ceres-solver / ceres-solver / 12263e28305a2a43b6c6a6b4f7f76814ab7ffa5f / . / docs / source / nnls_tutorial.rst

.. highlight:: c++ | |

.. default-domain:: cpp | |

.. _chapter-nnls_tutorial: | |

======================== | |

Non-linear Least Squares | |

======================== | |

Introduction | |

============ | |

Ceres can solve bounds constrained robustified non-linear least | |

squares problems of the form | |

.. math:: :label: ceresproblem | |

\min_{\mathbf{x}} &\quad \frac{1}{2}\sum_{i} \rho_i\left(\left\|f_i\left(x_{i_1}, ... ,x_{i_k}\right)\right\|^2\right) \\ | |

\text{s.t.} &\quad l_j \le x_j \le u_j | |

Problems of this form comes up in a broad range of areas across | |

science and engineering - from `fitting curves`_ in statistics, to | |

constructing `3D models from photographs`_ in computer vision. | |

.. _fitting curves: http://en.wikipedia.org/wiki/Nonlinear_regression | |

.. _3D models from photographs: http://en.wikipedia.org/wiki/Bundle_adjustment | |

In this chapter we will learn how to solve :eq:`ceresproblem` using | |

Ceres Solver. Full working code for all the examples described in this | |

chapter and more can be found in the `examples | |

<https://ceres-solver.googlesource.com/ceres-solver/+/master/examples/>`_ | |

directory. | |

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:`l_j` and | |

:math:`u_j` are bounds on the parameter block :math:`x_j`. | |

: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. | |

As a special case, when :math:`\rho_i(x) = x`, i.e., the identity | |

function, and :math:`l_j = -\infty` and :math:`u_j = \infty` we get | |

the more familiar `non-linear least squares problem | |

<http://en.wikipedia.org/wiki/Non-linear_least_squares>`_. | |

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

:label: ceresproblemnonrobust | |

.. _section-hello-world: | |

Hello World! | |

============ | |

To get started, consider the problem of finding the minimum of the | |

function | |

.. math:: \frac{1}{2}(10 -x)^2. | |

This is a trivial problem, whose minimum is located at :math:`x = 10`, | |

but it is a good place to start to illustrate the basics of solving a | |

problem with Ceres [#f1]_. | |

The first step is to write a functor that will evaluate this the | |

function :math:`f(x) = 10 - x`: | |

.. code-block:: c++ | |

struct CostFunctor { | |

template <typename T> | |

bool operator()(const T* const x, T* residual) const { | |

residual[0] = 10.0 - x[0]; | |

return true; | |

} | |

}; | |

The important thing to note here is that ``operator()`` is a templated | |

method, which assumes that all its inputs and outputs are of some type | |

``T``. The use of templating here allows Ceres to call | |

``CostFunctor::operator<T>()``, with ``T=double`` when just the value | |

of the residual is needed, and with a special type ``T=Jet`` when the | |

Jacobians are needed. In :ref:`section-derivatives` we will discuss the | |

various ways of supplying derivatives to Ceres in more detail. | |

Once we have a way of computing the residual function, it is now time | |

to construct a non-linear least squares problem using it and have | |

Ceres solve it. | |

.. code-block:: c++ | |

int main(int argc, char** argv) { | |

google::InitGoogleLogging(argv[0]); | |

// The variable to solve for with its initial value. | |

double initial_x = 5.0; | |

double x = initial_x; | |

// Build the problem. | |

Problem problem; | |

// Set up the only cost function (also known as residual). This uses | |

// auto-differentiation to obtain the derivative (jacobian). | |

CostFunction* cost_function = | |

new AutoDiffCostFunction<CostFunctor, 1, 1>(new CostFunctor); | |

problem.AddResidualBlock(cost_function, nullptr, &x); | |

// Run the solver! | |

Solver::Options options; | |

options.linear_solver_type = ceres::DENSE_QR; | |

options.minimizer_progress_to_stdout = true; | |

Solver::Summary summary; | |

Solve(options, &problem, &summary); | |

std::cout << summary.BriefReport() << "\n"; | |

std::cout << "x : " << initial_x | |

<< " -> " << x << "\n"; | |

return 0; | |

} | |

:class:`AutoDiffCostFunction` takes a ``CostFunctor`` as input, | |

automatically differentiates it and gives it a :class:`CostFunction` | |

interface. | |

Compiling and running `examples/helloworld.cc | |

<https://ceres-solver.googlesource.com/ceres-solver/+/master/examples/helloworld.cc>`_ | |

gives us | |

.. code-block:: bash | |

iter cost cost_change |gradient| |step| tr_ratio tr_radius ls_iter iter_time total_time | |

0 4.512500e+01 0.00e+00 9.50e+00 0.00e+00 0.00e+00 1.00e+04 0 5.33e-04 3.46e-03 | |

1 4.511598e-07 4.51e+01 9.50e-04 9.50e+00 1.00e+00 3.00e+04 1 5.00e-04 4.05e-03 | |

2 5.012552e-16 4.51e-07 3.17e-08 9.50e-04 1.00e+00 9.00e+04 1 1.60e-05 4.09e-03 | |

Ceres Solver Report: Iterations: 2, Initial cost: 4.512500e+01, Final cost: 5.012552e-16, Termination: CONVERGENCE | |

x : 0.5 -> 10 | |

Starting from a :math:`x=5`, the solver in two iterations goes to 10 | |

[#f2]_. The careful reader will note that this is a linear problem and | |

one linear solve should be enough to get the optimal value. The | |

default configuration of the solver is aimed at non-linear problems, | |

and for reasons of simplicity we did not change it in this example. It | |

is indeed possible to obtain the solution to this problem using Ceres | |

in one iteration. Also note that the solver did get very close to the | |

optimal function value of 0 in the very first iteration. We will | |

discuss these issues in greater detail when we talk about convergence | |

and parameter settings for Ceres. | |

.. rubric:: Footnotes | |

.. [#f1] `examples/helloworld.cc | |

<https://ceres-solver.googlesource.com/ceres-solver/+/master/examples/helloworld.cc>`_ | |

