ceres-solver/docs/source/modeling.rst

.. default-domain:: cpp

.. cpp:namespace:: ceres

.. _`chapter-modeling`:

============
Modeling API
============

Recall that Ceres solves robustified non-linear least squares problems
of the form

.. math:: \frac{1}{2}\sum_{i=1} \rho_i\left(\left\|f_i\left(x_{i_1}, ... ,x_{i_k}\right)\right\|^2\right).
   :label: ceresproblem3

The expression
:math:`\rho_i\left(\left\|f_i\left(x_{i_1},...,x_{i_k}\right)\right\|^2\right)`
is known as a ``ResidualBlock``, where :math:`f_i(\cdot)` is a
:class:`CostFunction` that depends on the parameter blocks
:math:`\left[x_{i_1},... , x_{i_k}\right]`. In most optimization
problems small groups of scalars occur together. For example the three
components of a translation vector and the four components of the
quaternion that define the pose of a camera. We refer to such a group
of small scalars as a ``ParameterBlock``. Of course a
``ParameterBlock`` can just be a single parameter. :math:`\rho_i` is a
:class:`LossFunction`. A :class:`LossFunction` is a scalar function
that is used to reduce the influence of outliers on the solution of
non-linear least squares problems.

In this chapter we will describe the various classes that are part of
Ceres Solver's modeling API, and how they can be used to construct
optimization.

Once a problem has been constructed, various methods for solving them
will be discussed in :ref:`chapter-solving`. It is by design that the
modeling and the solving APIs are orthogonal to each other. This
enables easy switching/tweaking of various solver parameters without
having to touch the problem once it has been successfuly modeling.

:class:`CostFunction`
---------------------

.. class:: CostFunction

   .. code-block:: c++

    class CostFunction {
     public:
      virtual bool Evaluate(double const* const* parameters,
                            double* residuals,
                            double** jacobians) = 0;
      const vector<int16>& parameter_block_sizes();
      int num_residuals() const;

     protected:
      vector<int16>* mutable_parameter_block_sizes();
      void set_num_residuals(int num_residuals);
    };

   Given parameter blocks :math:`\left[x_{i_1}, ... , x_{i_k}\right]`,
   a :class:`CostFunction` is responsible for computing a vector of
   residuals and if asked a vector of Jacobian matrices, i.e., given
   :math:`\left[x_{i_1}, ... , x_{i_k}\right]`, compute the vector
   :math:`f_i\left(x_{i_1},...,x_{i_k}\right)` and the matrices

   .. math:: J_{ij} = \frac{\partial}{\partial x_{i_j}}f_i\left(x_{i_1},...,x_{i_k}\right),\quad \forall j \in \{i_1,..., i_k\}

   The signature of the class:`CostFunction` (number and sizes of
   input parameter blocks and number of outputs) is stored in
   :member:`CostFunction::parameter_block_sizes_` and
   :member:`CostFunction::num_residuals_` respectively. User code
   inheriting from this class is expected to set these two members
   with the corresponding accessors. This information will be verified
   by the :class:`Problem` when added with
   :func:`Problem::AddResidualBlock`.

.. function:: bool CostFunction::Evaluate(double const* const* parameters, double* residuals, double** jacobians)

   This is the key methods. It implements the residual and Jacobian
   computation.

   ``parameters`` is an array of pointers to arrays containing the
   various parameter blocks. parameters has the same number of
   elements as :member:`CostFunction::parameter_block_sizes_`.
   Parameter blocks are in the same order as
   :member:`CostFunction::parameter_block_sizes_`.

   ``residuals`` is an array of size ``num_residuals_``.


   ``jacobians`` is an array of size
   :member:`CostFunction::parameter_block_sizes_` containing pointers
   to storage for Jacobian matrices corresponding to each parameter
   block. The Jacobian matrices are in the same order as
   :member:`CostFunction::parameter_block_sizes_`. ``jacobians[i]`` is
   an array that contains :member:`CostFunction::num_residuals_` x
   :member:`CostFunction::parameter_block_sizes_` ``[i]``
   elements. Each Jacobian matrix is stored in row-major order, i.e.,
   ``jacobians[i][r * parameter_block_size_[i] + c]`` =
   :math:`\frac{\partial residual[r]}{\partial parameters[i][c]}`


   If ``jacobians`` is ``NULL``, then no derivatives are returned;
   this is the case when computing cost only. If ``jacobians[i]`` is
   ``NULL``, then the Jacobian matrix corresponding to the
   :math:`i^{\textrm{th}}` parameter block must not be returned, this
   is the case when the a parameter block is marked constant.


:class:`SizedCostFunction`
--------------------------

.. class:: SizedCostFunction

   If the size of the parameter blocks and the size of the residual
   vector is known at compile time (this is the common case), Ceres
   provides :class:`SizedCostFunction`, where these values can be
   specified as template parameters. In this case the user only needs
   to implement the :func:`CostFunction::Evaluate`.

