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// Ceres Solver - A fast non-linear least squares minimizer
// Copyright 2010, 2011, 2012 Google Inc. All rights reserved.
// http://code.google.com/p/ceres-solver/
//
// Redistribution and use in source and binary forms, with or without
// modification, are permitted provided that the following conditions are met:
//
// * Redistributions of source code must retain the above copyright notice,
// this list of conditions and the following disclaimer.
// * Redistributions in binary form must reproduce the above copyright notice,
// this list of conditions and the following disclaimer in the documentation
// and/or other materials provided with the distribution.
// * Neither the name of Google Inc. nor the names of its contributors may be
// used to endorse or promote products derived from this software without
// specific prior written permission.
//
// THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
// AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
// IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
// ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE
// LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
// CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
// SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
// INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
// CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
// ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
// POSSIBILITY OF SUCH DAMAGE.
//
// 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_
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