.. [#f2] Actually the solver ran for three iterations, and it was | |

by looking at the value returned by the linear solver in the third | |

iteration, it observed that the update to the parameter block was too | |

small and declared convergence. Ceres only prints out the display at | |

the end of an iteration, and terminates as soon as it detects | |

convergence, which is why you only see two iterations here and not | |

three. | |

.. _section-derivatives: | |

Derivatives | |

=========== | |

Ceres Solver like most optimization packages, depends on being able to | |

evaluate the value and the derivatives of each term in the objective | |

function at arbitrary parameter values. Doing so correctly and | |

efficiently is essential to getting good results. Ceres Solver | |

provides a number of ways of doing so. You have already seen one of | |

them in action -- | |

Automatic Differentiation in `examples/helloworld.cc | |

<https://ceres-solver.googlesource.com/ceres-solver/+/master/examples/helloworld.cc>`_ | |

We now consider the other two possibilities. Analytic and numeric | |

derivatives. | |

Numeric Derivatives | |

------------------- | |

In some cases, its not possible to define a templated cost functor, | |

for example when the evaluation of the residual involves a call to a | |

library function that you do not have control over. In such a | |

situation, numerical differentiation can be used. The user defines a | |

functor which computes the residual value and construct a | |

:class:`NumericDiffCostFunction` using it. e.g., for :math:`f(x) = 10 - x` | |

the corresponding functor would be | |

.. code-block:: c++ | |

struct NumericDiffCostFunctor { | |

bool operator()(const double* const x, double* residual) const { | |

residual[0] = 10.0 - x[0]; | |

return true; | |

} | |

}; | |

Which is added to the :class:`Problem` as: | |

.. code-block:: c++ | |

CostFunction* cost_function = | |

new NumericDiffCostFunction<NumericDiffCostFunctor, ceres::CENTRAL, 1, 1>( | |

new NumericDiffCostFunctor); | |

problem.AddResidualBlock(cost_function, nullptr, &x); | |

Notice the parallel from when we were using automatic differentiation | |

.. code-block:: c++ | |

CostFunction* cost_function = | |

new AutoDiffCostFunction<CostFunctor, 1, 1>(new CostFunctor); | |

problem.AddResidualBlock(cost_function, nullptr, &x); | |

The construction looks almost identical to the one used for automatic | |

differentiation, except for an extra template parameter that indicates | |

the kind of finite differencing scheme to be used for computing the | |

numerical derivatives [#f3]_. For more details see the documentation | |

for :class:`NumericDiffCostFunction`. | |

**Generally speaking we recommend automatic differentiation instead of | |

numeric differentiation. The use of C++ templates makes automatic | |

differentiation efficient, whereas numeric differentiation is | |

expensive, prone to numeric errors, and leads to slower convergence.** | |

Analytic Derivatives | |

-------------------- | |

In some cases, using automatic differentiation is not possible. For | |

example, it may be the case that it is more efficient to compute the | |

derivatives in closed form instead of relying on the chain rule used | |

by the automatic differentiation code. | |

In such cases, it is possible to supply your own residual and jacobian | |

computation code. To do this, define a subclass of | |

:class:`CostFunction` or :class:`SizedCostFunction` if you know the | |

sizes of the parameters and residuals at compile time. Here for | |

example is ``SimpleCostFunction`` that implements :math:`f(x) = 10 - | |

x`. | |

.. code-block:: c++ | |

class QuadraticCostFunction : public ceres::SizedCostFunction<1, 1> { | |

public: | |

virtual ~QuadraticCostFunction() {} | |

virtual bool Evaluate(double const* const* parameters, | |

double* residuals, | |

double** jacobians) const { | |

const double x = parameters[0][0]; | |

residuals[0] = 10 - x; | |

// Compute the Jacobian if asked for. | |

if (jacobians != nullptr && jacobians[0] != nullptr) { | |

jacobians[0][0] = -1; | |

} | |

return true; | |

} | |

}; | |

``SimpleCostFunction::Evaluate`` is provided with an input array of | |

``parameters``, an output array ``residuals`` for residuals and an | |

output array ``jacobians`` for Jacobians. The ``jacobians`` array is | |

optional, ``Evaluate`` is expected to check when it is non-null, and | |

if it is the case then fill it with the values of the derivative of | |

the residual function. In this case since the residual function is | |

linear, the Jacobian is constant [#f4]_ . | |

As can be seen from the above code fragments, implementing | |

:class:`CostFunction` objects is a bit tedious. We recommend that | |

unless you have a good reason to manage the jacobian computation | |

yourself, you use :class:`AutoDiffCostFunction` or | |

:class:`NumericDiffCostFunction` to construct your residual blocks. | |

More About Derivatives | |

---------------------- | |

Computing derivatives is by far the most complicated part of using | |

Ceres, and depending on the circumstance the user may need more | |

sophisticated ways of computing derivatives. This section just | |

scratches the surface of how derivatives can be supplied to | |

Ceres. Once you are comfortable with using | |

:class:`NumericDiffCostFunction` and :class:`AutoDiffCostFunction` we | |

recommend taking a look at :class:`DynamicAutoDiffCostFunction`, | |

:class:`CostFunctionToFunctor`, :class:`NumericDiffFunctor` and | |

:class:`ConditionedCostFunction` for more advanced ways of | |

constructing and computing cost functions. | |

.. rubric:: Footnotes | |

.. [#f3] `examples/helloworld_numeric_diff.cc | |

<https://ceres-solver.googlesource.com/ceres-solver/+/master/examples/helloworld_numeric_diff.cc>`_. | |

