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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: 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_