   .. code-block:: c++

    template<int kNumResiduals,
             int N0 = 0, int N1 = 0, int N2 = 0, int N3 = 0, int N4 = 0,
             int N5 = 0, int N6 = 0, int N7 = 0, int N8 = 0, int N9 = 0>
    class SizedCostFunction : public CostFunction {
     public:
      virtual bool Evaluate(double const* const* parameters,
                            double* residuals,
                            double** jacobians) const = 0;
    };


:class:`AutoDiffCostFunction`
-----------------------------

.. class:: AutoDiffCostFunction

   But even defining the :class:`SizedCostFunction` can be a tedious
   affair if complicated derivative computations are involved. To this
   end Ceres provides automatic differentiation.

   To get an auto differentiated cost function, you must define a
   class with a templated ``operator()`` (a functor) that computes the
   cost function in terms of the template parameter ``T``. The
   autodiff framework substitutes appropriate ``Jet`` objects for
   ``T`` in order to compute the derivative when necessary, but this
   is hidden, and you should write the function as if ``T`` were a
   scalar type (e.g. a double-precision floating point number).

   The function must write the computed value in the last argument
   (the only non-``const`` one) and return true to indicate success.

   For example, consider a scalar error :math:`e = k - x^\top y`,
   where both :math:`x` and :math:`y` are two-dimensional vector
   parameters and :math:`k` is a constant. The form of this error,
   which is the difference between a constant and an expression, is a
   common pattern in least squares problems. For example, the value
   :math:`x^\top y` might be the model expectation for a series of
   measurements, where there is an instance of the cost function for
   each measurement :math:`k`.

   The actual cost added to the total problem is :math:`e^2`, or
   :math:`(k - x^\top y)^2`; however, the squaring is implicitly done
   by the optimization framework.

   To write an auto-differentiable cost function for the above model,
   first define the object

   .. code-block:: c++

    class MyScalarCostFunctor {
      MyScalarCostFunctor(double k): k_(k) {}

      template <typename T>
      bool operator()(const T* const x , const T* const y, T* e) const {
        e[0] = T(k_) - x[0] * y[0] - x[1] * y[1];
        return true;
      }

     private:
      double k_;
    };


   Note that in the declaration of ``operator()`` the input parameters
   ``x`` and ``y`` come first, and are passed as const pointers to arrays
   of ``T``. If there were three input parameters, then the third input
   parameter would come after ``y``. The output is always the last
   parameter, and is also a pointer to an array. In the example above,
   ``e`` is a scalar, so only ``e[0]`` is set.

   Then given this class definition, the auto differentiated cost
   function for it can be constructed as follows.

   .. code-block:: c++

    CostFunction* cost_function
        = new AutoDiffCostFunction<MyScalarCostFunctor, 1, 2, 2>(
            new MyScalarCostFunctor(1.0));              ^  ^  ^
                                                        |  |  |
                            Dimension of residual ------+  |  |
                            Dimension of x ----------------+  |
                            Dimension of y -------------------+


   In this example, there is usually an instance for each measurement
   of ``k``.

   In the instantiation above, the template parameters following
   ``MyScalarCostFunction``, ``<1, 2, 2>`` describe the functor as
   computing a 1-dimensional output from two arguments, both
   2-dimensional.

   The framework can currently accommodate cost functions of up to 6
   independent variables, and there is no limit on the dimensionality of
   each of them.

   **WARNING 1** Since the functor will get instantiated with
   different types for ``T``, you must convert from other numeric
   types to ``T`` before mixing computations with other variables
   oftype ``T``. In the example above, this is seen where instead of
   using ``k_`` directly, ``k_`` is wrapped with ``T(k_)``.

   **WARNING 2** A common beginner's error when first using
   :class:`AutoDiffCostFunction` is to get the sizing wrong. In particular,
   there is a tendency to set the template parameters to (dimension of
   residual, number of parameters) instead of passing a dimension
   parameter for *every parameter block*. In the example above, that
   would be ``<MyScalarCostFunction, 1, 2>``, which is missing the 2
   as the last template argument.


:class:`NumericDiffCostFunction`
--------------------------------

.. class:: NumericDiffCostFunction

   .. code-block:: c++

      template <typename CostFunctionNoJacobian,
                NumericDiffMethod method = CENTRAL, int M = 0,
                int N0 = 0, int N1 = 0, int N2 = 0, int N3 = 0, int N4 = 0,
                int N5 = 0, int N6 = 0, int N7 = 0, int N8 = 0, int N9 = 0>
      class NumericDiffCostFunction
        : public SizedCostFunction<M, N0, N1, N2, N3, N4, N5, N6, N7, N8, N9> {
      };


   Create a :class:`CostFunction` as needed by the least squares
   framework with jacobians computed via numeric (a.k.a. finite)
   differentiation. For more details see
   http://en.wikipedia.org/wiki/Numerical_differentiation.