.. [#f4] `examples/helloworld_analytic_diff.cc | |

<https://ceres-solver.googlesource.com/ceres-solver/+/master/examples/helloworld_analytic_diff.cc>`_. | |

.. _section-powell: | |

Powell's Function | |

================= | |

Consider now a slightly more complicated example -- the minimization | |

of Powell's function. Let :math:`x = \left[x_1, x_2, x_3, x_4 \right]` | |

and | |

.. math:: | |

\begin{align} | |

f_1(x) &= x_1 + 10x_2 \\ | |

f_2(x) &= \sqrt{5} (x_3 - x_4)\\ | |

f_3(x) &= (x_2 - 2x_3)^2\\ | |

f_4(x) &= \sqrt{10} (x_1 - x_4)^2\\ | |

F(x) &= \left[f_1(x),\ f_2(x),\ f_3(x),\ f_4(x) \right] | |

\end{align} | |

:math:`F(x)` is a function of four parameters, has four residuals | |

and we wish to find :math:`x` such that :math:`\frac{1}{2}\|F(x)\|^2` | |

is minimized. | |

Again, the first step is to define functors that evaluate of the terms | |

in the objective functor. Here is the code for evaluating | |

:math:`f_4(x_1, x_4)`: | |

.. code-block:: c++ | |

struct F4 { | |

template <typename T> | |

bool operator()(const T* const x1, const T* const x4, T* residual) const { | |

residual[0] = sqrt(10.0) * (x1[0] - x4[0]) * (x1[0] - x4[0]); | |

return true; | |

} | |

}; | |

Similarly, we can define classes ``F1``, ``F2`` and ``F3`` to evaluate | |

:math:`f_1(x_1, x_2)`, :math:`f_2(x_3, x_4)` and :math:`f_3(x_2, x_3)` | |

respectively. Using these, the problem can be constructed as follows: | |

.. code-block:: c++ | |

double x1 = 3.0; double x2 = -1.0; double x3 = 0.0; double x4 = 1.0; | |

Problem problem; | |

// Add residual terms to the problem using the autodiff | |

// wrapper to get the derivatives automatically. | |

problem.AddResidualBlock( | |

new AutoDiffCostFunction<F1, 1, 1, 1>(new F1), nullptr, &x1, &x2); | |

problem.AddResidualBlock( | |

new AutoDiffCostFunction<F2, 1, 1, 1>(new F2), nullptr, &x3, &x4); | |

problem.AddResidualBlock( | |

new AutoDiffCostFunction<F3, 1, 1, 1>(new F3), nullptr, &x2, &x3); | |

problem.AddResidualBlock( | |

new AutoDiffCostFunction<F4, 1, 1, 1>(new F4), nullptr, &x1, &x4); | |

Note that each ``ResidualBlock`` only depends on the two parameters | |

that the corresponding residual object depends on and not on all four | |

parameters. Compiling and running `examples/powell.cc | |

<https://ceres-solver.googlesource.com/ceres-solver/+/master/examples/powell.cc>`_ | |

gives us: | |

.. code-block:: bash | |

Initial x1 = 3, x2 = -1, x3 = 0, x4 = 1 | |

iter cost cost_change |gradient| |step| tr_ratio tr_radius ls_iter iter_time total_time | |

0 1.075000e+02 0.00e+00 1.55e+02 0.00e+00 0.00e+00 1.00e+04 0 4.95e-04 2.30e-03 | |

1 5.036190e+00 1.02e+02 2.00e+01 2.16e+00 9.53e-01 3.00e+04 1 4.39e-05 2.40e-03 | |

2 3.148168e-01 4.72e+00 2.50e+00 6.23e-01 9.37e-01 9.00e+04 1 9.06e-06 2.43e-03 | |

3 1.967760e-02 2.95e-01 3.13e-01 3.08e-01 9.37e-01 2.70e+05 1 8.11e-06 2.45e-03 | |

4 1.229900e-03 1.84e-02 3.91e-02 1.54e-01 9.37e-01 8.10e+05 1 6.91e-06 2.48e-03 | |

5 7.687123e-05 1.15e-03 4.89e-03 7.69e-02 9.37e-01 2.43e+06 1 7.87e-06 2.50e-03 | |

6 4.804625e-06 7.21e-05 6.11e-04 3.85e-02 9.37e-01 7.29e+06 1 5.96e-06 2.52e-03 | |

7 3.003028e-07 4.50e-06 7.64e-05 1.92e-02 9.37e-01 2.19e+07 1 5.96e-06 2.55e-03 | |

8 1.877006e-08 2.82e-07 9.54e-06 9.62e-03 9.37e-01 6.56e+07 1 5.96e-06 2.57e-03 | |

9 1.173223e-09 1.76e-08 1.19e-06 4.81e-03 9.37e-01 1.97e+08 1 7.87e-06 2.60e-03 | |

10 7.333425e-11 1.10e-09 1.49e-07 2.40e-03 9.37e-01 5.90e+08 1 6.20e-06 2.63e-03 | |

11 4.584044e-12 6.88e-11 1.86e-08 1.20e-03 9.37e-01 1.77e+09 1 6.91e-06 2.65e-03 | |

12 2.865573e-13 4.30e-12 2.33e-09 6.02e-04 9.37e-01 5.31e+09 1 5.96e-06 2.67e-03 | |

