loss_function.h

// 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)
//
// The LossFunction interface is the way users describe how residuals
// are converted to cost terms for the overall problem cost function.
// For the exact manner in which loss functions are converted to the
// overall cost for a problem, see problem.h.
//
// For least squares problem where there are no outliers and standard
// squared loss is expected, it is not necessary to create a loss
// function; instead passing a NULL to the problem when adding
// residuals implies a standard squared loss.
//
// For least squares problems where the minimization may encounter
// input terms that contain outliers, that is, completely bogus
// measurements, it is important to use a loss function that reduces
// their associated penalty.
//
// Consider a structure from motion problem. The unknowns are 3D
// points and camera parameters, and the measurements are image
// coordinates describing the expected reprojected position for a
// point in a camera. For example, we want to model the geometry of a
// street scene with fire hydrants and cars, observed by a moving
// camera with unknown parameters, and the only 3D points we care
// about are the pointy tippy-tops of the fire hydrants. Our magic
// image processing algorithm, which is responsible for producing the
// measurements that are input to Ceres, has found and matched all
// such tippy-tops in all image frames, except that in one of the
// frame it mistook a car's headlight for a hydrant. If we didn't do
// anything special (i.e. if we used a basic quadratic loss), the
// residual for the erroneous measurement will result in extreme error
// due to the quadratic nature of squared loss. This results in the
// entire solution getting pulled away from the optimimum to reduce
// the large error that would otherwise be attributed to the wrong
// measurement.
//
// Using a robust loss function, the cost for large residuals is
// reduced. In the example above, this leads to outlier terms getting
// downweighted so they do not overly influence the final solution.
//
// What cost function is best?
//
// In general, there isn't a principled way to select a robust loss
// function. The authors suggest starting with a non-robust cost, then
// only experimenting with robust loss functions if standard squared
// loss doesn't work.

#ifndef CERES_PUBLIC_LOSS_FUNCTION_H_
#define CERES_PUBLIC_LOSS_FUNCTION_H_

#include <glog/logging.h>
#include "ceres/internal/macros.h"
#include "ceres/internal/scoped_ptr.h"
#include "ceres/types.h"

namespace ceres {

class LossFunction {
 public:
  virtual ~LossFunction() {}

  // For a residual vector with squared 2-norm 'sq_norm', this method
  // is required to fill in the value and derivatives of the loss
  // function (rho in this example):
  //
  //   out[0] = rho(sq_norm),
  //   out[1] = rho'(sq_norm),
  //   out[2] = rho''(sq_norm),
  //
  // Here the convention is that the contribution of a term to the
  // cost function is given by 1/2 rho(s),  where
  //
  //   s = ||residuals||^2.
  //
  // Calling the method with a negative value of 's' is an error and
  // the implementations are not required to handle that case.
  //
  // Most sane choices of rho() satisfy:
  //
  //   rho(0) = 0,
  //   rho'(0) = 1,
  //   rho'(s) < 1 in outlier region,
  //   rho''(s) < 0 in outlier region,
  //
  // so that they mimic the least squares cost for small residuals.
  virtual void Evaluate(double sq_norm, double out[3]) const = 0;
};

// Some common implementations follow below.
//
// Note: in the region of interest (i.e. s < 3) we have:
//   TrivialLoss >= HuberLoss >= SoftLOneLoss >= CauchyLoss


// This corresponds to no robustification.
//
//   rho(s) = s
//
// At s = 0: rho = [0, 1, 0].
//
// It is not normally necessary to use this, as passing NULL for the
// loss function when building the problem accomplishes the same
// thing.
class TrivialLoss : public LossFunction {
 public:
  virtual void Evaluate(double, double*) const;
};

// Scaling
// -------
// Given one robustifier
//   s -> rho(s)
// one can change the length scale at which robustification takes
// place, by adding a scale factor 'a' as follows:
//
//   s -> a^2 rho(s / a^2).
//
// The first and second derivatives are:
//
//   s -> rho'(s / a^2),
//   s -> (1 / a^2) rho''(s / a^2),
//
// but the behaviour near s = 0 is the same as the original function,
// i.e.
//
//   rho(s) = s + higher order terms,
//   a^2 rho(s / a^2) = s + higher order terms.
//
// The scalar 'a' should be positive.
//
// The reason for the appearance of squaring is that 'a' is in the
// units of the residual vector norm whereas 's' is a squared
// norm. For applications it is more convenient to specify 'a' than
// its square. The commonly used robustifiers below are described in
// un-scaled format (a = 1) but their implementations work for any
// non-zero value of 'a'.

