref * (r_ref.transpose() * j_ref));
}

}  // namespace internal
}  // namespace ceres