   To get an numerically differentiated :class:`CostFunction`, you
   must define a class with a ``operator()`` (a functor) that computes
   the residuals. The functor must write the computed value in the
   last argument (the only non-``const`` one) and return ``true`` to
   indicate success. e.g., an object of the form

   .. code-block:: c++

     struct ScalarFunctor {
      public:
       bool operator()(const double* const x1,
                       const double* const x2,
                       double* residuals) const;
     }

   For example, consider a scalar error :math:`e = k - x'y`, where
   both :math:`x` and :math:`y` are two-dimensional column vector
   parameters, the prime sign indicates transposition, and :math:`k`
   is a constant. The form of this error, which is the difference
   between a constant and an expression, is a common pattern in least
   squares problems. For example, the value :math:`x'y` might be the
   model expectation for a series of measurements, where there is an
   instance of the cost function for each measurement :math:`k`.

   To write an numerically-differentiable class:`CostFunction` for the
   above model, first define the object

   .. code-block::  c++

     class MyScalarCostFunctor {
       MyScalarCostFunctor(double k): k_(k) {}

       bool operator()(const double* const x,
                       const double* const y,
                       double* residuals) const {
         residuals[0] = k_ - x[0] * y[0] + x[1] * y[1];
         return true;
       }

      private:
       double k_;
     };

   Note that in the declaration of ``operator()`` the input parameters
   ``x`` and ``y`` come first, and are passed as const pointers to
   arrays of ``double`` s. If there were three input parameters, then
   the third input parameter would come after ``y``. The output is
   always the last parameter, and is also a pointer to an array. In
   the example above, the residual is a scalar, so only
   ``residuals[0]`` is set.

   Then given this class definition, the numerically differentiated
   :class:`CostFunction` with central differences used for computing
   the derivative can be constructed as follows.

   .. code-block:: c++

     CostFunction* cost_function
         = new NumericDiffCostFunction<MyScalarCostFunctor, CENTRAL, 1, 2, 2>(
             new MyScalarCostFunctor(1.0));                          ^  ^  ^
                                                                 |   |  |  |
                                     Finite Differencing Scheme -+   |  |  |
                                     Dimension of residual ----------+  |  |
                                     Dimension of x --------------------+  |
                                     Dimension of y -----------------------+

   In this example, there is usually an instance for each measumerent of `k`.

   In the instantiation above, the template parameters following
   ``MyScalarCostFunctor``, ``1, 2, 2``, describe the functor as
   computing a 1-dimensional output from two arguments, both
   2-dimensional.

   The framework can currently accommodate cost functions of up to 10
   independent variables, and there is no limit on the dimensionality
   of each of them.

   The ``CENTRAL`` difference method is considerably more accurate at
   the cost of twice as many function evaluations than forward
   difference. Consider using central differences begin with, and only
   after that works, trying forward difference to improve performance.

   **WARNING** A common beginner's error when first using
   NumericDiffCostFunction is to get the sizing wrong. In particular,
   there is a tendency to set the template parameters to (dimension of
   residual, number of parameters) instead of passing a dimension
   parameter for *every parameter*. In the example above, that would
   be ``<MyScalarCostFunctor, 1, 2>``, which is missing the last ``2``
   argument. Please be careful when setting the size parameters.


   **Alternate Interface**

   For a variety of reason, including compatibility with legacy code,
   :class:`NumericDiffCostFunction` can also take
   :class:`CostFunction` objects as input. The following describes
   how.

   To get a numerically differentiated cost function, define a
   subclass of :class:`CostFunction` such that the
   :func:`CostFunction::Evaluate` function ignores the ``jacobians``
   parameter. The numeric differentiation wrapper will fill in the
   jacobian parameter if nececssary by repeatedly calling the
   :func:`CostFunction::Evaluate` with small changes to the
   appropriate parameters, and computing the slope. For performance,
   the numeric differentiation wrapper class is templated on the
   concrete cost function, even though it could be implemented only in
   terms of the :class:`CostFunction` interface.

   The numerically differentiated version of a cost function for a
   cost function can be constructed as follows:

   .. code-block:: c++

     CostFunction* cost_function
         = new NumericDiffCostFunction<MyCostFunction, CENTRAL, 1, 4, 8>(
             new MyCostFunction(...), TAKE_OWNERSHIP);

   where ``MyCostFunction`` has 1 residual and 2 parameter blocks with
   sizes 4 and 8 respectively. Look at the tests for a more detailed
   example.