// Huber.
//
//   rho(s) = s               for s <= 1,
//   rho(s) = 2 sqrt(s) - 1   for s >= 1.
//
// At s = 0: rho = [0, 1, 0].
//
// The scaling parameter 'a' corresponds to 'delta' on this page:
//   http://en.wikipedia.org/wiki/Huber_Loss_Function
class HuberLoss : public LossFunction {
 public:
  explicit HuberLoss(double a) : a_(a), b_(a * a) { }
  virtual void Evaluate(double, double*) const;
 private:
  const double a_;
  // b = a^2.
  const double b_;
};

// Soft L1, similar to Huber but smooth.
//
//   rho(s) = 2 (sqrt(1 + s) - 1).
//
// At s = 0: rho = [0, 1, -1/2].
class SoftLOneLoss : public LossFunction {
 public:
  explicit SoftLOneLoss(double a) : b_(a * a), c_(1 / b_) { }
  virtual void Evaluate(double, double*) const;
 private:
  // b = a^2.
  const double b_;
  // c = 1 / a^2.
  const double c_;
};

// Inspired by the Cauchy distribution
//
//   rho(s) = log(1 + s).
//
// At s = 0: rho = [0, 1, -1].
class CauchyLoss : public LossFunction {
 public:
  explicit CauchyLoss(double a) : b_(a * a), c_(1 / b_) { }
  virtual void Evaluate(double, double*) const;
 private:
  // b = a^2.
  const double b_;
  // c = 1 / a^2.
  const double c_;
};

// The discussion above has to do with length scaling: it affects the space
// in which s is measured. Sometimes you want to simply scale the output
// value of the robustifier. For example, you might want to weight
// different error terms differently (e.g., weight pixel reprojection
// errors differently from terrain errors).
//
// If rho is the wrapped robustifier, then this simply outputs
// s -> a * rho(s)
//
// The first and second derivatives are, not surprisingly
// s -> a * rho'(s)
// s -> a * rho''(s)
//
// Since we treat the a NULL Loss function as the Identity loss
// function, rho = NULL is a valid input and will result in the input
// being scaled by a. This provides a simple way of implementing a
// scaled ResidualBlock.
class ScaledLoss : public LossFunction {
 public:
  // Constructs a ScaledLoss wrapping another loss function. Takes
  // ownership of the wrapped loss function or not depending on the
  // ownership parameter.
  ScaledLoss(const LossFunction* rho, double a, Ownership ownership) :
      rho_(rho), a_(a), ownership_(ownership) { }

  virtual ~ScaledLoss() {
    if (ownership_ == DO_NOT_TAKE_OWNERSHIP) {
      rho_.release();
    }
  }
  virtual void Evaluate(double, double*) const;

 private:
  internal::scoped_ptr<const LossFunction> rho_;
  const double a_;
  const Ownership ownership_;
  DISALLOW_COPY_AND_ASSIGN(ScaledLoss);
};

// Sometimes after the optimization problem has been constructed, we
// wish to mutate the scale of the loss function. For example, when
// performing estimation from data which has substantial outliers,
// convergence can be improved by starting out with a large scale,
// optimizing the problem and then reducing the scale. This can have
// better convergence behaviour than just using a loss function with a
// small scale.
//
// This templated class allows the user to implement a loss function
// whose scale can be mutated after an optimization problem has been
// constructed.
//
// Example usage
//
//  Problem problem;
//
//  // Add parameter blocks
//
//  CostFunction* cost_function =
//    new AutoDiffCostFunction < UW_Camera_Mapper, 2, 9, 3>(
//      new UW_Camera_Mapper(data->observations[2*i + 0],
//                           data->observations[2*i + 1]));
//
//  LossFunctionWrapper* loss_function(new HuberLoss(1.0), TAKE_OWNERSHIP);
//
//  problem.AddResidualBlock(cost_function, loss_function, parameters);
//
//  Solver::Options options;
//  scoped_ptr<Solver::Summary> summary1(Solve(problem, options));
//
//  loss_function->Reset(new HuberLoss(1.0), TAKE_OWNERSHIP);
//
//  scoped_ptr<Solver::Summary> summary2(Solve(problem, options));
//
class LossFunctionWrapper : public LossFunction {
 public:
  LossFunctionWrapper(LossFunction* rho, Ownership ownership)
      : rho_(rho), ownership_(ownership) {
  }

  virtual ~LossFunctionWrapper() {
    if (ownership_ == DO_NOT_TAKE_OWNERSHIP) {
      rho_.release();
    }
  }

  virtual void Evaluate(double sq_norm, double out[3]) const {
    CHECK_NOTNULL(rho_.get());
    rho_->Evaluate(sq_norm, out);
  }

  void Reset(LossFunction* rho, Ownership ownership) {
    if (ownership_ == DO_NOT_TAKE_OWNERSHIP) {
      rho_.release();
    }
    rho_.reset(rho);
    ownership_ = ownership;
  }

 private:
  internal::scoped_ptr<const LossFunction> rho_;
  Ownership ownership_;
  DISALLOW_COPY_AND_ASSIGN(LossFunctionWrapper);
};

}  // namespace ceres

#endif  // CERES_PUBLIC_LOSS_FUNCTION_H_