13 1.791438e-14 2.69e-13 2.91e-10 3.01e-04 9.37e-01 1.59e+10 1 7.15e-06 2.69e-03 | |

Ceres Solver v1.12.0 Solve Report | |

---------------------------------- | |

Original Reduced | |

Parameter blocks 4 4 | |

Parameters 4 4 | |

Residual blocks 4 4 | |

Residual 4 4 | |

Minimizer TRUST_REGION | |

Dense linear algebra library EIGEN | |

Trust region strategy LEVENBERG_MARQUARDT | |

Given Used | |

Linear solver DENSE_QR DENSE_QR | |

Threads 1 1 | |

Linear solver threads 1 1 | |

Cost: | |

Initial 1.075000e+02 | |

Final 1.791438e-14 | |

Change 1.075000e+02 | |

Minimizer iterations 14 | |

Successful steps 14 | |

Unsuccessful steps 0 | |

Time (in seconds): | |

Preprocessor 0.002 | |

Residual evaluation 0.000 | |

Jacobian evaluation 0.000 | |

Linear solver 0.000 | |

Minimizer 0.001 | |

Postprocessor 0.000 | |

Total 0.005 | |

Termination: CONVERGENCE (Gradient tolerance reached. Gradient max norm: 3.642190e-11 <= 1.000000e-10) | |

Final x1 = 0.000292189, x2 = -2.92189e-05, x3 = 4.79511e-05, x4 = 4.79511e-05 | |

It is easy to see that the optimal solution to this problem is at | |

:math:`x_1=0, x_2=0, x_3=0, x_4=0` with an objective function value of | |

:math:`0`. In 10 iterations, Ceres finds a solution with an objective | |

function value of :math:`4\times 10^{-12}`. | |

.. rubric:: Footnotes | |

.. [#f5] `examples/powell.cc | |

<https://ceres-solver.googlesource.com/ceres-solver/+/master/examples/powell.cc>`_. | |

.. _section-fitting: | |

Curve Fitting | |

============= | |

The examples we have seen until now are simple optimization problems | |

with no data. The original purpose of least squares and non-linear | |

least squares analysis was fitting curves to data. It is only | |

appropriate that we now consider an example of such a problem | |

[#f6]_. It contains data generated by sampling the curve :math:`y = | |

e^{0.3x + 0.1}` and adding Gaussian noise with standard deviation | |

:math:`\sigma = 0.2`. Let us fit some data to the curve | |

.. math:: y = e^{mx + c}. | |

We begin by defining a templated object to evaluate the | |

residual. There will be a residual for each observation. | |

.. code-block:: c++ | |

struct ExponentialResidual { | |

ExponentialResidual(double x, double y) | |

: x_(x), y_(y) {} | |

template <typename T> | |

bool operator()(const T* const m, const T* const c, T* residual) const { | |

residual[0] = y_ - exp(m[0] * x_ + c[0]); | |

return true; | |

} | |

private: | |

// Observations for a sample. | |

const double x_; | |

const double y_; | |

}; | |

Assuming the observations are in a :math:`2n` sized array called | |

``data`` the problem construction is a simple matter of creating a | |

:class:`CostFunction` for every observation. | |

.. code-block:: c++ | |

double m = 0.0; | |

double c = 0.0; | |

Problem problem; | |

for (int i = 0; i < kNumObservations; ++i) { | |

CostFunction* cost_function = | |

new AutoDiffCostFunction<ExponentialResidual, 1, 1, 1>( | |

new ExponentialResidual(data[2 * i], data[2 * i + 1])); | |

problem.AddResidualBlock(cost_function, nullptr, &m, &c); | |

} | |

Compiling and running `examples/curve_fitting.cc | |

<https://ceres-solver.googlesource.com/ceres-solver/+/master/examples/curve_fitting.cc>`_ | |

gives us: | |

.. code-block:: bash | |

iter cost cost_change |gradient| |step| tr_ratio tr_radius ls_iter iter_time total_time | |

0 1.211734e+02 0.00e+00 3.61e+02 0.00e+00 0.00e+00 1.00e+04 0 5.34e-04 2.56e-03 | |

1 1.211734e+02 -2.21e+03 0.00e+00 7.52e-01 -1.87e+01 5.00e+03 1 4.29e-05 3.25e-03 | |

2 1.211734e+02 -2.21e+03 0.00e+00 7.51e-01 -1.86e+01 1.25e+03 1 1.10e-05 3.28e-03 | |

3 1.211734e+02 -2.19e+03 0.00e+00 7.48e-01 -1.85e+01 1.56e+02 1 1.41e-05 3.31e-03 | |

4 1.211734e+02 -2.02e+03 0.00e+00 7.22e-01 -1.70e+01 9.77e+00 1 1.00e-05 3.34e-03 | |

5 1.211734e+02 -7.34e+02 0.00e+00 5.78e-01 -6.32e+00 3.05e-01 1 1.00e-05 3.36e-03 | |

6 3.306595e+01 8.81e+01 4.10e+02 3.18e-01 1.37e+00 9.16e-01 1 2.79e-05 3.41e-03 | |

7 6.426770e+00 2.66e+01 1.81e+02 1.29e-01 1.10e+00 2.75e+00 1 2.10e-05 3.45e-03 | |

8 3.344546e+00 3.08e+00 5.51e+01 3.05e-02 1.03e+00 8.24e+00 1 2.10e-05 3.48e-03 | |

9 1.987485e+00 1.36e+00 2.33e+01 8.87e-02 9.94e-01 2.47e+01 1 2.10e-05 3.52e-03 | |

10 1.211585e+00 7.76e-01 8.22e+00 1.05e-01 9.89e-01 7.42e+01 1 2.10e-05 3.56e-03 | |