:class:`NormalPrior`
--------------------

.. class:: NormalPrior

   .. code-block:: c++

     class NormalPrior: public CostFunction {
      public:
       // Check that the number of rows in the vector b are the same as the
       // number of columns in the matrix A, crash otherwise.
       NormalPrior(const Matrix& A, const Vector& b);

       virtual bool Evaluate(double const* const* parameters,
                             double* residuals,
                             double** jacobians) const;
      };

   Implements a cost function of the form

   .. math::  cost(x) = ||A(x - b)||^2

   where, the matrix A and the vector b are fixed and x is the
   variable. In case the user is interested in implementing a cost
   function of the form

  .. math::  cost(x) = (x - \mu)^T S^{-1} (x - \mu)

  where, :math:`\mu` is a vector and :math:`S` is a covariance matrix,
  then, :math:`A = S^{-1/2}`, i.e the matrix :math:`A` is the square
  root of the inverse of the covariance, also known as the stiffness
  matrix. There are however no restrictions on the shape of
  :math:`A`. It is free to be rectangular, which would be the case if
  the covariance matrix :math:`S` is rank deficient.


:class:`ConditionedCostFunction`
--------------------------------

.. class:: ConditionedCostFunction

   This class allows you to apply different conditioning to the residual
   values of a wrapped cost function. An example where this is useful is
   where you have an existing cost function that produces N values, but you
   want the total cost to be something other than just the sum of these
   squared values - maybe you want to apply a different scaling to some
   values, to change their contribution to the cost.

   Usage:

   .. code-block:: c++

       //  my_cost_function produces N residuals
       CostFunction* my_cost_function = ...
       CHECK_EQ(N, my_cost_function->num_residuals());
       vector<CostFunction*> conditioners;

       //  Make N 1x1 cost functions (1 parameter, 1 residual)
       CostFunction* f_1 = ...
       conditioners.push_back(f_1);

       CostFunction* f_N = ...
       conditioners.push_back(f_N);
       ConditionedCostFunction* ccf =
         new ConditionedCostFunction(my_cost_function, conditioners);


   Now ``ccf`` 's ``residual[i]`` (i=0..N-1) will be passed though the
   :math:`i^{\text{th}}` conditioner.

   .. code-block:: c++

      ccf_residual[i] = f_i(my_cost_function_residual[i])

   and the Jacobian will be affected appropriately.

:class:`CostFunctionToFunctor`
------------------------------

.. class:: CostFunctionToFunctor

   :class:`CostFunctionToFunctor` is an adapter class that allows users to use
   :class:`CostFunction` objects in templated functors which are to be used for
   automatic differentiation.  This allows the user to seamlessly mix
   analytic, numeric and automatic differentiation.

   For example, let us assume that

   .. code-block:: c++

     class IntrinsicProjection : public SizedCostFunction<2, 5, 3> {
       public:
         IntrinsicProjection(const double* observations);
         virtual bool Evaluate(double const* const* parameters,
                               double* residuals,
                               double** jacobians) const;
     };

   is a :class:`CostFunction` that implements the projection of a
   point in its local coordinate system onto its image plane and
   subtracts it from the observed point projection. It can compute its
   residual and either via analytic or numerical differentiation can
   compute its jacobians.

   Now we would like to compose the action of this
   :class:`CostFunction` with the action of camera extrinsics, i.e.,
   rotation and translation. Say we have a templated function

   .. code-block:: c++

      template<typename T>
      void RotateAndTranslatePoint(const T* rotation,
                                   const T* translation,
                                   const T* point,
                                   T* result);


   Then we can now do the following,

   .. code-block:: c++

    struct CameraProjection {
      CameraProjection(double* observation) {
        intrinsic_projection_.reset(
            new CostFunctionToFunctor<2, 5, 3>(new IntrinsicProjection(observation_)));
      }
      template <typename T>
      bool operator(const T* rotation,
                    const T* translation,
                    const T* intrinsics,
                    const T* point,
                    T* residual) const {
        T transformed_point[3];
        RotateAndTranslatePoint(rotation, translation, point, transformed_point);

        //   Note that we call intrinsic_projection_, just like it was
        //   any other templated functor.
        return (*intrinsic_projection_)(intrinsics, transformed_point, residual);
      }

     private:
      scoped_ptr<CostFunctionToFunctor<2,5,3> > intrinsic_projection_;
    };


:class:`NumericDiffFunctor`
---------------------------

.. class:: NumericDiffFunctor

   A wrapper class that takes a variadic functor evaluating a
   function, numerically differentiates it and makes it available as a
   templated functor so that it can be easily used as part of Ceres'
   automatic differentiation framework.

   For example, let us assume that

   .. code-block:: c++

     struct IntrinsicProjection
       IntrinsicProjection(const double* observations);
       bool operator()(const double* calibration,
                       const double* point,
                       double* residuals);
     };

   is a functor that implements the projection of a point in its local
   coordinate system onto its image plane and subtracts it from the
   observed point projection.

