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.. 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