11 1.063265e+00 1.48e-01 1.44e+00 6.06e-02 9.97e-01 2.22e+02 1 2.60e-05 3.61e-03 | |

12 1.056795e+00 6.47e-03 1.18e-01 1.47e-02 1.00e+00 6.67e+02 1 2.10e-05 3.64e-03 | |

13 1.056751e+00 4.39e-05 3.79e-03 1.28e-03 1.00e+00 2.00e+03 1 2.10e-05 3.68e-03 | |

Ceres Solver Report: Iterations: 13, Initial cost: 1.211734e+02, Final cost: 1.056751e+00, Termination: CONVERGENCE | |

Initial m: 0 c: 0 | |

Final m: 0.291861 c: 0.131439 | |

Starting from parameter values :math:`m = 0, c=0` with an initial | |

objective function value of :math:`121.173` Ceres finds a solution | |

:math:`m= 0.291861, c = 0.131439` with an objective function value of | |

:math:`1.05675`. These values are a bit different than the | |

parameters of the original model :math:`m=0.3, c= 0.1`, but this is | |

expected. When reconstructing a curve from noisy data, we expect to | |

see such deviations. Indeed, if you were to evaluate the objective | |

function for :math:`m=0.3, c=0.1`, the fit is worse with an objective | |

function value of :math:`1.082425`. The figure below illustrates the fit. | |

.. figure:: least_squares_fit.png | |

:figwidth: 500px | |

:height: 400px | |

:align: center | |

Least squares curve fitting. | |

.. rubric:: Footnotes | |

.. [#f6] `examples/curve_fitting.cc | |

<https://ceres-solver.googlesource.com/ceres-solver/+/master/examples/curve_fitting.cc>`_ | |

Robust Curve Fitting | |

===================== | |

Now suppose the data we are given has some outliers, i.e., we have | |

some points that do not obey the noise model. If we were to use the | |

code above to fit such data, we would get a fit that looks as | |

below. Notice how the fitted curve deviates from the ground truth. | |

.. figure:: non_robust_least_squares_fit.png | |

:figwidth: 500px | |

:height: 400px | |

:align: center | |

To deal with outliers, a standard technique is to use a | |

:class:`LossFunction`. Loss functions reduce the influence of | |

residual blocks with high residuals, usually the ones corresponding to | |

outliers. To associate a loss function with a residual block, we change | |

.. code-block:: c++ | |

problem.AddResidualBlock(cost_function, nullptr , &m, &c); | |

to | |

.. code-block:: c++ | |

problem.AddResidualBlock(cost_function, new CauchyLoss(0.5) , &m, &c); | |

:class:`CauchyLoss` is one of the loss functions that ships with Ceres | |

Solver. The argument :math:`0.5` specifies the scale of the loss | |

function. As a result, we get the fit below [#f7]_. Notice how the | |

fitted curve moves back closer to the ground truth curve. | |

.. figure:: robust_least_squares_fit.png | |

:figwidth: 500px | |

:height: 400px | |

:align: center | |

Using :class:`LossFunction` to reduce the effect of outliers on a | |

least squares fit. | |

.. rubric:: Footnotes | |

.. [#f7] `examples/robust_curve_fitting.cc | |

<https://ceres-solver.googlesource.com/ceres-solver/+/master/examples/robust_curve_fitting.cc>`_ | |