   Now we would like to compose the action of this functor with the
   action of camera extrinsics, i.e., rotation and translation, which
   is given by the following templated function

   .. code-block:: c++

     template<typename T>
     void RotateAndTranslatePoint(const T* rotation,
                                  const T* translation,
                                  const T* point,
                                  T* result);

   To compose the extrinsics and intrinsics, we can construct a
   ``CameraProjection`` functor as follows.

   .. code-block:: c++

    struct CameraProjection {
       typedef NumericDiffFunctor<IntrinsicProjection, CENTRAL, 2, 5, 3>
          IntrinsicProjectionFunctor;

      CameraProjection(double* observation) {
        intrinsic_projection_.reset(
            new IntrinsicProjectionFunctor(observation)) {
      }

      template <typename T>
      bool operator(const T* rotation,
                    const T* translation,
                    const T* intrinsics,
                    const T* point,
                    T* residuals) const {
        T transformed_point[3];
        RotateAndTranslatePoint(rotation, translation, point, transformed_point);
        return (*intrinsic_projection_)(intrinsics, transformed_point, residual);
      }

     private:
      scoped_ptr<IntrinsicProjectionFunctor> intrinsic_projection_;
    };

   Here, we made the choice of using ``CENTRAL`` differences to compute
   the jacobian of ``IntrinsicProjection``.

   Now, we are ready to construct an automatically differentiated cost
   function as

   .. code-block:: c++

    CostFunction* cost_function =
        new AutoDiffCostFunction<CameraProjection, 2, 3, 3, 5>(
           new CameraProjection(observations));

   ``cost_function`` now seamlessly integrates automatic
   differentiation of ``RotateAndTranslatePoint`` with a numerically
   differentiated version of ``IntrinsicProjection``.


:class:`LossFunction`
---------------------

.. class:: LossFunction

   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 influence.

   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 the residual for the erroneous measurement will
   result in the entire solution getting pulled away from the optimum
   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
   down-weighted so they do not overly influence the final solution.

   .. code-block:: c++

    class LossFunction {
     public:
      virtual void Evaluate(double s, double out[3]) const = 0;
    };


   The key method is :func:`LossFunction::Evaluate`, which given a
   non-negative scalar ``s``, computes

   .. math:: out = \begin{bmatrix}\rho(s), & \rho'(s), & \rho''(s)\end{bmatrix}

   Here the convention is that the contribution of a term to the cost
   function is given by :math:`\frac{1}{2}\rho(s)`, where :math:`s
   =\|f_i\|^2`. Calling the method with a negative value of :math:`s`
   is an error and the implementations are not required to handle that
   case.

   Most sane choices of :math:`\rho` satisfy:

   .. math::

      \rho(0) &= 0\\
      \rho'(0) &= 1\\
      \rho'(s) &< 1 \text{ in the outlier region}\\
      \rho''(s) &< 0 \text{ in the outlier region}

   so that they mimic the squared cost for small residuals.

   **Scaling**


   Given one robustifier :math:`\rho(s)` one can change the length
   scale at which robustification takes place, by adding a scale
   factor :math:`a > 0` which gives us :math:`\rho(s,a) = a^2 \rho(s /
   a^2)` and the first and second derivatives as :math:`\rho'(s /
   a^2)` and :math:`(1 / a^2) \rho''(s / a^2)` respectively.


   The reason for the appearance of squaring is that :math:`a` is in
   the units of the residual vector norm whereas :math:`s` is a squared
   norm. For applications it is more convenient to specify :math:`a` than
   its square.

Instances
^^^^^^^^^

Ceres includes a number of other loss functions. For simplicity we
described their unscaled versions. The figure below illustrates their
shape graphically. More details can be found in
``include/ceres/loss_function.h``.

.. figure:: loss.png
   :figwidth: 500px
   :height: 400px
   :align: center

   Shape of the various common loss functions.

.. class:: TrivialLoss

      .. math:: \rho(s) = s

.. class:: HuberLoss

   .. math:: \rho(s) = \begin{cases} s & s \le 1\\ 2 \sqrt{s} - 1 & s > 1 \end{cases}

.. class:: SoftLOneLoss

   .. math:: \rho(s) = 2 (\sqrt{1+s} - 1)

.. class:: CauchyLoss

   .. math:: \rho(s) = \log(1 + s)

.. class:: ArctanLoss

   .. math:: \rho(s) = \arctan(s)

.. class:: TolerantLoss

   .. math:: \rho(s,a,b) = b \log(1 + e^{(s - a) / b}) - b \log(1 + e^{-a / b})

.. class:: ComposedLoss

.. class:: ScaledLoss

.. class:: LossFunctionWrapper


Theory
^^^^^^

Let us consider a problem with a single problem and a single parameter
block.

