include/ceres/local_parameterization.h

// 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.
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//   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
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// LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
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//
// Author: keir@google.com (Keir Mierle)
//         sameeragarwal@google.com (Sameer Agarwal)

#ifndef CERES_PUBLIC_LOCAL_PARAMETERIZATION_H_
#define CERES_PUBLIC_LOCAL_PARAMETERIZATION_H_

#include <vector>
#include "ceres/internal/port.h"

namespace ceres {

// Purpose: Sometimes parameter blocks x can overparameterize a problem
//
//   min f(x)
//    x
//
// In that case it is desirable to choose a parameterization for the
// block itself to remove the null directions of the cost. More
// generally, if x lies on a manifold of a smaller dimension than the
// ambient space that it is embedded in, then it is numerically and
// computationally more effective to optimize it using a
// parameterization that lives in the tangent space of that manifold
// at each point.
//
// For example, a sphere in three dimensions is a 2 dimensional
// manifold, embedded in a three dimensional space. At each point on
// the sphere, the plane tangent to it defines a two dimensional
// tangent space. For a cost function defined on this sphere, given a
// point x, moving in the direction normal to the sphere at that point
// is not useful. Thus a better way to do a local optimization is to
// optimize over two dimensional vector delta in the tangent space at
// that point and then "move" to the point x + delta, where the move
// operation involves projecting back onto the sphere. Doing so
// removes a redundent dimension from the optimization, making it
// numerically more robust and efficient.
//
// More generally we can define a function
//
//   x_plus_delta = Plus(x, delta),
//
// where x_plus_delta has the same size as x, and delta is of size
// less than or equal to x. The function Plus, generalizes the
// definition of vector addition. Thus it satisfies the identify
//
//   Plus(x, 0) = x, for all x.
//
// A trivial version of Plus is when delta is of the same size as x
// and
//
//   Plus(x, delta) = x + delta
//
// A more interesting case if x is two dimensional vector, and the
// user wishes to hold the first coordinate constant. Then, delta is a
// scalar and Plus is defined as
//
//   Plus(x, delta) = x + [0] * delta
//                        [1]
//
// An example that occurs commonly in Structure from Motion problems
// is when camera rotations are parameterized using Quaternion. There,
// it is useful only make updates orthogonal to that 4-vector defining
// the quaternion. One way to do this is to let delta be a 3
// dimensional vector and define Plus to be
//
//   Plus(x, delta) = [cos(|delta|), sin(|delta|) delta / |delta|] * x
//
// The multiplication between the two 4-vectors on the RHS is the
// standard quaternion product.
//
// Given g and a point x, optimizing f can now be restated as
//
//     min  f(Plus(x, delta))
//    delta
//
// Given a solution delta to this problem, the optimal value is then
// given by
//
//   x* = Plus(x, delta)
//
// The class LocalParameterization defines the function Plus and its
// Jacobian which is needed to compute the Jacobian of f w.r.t delta.
class LocalParameterization {
 public:
  virtual ~LocalParameterization() {}

  // Generalization of the addition operation,
  //
  //   x_plus_delta = Plus(x, delta)
  //
  // with the condition that Plus(x, 0) = x.
  virtual bool Plus(const double* x,
                    const double* delta,
                    double* x_plus_delta) const = 0;

  // The jacobian of Plus(x, delta) w.r.t delta at delta = 0.
  virtual bool ComputeJacobian(const double* x, double* jacobian) const = 0;

  // Size of x.
  virtual int GlobalSize() const = 0;

  // Size of delta.
  virtual int LocalSize() const = 0;
};

// Some basic parameterizations

// Identity Parameterization: Plus(x, delta) = x + delta
class IdentityParameterization : public LocalParameterization {
 public:
  explicit IdentityParameterization(int size);
  virtual ~IdentityParameterization() {}
  virtual bool Plus(const double* x,
                    const double* delta,
                    double* x_plus_delta) const;
  virtual bool ComputeJacobian(const double* x,
                               double* jacobian) const;
  virtual int GlobalSize() const { return size_; }
  virtual int LocalSize() const { return size_; }

 private:
  const int size_;
};

// Hold a subset of the parameters inside a parameter block constant.
class SubsetParameterization : public LocalParameterization {
 public:
  explicit SubsetParameterization(int size,
                                  const vector<int>& constant_parameters);
  virtual ~SubsetParameterization() {}
  virtual bool Plus(const double* x,
                    const double* delta,
                    double* x_plus_delta) const;
  virtual bool ComputeJacobian(const double* x,
                               double* jacobian) const;
  virtual int GlobalSize() const { return constancy_mask_.size(); }
  virtual int LocalSize() const { return local_size_; }

 private:
  const int local_size_;
  vector<int> constancy_mask_;
};

// Plus(x, delta) = [cos(|delta|), sin(|delta|) delta / |delta|] * x
// with * being the quaternion multiplication operator. Here we assume
// that the first element of the quaternion vector is the real (cos
// theta) part.
class QuaternionParameterization : public LocalParameterization {
 public:
  virtual ~QuaternionParameterization() {}
  virtual bool Plus(const double* x,
                    const double* delta,
                    double* x_plus_delta) const;
  virtual bool ComputeJacobian(const double* x,
                               double* jacobian) const;
  virtual int GlobalSize() const { return 4; }
  virtual int LocalSize() const { return 3; }
};

}  // namespace ceres

#endif  // CERES_PUBLIC_LOCAL_PARAMETERIZATION_H_