Bundle Adjustment | |

================= | |

One of the main reasons for writing Ceres was our need to solve large | |

scale bundle adjustment problems [HartleyZisserman]_, [Triggs]_. | |

Given a set of measured image feature locations and correspondences, | |

the goal of bundle adjustment is to find 3D point positions and camera | |

parameters that minimize the reprojection error. This optimization | |

problem is usually formulated as a non-linear least squares problem, | |

where the error is the squared :math:`L_2` norm of the difference between | |

the observed feature location and the projection of the corresponding | |

3D point on the image plane of the camera. Ceres has extensive support | |

for solving bundle adjustment problems. | |

Let us solve a problem from the `BAL | |

<http://grail.cs.washington.edu/projects/bal/>`_ dataset [#f8]_. | |

The first step as usual is to define a templated functor that computes | |

the reprojection error/residual. The structure of the functor is | |

similar to the ``ExponentialResidual``, in that there is an | |

instance of this object responsible for each image observation. | |

Each residual in a BAL problem depends on a three dimensional point | |

and a nine parameter camera. The nine parameters defining the camera | |

are: three for rotation as a Rodrigues' axis-angle vector, three | |

for translation, one for focal length and two for radial distortion. | |

The details of this camera model can be found the `Bundler homepage | |

<http://phototour.cs.washington.edu/bundler/>`_ and the `BAL homepage | |

<http://grail.cs.washington.edu/projects/bal/>`_. | |

.. code-block:: c++ | |

struct SnavelyReprojectionError { | |

SnavelyReprojectionError(double observed_x, double observed_y) | |

: observed_x(observed_x), observed_y(observed_y) {} | |

template <typename T> | |

bool operator()(const T* const camera, | |

const T* const point, | |

T* residuals) const { | |

// camera[0,1,2] are the angle-axis rotation. | |

T p[3]; | |

ceres::AngleAxisRotatePoint(camera, point, p); | |

// camera[3,4,5] are the translation. | |

p[0] += camera[3]; p[1] += camera[4]; p[2] += camera[5]; | |

// Compute the center of distortion. The sign change comes from | |

// the camera model that Noah Snavely's Bundler assumes, whereby | |

// the camera coordinate system has a negative z axis. | |

T xp = - p[0] / p[2]; | |

T yp = - p[1] / p[2]; | |

// Apply second and fourth order radial distortion. | |

const T& l1 = camera[7]; | |

const T& l2 = camera[8]; | |

T r2 = xp*xp + yp*yp; | |

T distortion = 1.0 + r2 * (l1 + l2 * r2); | |

// Compute final projected point position. | |

const T& focal = camera[6]; | |

T predicted_x = focal * distortion * xp; | |

T predicted_y = focal * distortion * yp; | |

// The error is the difference between the predicted and observed position. | |

residuals[0] = predicted_x - T(observed_x); | |

residuals[1] = predicted_y - T(observed_y); | |

return true; | |

} | |

// Factory to hide the construction of the CostFunction object from | |

// the client code. | |

static ceres::CostFunction* Create(const double observed_x, | |

const double observed_y) { | |

return (new ceres::AutoDiffCostFunction<SnavelyReprojectionError, 2, 9, 3>( | |

new SnavelyReprojectionError(observed_x, observed_y))); | |

} | |

double observed_x; | |

double observed_y; | |

}; | |

Note that unlike the examples before, this is a non-trivial function | |

and computing its analytic Jacobian is a bit of a pain. Automatic | |

differentiation makes life much simpler. The function | |

:func:`AngleAxisRotatePoint` and other functions for manipulating | |

rotations can be found in ``include/ceres/rotation.h``. | |

Given this functor, the bundle adjustment problem can be constructed | |

as follows: | |

.. code-block:: c++ | |

ceres::Problem problem; | |

for (int i = 0; i < bal_problem.num_observations(); ++i) { | |

ceres::CostFunction* cost_function = | |

SnavelyReprojectionError::Create( | |

bal_problem.observations()[2 * i + 0], | |

bal_problem.observations()[2 * i + 1]); | |

problem.AddResidualBlock(cost_function, | |

nullptr /* squared loss */, | |

bal_problem.mutable_camera_for_observation(i), | |

bal_problem.mutable_point_for_observation(i)); | |

} | |

Notice that the problem construction for bundle adjustment is very | |

similar to the curve fitting example -- one term is added to the | |

objective function per observation. | |

Since this is a large sparse problem (well large for ``DENSE_QR`` | |

anyways), one way to solve this problem is to set | |

:member:`Solver::Options::linear_solver_type` to | |

``SPARSE_NORMAL_CHOLESKY`` and call :func:`Solve`. And while this is | |

a reasonable thing to do, bundle adjustment problems have a special | |

sparsity structure that can be exploited to solve them much more | |

efficiently. Ceres provides three specialized solvers (collectively | |

known as Schur-based solvers) for this task. The example code uses the | |

simplest of them ``DENSE_SCHUR``. | |

.. code-block:: c++ | |

ceres::Solver::Options options; | |

options.linear_solver_type = ceres::DENSE_SCHUR; | |

options.minimizer_progress_to_stdout = true; | |

ceres::Solver::Summary summary; | |

ceres::Solve(options, &problem, &summary); | |

std::cout << summary.FullReport() << "\n"; | |

For a more sophisticated bundle adjustment example which demonstrates | |

the use of Ceres' more advanced features including its various linear | |

solvers, robust loss functions and manifolds see | |

`examples/bundle_adjuster.cc | |

<https://ceres-solver.googlesource.com/ceres-solver/+/master/examples/bundle_adjuster.cc>`_ | |

.. rubric:: Footnotes | |

.. [#f8] `examples/simple_bundle_adjuster.cc | |

<https://ceres-solver.googlesource.com/ceres-solver/+/master/examples/simple_bundle_adjuster.cc>`_ | |

Other Examples | |

============== | |

Besides the examples in this chapter, the `example | |

<https://ceres-solver.googlesource.com/ceres-solver/+/master/examples/>`_ | |

directory contains a number of other examples: | |

#. `bundle_adjuster.cc | |

<https://ceres-solver.googlesource.com/ceres-solver/+/master/examples/bundle_adjuster.cc>`_ | |

shows how to use the various features of Ceres to solve bundle | |

adjustment problems. | |

#. `circle_fit.cc | |

<https://ceres-solver.googlesource.com/ceres-solver/+/master/examples/circle_fit.cc>`_ | |

shows how to fit data to a circle. | |

#. `ellipse_approximation.cc | |

<https://ceres-solver.googlesource.com/ceres-solver/+/master/examples/ellipse_approximation.cc>`_ | |

fits points randomly distributed on an ellipse with an approximate | |

line segment contour. This is done by jointly optimizing the | |

control points of the line segment contour along with the preimage | |

positions for the data points. The purpose of this example is to | |

show an example use case for ``Solver::Options::dynamic_sparsity``, | |

and how it can benefit problems which are numerically dense but | |

dynamically sparse. | |

#. `denoising.cc | |

<https://ceres-solver.googlesource.com/ceres-solver/+/master/examples/denoising.cc>`_ | |

implements image denoising using the `Fields of Experts | |

<http://www.gris.informatik.tu-darmstadt.de/~sroth/research/foe/index.html>`_ | |

model. | |

#. `nist.cc | |

<https://ceres-solver.googlesource.com/ceres-solver/+/master/examples/nist.cc>`_ | |

implements and attempts to solves the `NIST | |

<http://www.itl.nist.gov/div898/strd/nls/nls_main.shtml>`_ | |

non-linear regression problems. | |

#. `more_garbow_hillstrom.cc | |

<https://ceres-solver.googlesource.com/ceres-solver/+/master/examples/more_garbow_hillstrom.cc>`_ | |