.. math::

 \min_x \frac{1}{2}\rho(f^2(x))


Then, the robustified gradient and the Gauss-Newton Hessian are

.. math::

	g(x) &= \rho'J^\top(x)f(x)\\
	H(x) &= J^\top(x)\left(\rho' + 2 \rho''f(x)f^\top(x)\right)J(x)

where the terms involving the second derivatives of :math:`f(x)` have
been ignored. Note that :math:`H(x)` is indefinite if
:math:`\rho''f(x)^\top f(x) + \frac{1}{2}\rho' < 0`. If this is not
the case, then its possible to re-weight the residual and the Jacobian
matrix such that the corresponding linear least squares problem for
the robustified Gauss-Newton step.


Let :math:`\alpha` be a root of

.. math:: \frac{1}{2}\alpha^2 - \alpha - \frac{\rho''}{\rho'}\|f(x)\|^2 = 0.


Then, define the rescaled residual and Jacobian as

.. math::

	\tilde{f}(x) &= \frac{\sqrt{\rho'}}{1 - \alpha} f(x)\\
	\tilde{J}(x) &= \sqrt{\rho'}\left(1 - \alpha
 	                \frac{f(x)f^\top(x)}{\left\|f(x)\right\|^2} \right)J(x)


In the case :math:`2 \rho''\left\|f(x)\right\|^2 + \rho' \lesssim 0`,
we limit :math:`\alpha \le 1- \epsilon` for some small
:math:`\epsilon`. For more details see [Triggs]_.

With this simple rescaling, one can use any Jacobian based non-linear
least squares algorithm to robustifed non-linear least squares
problems.


:class:`LocalParameterization`
------------------------------

.. class:: LocalParameterization

   .. code-block:: c++

     class LocalParameterization {
      public:
       virtual ~LocalParameterization() {}
       virtual bool Plus(const double* x,
                         const double* delta,
                         double* x_plus_delta) const = 0;
       virtual bool ComputeJacobian(const double* x, double* jacobian) const = 0;
       virtual int GlobalSize() const = 0;
       virtual int LocalSize() const = 0;
     };

   Sometimes the parameters :math:`x` can overparameterize a
   problem. In that case it is desirable to choose a parameterization
   to remove the null directions of the cost. More generally, if
   :math:`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 two 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 :math:`x`, moving in the direction normal to the sphere at
   that point is not useful. Thus a better way to parameterize a point
   on a sphere is to optimize over two dimensional vector
   :math:`\Delta x` in the tangent space at the point on the sphere
   point and then "move" to the point :math:`x + \Delta x`, where the
   move operation involves projecting back onto the sphere. Doing so
   removes a redundant dimension from the optimization, making it
   numerically more robust and efficient.

   More generally we can define a function

   .. math:: x' = \boxplus(x, \Delta x),

   where :math:`x` has the same size as :math:`x`, and :math:`\Delta
   x` is of size less than or equal to :math:`x`. The function
   :math:`\boxplus`, generalizes the definition of vector
   addition. Thus it satisfies the identity

   .. math:: \boxplus(x, 0) = x,\quad \forall x.

   Instances of :class:`LocalParameterization` implement the
   :math:`\boxplus` operation and its derivative with respect to
   :math:`\Delta x` at :math:`\Delta x = 0`.


.. function:: int LocalParameterization::GlobalSize()

   The dimension of the ambient space in which the parameter block
   :math:`x` lives.

.. function:: int LocalParamterization::LocaLocalSize()

   The size of the tangent space
   that :math:`\Delta x` lives in.

.. function:: bool LocalParameterization::Plus(const double* x, const double* delta, double* x_plus_delta) const

    :func:`LocalParameterization::Plus` implements :math:`\boxplus(x,\Delta x)`.

.. function:: bool LocalParameterization::ComputeJacobian(const double* x, double* jacobian) const

   Computes the Jacobian matrix

   .. math:: J = \left . \frac{\partial }{\partial \Delta x} \boxplus(x,\Delta x)\right|_{\Delta x = 0}

   in row major form.

Instances
^^^^^^^^^

.. class:: IdentityParameterization

   A trivial version of :math:`\boxplus` is when :math:`\Delta x` is
   of the same size as :math:`x` and

   .. math::  \boxplus(x, \Delta x) = x + \Delta x

.. class:: SubsetParameterization

   A more interesting case if :math:`x` is a two dimensional vector,
   and the user wishes to hold the first coordinate constant. Then,
   :math:`\Delta x` is a scalar and :math:`\boxplus` is defined as

   .. math::

      \boxplus(x, \Delta x) = x + \left[ \begin{array}{c} 0 \\ 1
                                  \end{array} \right] \Delta x

   :class:`SubsetParameterization` generalizes this construction to
   hold any part of a parameter block constant.

.. class:: QuaternionParameterization

   Another example that occurs commonly in Structure from Motion
   problems is when camera rotations are parameterized using a
   quaternion. There, it is useful only to make updates orthogonal to
   that 4-vector defining the quaternion. One way to do this is to let
   :math:`\Delta x` be a 3 dimensional vector and define
   :math:`\boxplus` to be

    .. math:: \boxplus(x, \Delta x) = \left[ \cos(|\Delta x|), \frac{\sin\left(|\Delta x|\right)}{|\Delta x|} \Delta x \right] * x
      :label: quaternion

   The multiplication between the two 4-vectors on the right hand side
   is the standard quaternion
   product. :class:`QuaternionParameterization` is an implementation
   of :eq:`quaternion`.


:class:`Problem`
----------------

.. class:: Problem

   :class:`Problem` holds the robustified non-linear least squares
   problem :eq:`ceresproblem`. To create a least squares problem, use
   the :func:`Problem::AddResidualBlock` and
   :func:`Problem::AddParameterBlock` methods.

   For example a problem containing 3 parameter blocks of sizes 3, 4
   and 5 respectively and two residual blocks of size 2 and 6:

   .. code-block:: c++

     double x1[] = { 1.0, 2.0, 3.0 };
     double x2[] = { 1.0, 2.0, 3.0, 5.0 };
     double x3[] = { 1.0, 2.0, 3.0, 6.0, 7.0 };

     Problem problem;
     problem.AddResidualBlock(new MyUnaryCostFunction(...), x1);
     problem.AddResidualBlock(new MyBinaryCostFunction(...), x2, x3);

   :func:`Problem::AddResidualBlock` as the name implies, adds a
   residual block to the problem. It adds a :class:`CostFunction` , an
   optional :class:`LossFunction` and connects the
   :class:`CostFunction` to a set of parameter block.

   The cost function carries with it information about the sizes of
   the parameter blocks it expects. The function checks that these
   match the sizes of the parameter blocks listed in
   ``parameter_blocks``. The program aborts if a mismatch is
   detected. ``loss_function`` can be ``NULL``, in which case the cost
   of the term is just the squared norm of the residuals.

   The user has the option of explicitly adding the parameter blocks
   using :func:`Problem::AddParameterBlock`. This causes additional correctness
   checking; however, :func:`Problem::AddResidualBlock` implicitly adds the
   parameter blocks if they are not present, so calling
   :func:`Problem::AddParameterBlock` explicitly is not required.


   :class:`Problem` by default takes ownership of the ``cost_function`` and
   ``loss_function`` pointers. These objects remain live for the life of
   the :class:`Problem` object. If the user wishes to keep control over the
   destruction of these objects, then they can do this by setting the
   corresponding enums in the ``Problem::Options`` struct.


   Note that even though the Problem takes ownership of ``cost_function``
   and ``loss_function``, it does not preclude the user from re-using
   them in another residual block. The destructor takes care to call
   delete on each ``cost_function`` or ``loss_function`` pointer only
   once, regardless of how many residual blocks refer to them.

   :func:`Problem::AddParameterBlock` explicitly adds a parameter
   block to the :class:`Problem`. Optionally it allows the user to
   associate a :class:`LocalParameterization` object with the parameter
   block too. Repeated calls with the same arguments are
   ignored. Repeated calls with the same double pointer but a
   different size results in undefined behaviour.

   You can set any parameter block to be constant using
   :func:`Problem::SetParameterBlockConstant` and undo this using
   :func:`SetParameterBlockVariable`.

   In fact you can set any number of parameter blocks to be constant,
   and Ceres is smart enough to figure out what part of the problem
   you have constructed depends on the parameter blocks that are free
   to change and only spends time solving it. So for example if you
   constructed a problem with a million parameter blocks and 2 million
   residual blocks, but then set all but one parameter blocks to be
   constant and say only 10 residual blocks depend on this one
   non-constant parameter block. Then the computational effort Ceres
   spends in solving this problem will be the same if you had defined
   a problem with one parameter block and 10 residual blocks.

   **Ownership**

   :class:`Problem` by default takes ownership of the
   ``cost_function``, ``loss_function`` and ``local_parameterization``
   pointers. These objects remain live for the life of the
   :class:`Problem`. If the user wishes to keep control over the
   destruction of these objects, then they can do this by setting the
   corresponding enums in the :class:`Problem::Options` struct.

   Even though :class:`Problem` takes ownership of these pointers, it
   does not preclude the user from re-using them in another residual
   or parameter block. The destructor takes care to call delete on
   each pointer only once.