A subset of the test problems from the paper | |

Testing Unconstrained Optimization Software | |

Jorge J. More, Burton S. Garbow and Kenneth E. Hillstrom | |

ACM Transactions on Mathematical Software, 7(1), pp. 17-41, 1981 | |

which were augmented with bounds and used for testing bounds | |

constrained optimization algorithms by | |

A Trust Region Approach to Linearly Constrained Optimization | |

David M. Gay | |

Numerical Analysis (Griffiths, D.F., ed.), pp. 72-105 | |

Lecture Notes in Mathematics 1066, Springer Verlag, 1984. | |

#. `libmv_bundle_adjuster.cc | |

<https://ceres-solver.googlesource.com/ceres-solver/+/master/examples/libmv_bundle_adjuster.cc>`_ | |

is the bundle adjustment algorithm used by `Blender <www.blender.org>`_/libmv. | |

#. `libmv_homography.cc | |

<https://ceres-solver.googlesource.com/ceres-solver/+/master/examples/libmv_homography.cc>`_ | |

This file demonstrates solving for a homography between two sets of | |

points and using a custom exit criterion by having a callback check | |

for image-space error. | |

#. `robot_pose_mle.cc | |

<https://ceres-solver.googlesource.com/ceres-solver/+/master/examples/robot_pose_mle.cc>`_ | |

This example demonstrates how to use the ``DynamicAutoDiffCostFunction`` | |

variant of CostFunction. The ``DynamicAutoDiffCostFunction`` is meant to | |

be used in cases where the number of parameter blocks or the sizes are not | |

known at compile time. | |

This example simulates a robot traversing down a 1-dimension hallway with | |

noise odometry readings and noisy range readings of the end of the hallway. | |

By fusing the noisy odometry and sensor readings this example demonstrates | |

how to compute the maximum likelihood estimate (MLE) of the robot's pose at | |

each timestep. | |

#. `slam/pose_graph_2d/pose_graph_2d.cc | |

<https://ceres-solver.googlesource.com/ceres-solver/+/master/examples/slam/pose_graph_2d/pose_graph_2d.cc>`_ | |

The Simultaneous Localization and Mapping (SLAM) problem consists of building | |

a map of an unknown environment while simultaneously localizing against this | |

map. The main difficulty of this problem stems from not having any additional | |

external aiding information such as GPS. SLAM has been considered one of the | |

fundamental challenges of robotics. There are many resources on SLAM | |

[#f9]_. A pose graph optimization problem is one example of a SLAM | |

problem. The following explains how to formulate the pose graph based SLAM | |

problem in 2-Dimensions with relative pose constraints. | |

Consider a robot moving in a 2-Dimensional plane. The robot has access to a | |

set of sensors such as wheel odometry or a laser range scanner. From these | |

raw measurements, we want to estimate the trajectory of the robot as well as | |

build a map of the environment. In order to reduce the computational | |

complexity of the problem, the pose graph approach abstracts the raw | |

measurements away. Specifically, it creates a graph of nodes which represent | |

the pose of the robot, and edges which represent the relative transformation | |

(delta position and orientation) between the two nodes. The edges are virtual | |

measurements derived from the raw sensor measurements, e.g. by integrating | |

the raw wheel odometry or aligning the laser range scans acquired from the | |

robot. A visualization of the resulting graph is shown below. | |

.. figure:: slam2d.png | |

:figwidth: 500px | |

:height: 400px | |

:align: center | |

Visual representation of a graph SLAM problem. | |

The figure depicts the pose of the robot as the triangles, the measurements | |

are indicated by the connecting lines, and the loop closure measurements are | |

shown as dotted lines. Loop closures are measurements between non-sequential | |

robot states and they reduce the accumulation of error over time. The | |

following will describe the mathematical formulation of the pose graph | |

problem. | |

The robot at timestamp :math:`t` has state :math:`x_t = [p^T, \psi]^T` where | |

:math:`p` is a 2D vector that represents the position in the plane and | |

:math:`\psi` is the orientation in radians. The measurement of the relative | |

transform between the robot state at two timestamps :math:`a` and :math:`b` | |

is given as: :math:`z_{ab} = [\hat{p}_{ab}^T, \hat{\psi}_{ab}]`. The residual | |

implemented in the Ceres cost function which computes the error between the | |

measurement and the predicted measurement is: | |

.. math:: r_{ab} = | |

\left[ | |

\begin{array}{c} | |

R_a^T\left(p_b - p_a\right) - \hat{p}_{ab} \\ | |

\mathrm{Normalize}\left(\psi_b - \psi_a - \hat{\psi}_{ab}\right) | |

\end{array} | |

\right] | |

where the function :math:`\mathrm{Normalize}()` normalizes the angle in the range | |