.. function:: ResidualBlockId Problem::AddResidualBlock(CostFunction* cost_function, LossFunction* loss_function, const vector<double*> parameter_blocks)


.. function:: void Problem::AddParameterBlock(double* values, int size, LocalParameterization* local_parameterization)
              void Problem::AddParameterBlock(double* values, int size)


.. function:: void Problem::SetParameterBlockConstant(double* values)

.. function:: void Problem::SetParameterBlockVariable(double* values)


.. function:: void Problem::SetParameterization(double* values, LocalParameterization* local_parameterization)


.. function:: int Problem::NumParameterBlocks() const


.. function:: int Problem::NumParameters() const


.. function:: int Problem::NumResidualBlocks() const


.. function:: int Problem::NumResiduals() const



``rotation.h``
--------------

Many applications of Ceres Solver involve optimization problems where
some of the variables correspond to rotations. To ease the pain of
work with the various representations of rotations (angle-axis,
quaternion and matrix) we provide a handy set of templated
functions. These functions are templated so that the user can use them
within Ceres Solver's automatic differentiation framework.

.. function:: void AngleAxisToQuaternion<T>(T const* angle_axis, T* quaternion)

   Convert a value in combined axis-angle representation to a
   quaternion.

   The value ``angle_axis`` is a triple whose norm is an angle in radians,
   and whose direction is aligned with the axis of rotation, and
   ``quaternion`` is a 4-tuple that will contain the resulting quaternion.

.. function:: void QuaternionToAngleAxis<T>(T const* quaternion, T* angle_axis)

   Convert a quaternion to the equivalent combined axis-angle
   representation.

   The value ``quaternion`` must be a unit quaternion - it is not
   normalized first, and ``angle_axis`` will be filled with a value
   whose norm is the angle of rotation in radians, and whose direction
   is the axis of rotation.

.. function:: void RotationMatrixToAngleAxis<T>(T const * R, T * angle_axis)
.. function:: void AngleAxisToRotationMatrix<T>(T const * angle_axis, T * R)

   Conversions between 3x3 rotation matrix (in column major order) and
   axis-angle rotation representations.

.. function:: void EulerAnglesToRotationMatrix<T>(const T* euler, int row_stride, T* R)

   Conversions between 3x3 rotation matrix (in row major order) and
   Euler angle (in degrees) rotation representations.

   The {pitch,roll,yaw} Euler angles are rotations around the {x,y,z}
   axes, respectively.  They are applied in that same order, so the
   total rotation R is Rz * Ry * Rx.

.. function:: void QuaternionToScaledRotation<T>(const T q[4], T R[3 * 3])

   Convert a 4-vector to a 3x3 scaled rotation matrix.

   The choice of rotation is such that the quaternion
   :math:`\begin{bmatrix} 1 &0 &0 &0\end{bmatrix}` goes to an identity
   matrix and for small :math:`a, b, c` the quaternion
   :math:`\begin{bmatrix}1 &a &b &c\end{bmatrix}` goes to the matrix

   .. math::

     I + 2 \begin{bmatrix} 0 & -c & b \\ c & 0 & -a\\ -b & a & 0
           \end{bmatrix} + O(q^2)

   which corresponds to a Rodrigues approximation, the last matrix
   being the cross-product matrix of :math:`\begin{bmatrix} a& b&
   c\end{bmatrix}`. Together with the property that :math:`R(q1 * q2)
   = R(q1) * R(q2)` this uniquely defines the mapping from :math:`q` to
   :math:`R`.

   The rotation matrix ``R`` is row-major.

   No normalization of the quaternion is performed, i.e.
   :math:`R = \|q\|^2  Q`, where :math:`Q` is an orthonormal matrix
   such that :math:`\det(Q) = 1` and :math:`Q*Q' = I`.


.. function:: void QuaternionToRotation<T>(const T q[4], T R[3 * 3])

   Same as above except that the rotation matrix is normalized by the
   Frobenius norm, so that :math:`R R' = I` (and :math:`\det(R) = 1`).

.. function:: void UnitQuaternionRotatePoint<T>(const T q[4], const T pt[3], T result[3])

   Rotates a point pt by a quaternion q:

   .. math:: \text{result} = R(q)  \text{pt}

   Assumes the quaternion is unit norm. If you pass in a quaternion
   with :math:`|q|^2 = 2` then you WILL NOT get back 2 times the
   result you get for a unit quaternion.


.. function:: void QuaternionRotatePoint<T>(const T q[4], const T pt[3], T result[3])

   With this function you do not need to assume that q has unit norm.
   It does assume that the norm is non-zero.

.. function:: void QuaternionProduct<T>(const T z[4], const T w[4], T zw[4])

   .. math:: zw = z * w

   where :math:`*` is the Quaternion product between 4-vectors.


.. function:: void CrossProduct<T>(const T x[3], const T y[3], T x_cross_y[3])

   .. math:: \text{x_cross_y} = x \times y

.. function:: void AngleAxisRotatePoint<T>(const T angle_axis[3], const T pt[3], T result[3])

   .. math:: y = R(\text{angle_axis}) x