:math:`[-\pi,\pi)`, and :math:`R` is the rotation matrix given by | |

.. math:: R_a = | |

\left[ | |

\begin{array}{cc} | |

\cos \psi_a & -\sin \psi_a \\ | |

\sin \psi_a & \cos \psi_a \\ | |

\end{array} | |

\right] | |

To finish the cost function, we need to weight the residual by the | |

uncertainty of the measurement. Hence, we pre-multiply the residual by the | |

inverse square root of the covariance matrix for the measurement, | |

i.e. :math:`\Sigma_{ab}^{-\frac{1}{2}} r_{ab}` where :math:`\Sigma_{ab}` is | |

the covariance. | |

Lastly, we use a manifold to normalize the orientation in the range | |

:math:`[-\pi,\pi)`. Specially, we define the | |

:member:`AngleManifold::Plus()` function to be: | |

:math:`\mathrm{Normalize}(\psi + \Delta)` and | |

::member::`AngleManifold::Minus()` function to be | |

:math:`\mathrm{Normalize}(y) - \mathrm{Normalize}(x)`. | |

This package includes an executable :member:`pose_graph_2d` that will read a | |

problem definition file. This executable can work with any 2D problem | |

definition that uses the g2o format. It would be relatively straightforward | |

to implement a new reader for a different format such as TORO or | |

others. :member:`pose_graph_2d` will print the Ceres solver full summary and | |

then output to disk the original and optimized poses (``poses_original.txt`` | |

and ``poses_optimized.txt``, respectively) of the robot in the following | |

format: | |

.. code-block:: bash | |

pose_id x y yaw_radians | |

pose_id x y yaw_radians | |

pose_id x y yaw_radians | |

where ``pose_id`` is the corresponding integer ID from the file | |

definition. Note, the file will be sorted in ascending order for the | |

``pose_id``. | |

The executable :member:`pose_graph_2d` expects the first argument to be | |

the path to the problem definition. To run the executable, | |

.. code-block:: bash | |

/path/to/bin/pose_graph_2d /path/to/dataset/dataset.g2o | |

A python script is provided to visualize the resulting output files. | |

.. code-block:: bash | |

/path/to/repo/examples/slam/pose_graph_2d/plot_results.py --optimized_poses ./poses_optimized.txt --initial_poses ./poses_original.txt | |

As an example, a standard synthetic benchmark dataset [#f10]_ created by | |

Edwin Olson which has 3500 nodes in a grid world with a total of 5598 edges | |

was solved. Visualizing the results with the provided script produces: | |

.. figure:: manhattan_olson_3500_result.png | |

:figwidth: 600px | |

:height: 600px | |

:align: center | |

with the original poses in green and the optimized poses in blue. As shown, | |

the optimized poses more closely match the underlying grid world. Note, the | |

left side of the graph has a small yaw drift due to a lack of relative | |

constraints to provide enough information to reconstruct the trajectory. | |

.. rubric:: Footnotes | |

.. [#f9] Giorgio Grisetti, Rainer Kummerle, Cyrill Stachniss, Wolfram | |

Burgard. A Tutorial on Graph-Based SLAM. IEEE Intelligent Transportation | |

Systems Magazine, 52(3):199-222, 2010. | |

.. [#f10] E. Olson, J. Leonard, and S. Teller, “Fast iterative optimization of | |

pose graphs with poor initial estimates,” in Robotics and Automation | |

(ICRA), IEEE International Conference on, 2006, pp. 2262-2269. | |

#. `slam/pose_graph_3d/pose_graph_3d.cc | |

<https://ceres-solver.googlesource.com/ceres-solver/+/master/examples/slam/pose_graph_3d/pose_graph_3d.cc>`_ | |

The following explains how to formulate the pose graph based SLAM problem in | |

3-Dimensions with relative pose constraints. The example also illustrates how | |

to use Eigen's geometry module with Ceres's automatic differentiation | |

functionality. | |

The robot at timestamp :math:`t` has state :math:`x_t = [p^T, q^T]^T` where | |

:math:`p` is a 3D vector that represents the position and :math:`q` is the | |

orientation represented as an Eigen quaternion. The measurement of the | |

relative transform between the robot state at two timestamps :math:`a` and | |

:math:`b` is given as: :math:`z_{ab} = [\hat{p}_{ab}^T, \hat{q}_{ab}^T]^T`. | |

The residual implemented in the Ceres cost function which computes the error | |

between the measurement and the predicted measurement is: | |

.. math:: r_{ab} = | |

\left[ | |

\begin{array}{c} | |

R(q_a)^{T} (p_b - p_a) - \hat{p}_{ab} \\ | |

2.0 \mathrm{vec}\left((q_a^{-1} q_b) \hat{q}_{ab}^{-1}\right) | |

\end{array} | |

\right] | |

where the function :math:`\mathrm{vec}()` returns the vector part of the | |

quaternion, i.e. :math:`[q_x, q_y, q_z]`, and :math:`R(q)` is the rotation | |

matrix for the quaternion. | |

To finish the cost function, we need to weight the residual by the | |

uncertainty of the measurement. Hence, we pre-multiply the residual by the | |

inverse square root of the covariance matrix for the measurement, | |

i.e. :math:`\Sigma_{ab}^{-\frac{1}{2}} r_{ab}` where :math:`\Sigma_{ab}` is | |

the covariance. | |

Given that we are using a quaternion to represent the orientation, | |

we need to use a manifold (:class:`EigenQuaternionManifold`) to | |

only apply updates orthogonal to the 4-vector defining the | |

quaternion. Eigen's quaternion uses a different internal memory | |

layout for the elements of the quaternion than what is commonly | |

used. Specifically, Eigen stores the elements in memory as | |

:math:`[x, y, z, w]` where the real part is last whereas it is | |

typically stored first. Note, when creating an Eigen quaternion | |

through the constructor the elements are accepted in :math:`w`, | |

:math:`x`, :math:`y`, :math:`z` order. Since Ceres operates on | |

parameter blocks which are raw double pointers this difference is | |

important and requires a different parameterization. | |

This package includes an executable :member:`pose_graph_3d` that will read a | |

problem definition file. This executable can work with any 3D problem | |

definition that uses the g2o format with quaternions used for the orientation | |

representation. It would be relatively straightforward to implement a new | |

reader for a different format such as TORO or others. :member:`pose_graph_3d` | |

will print the Ceres solver full summary and then output to disk the original | |

and optimized poses (``poses_original.txt`` and ``poses_optimized.txt``, | |

respectively) of the robot in the following format: | |

.. code-block:: bash | |

pose_id x y z q_x q_y q_z q_w | |

pose_id x y z q_x q_y q_z q_w | |

pose_id x y z q_x q_y q_z q_w | |

... | |

where ``pose_id`` is the corresponding integer ID from the file | |

definition. Note, the file will be sorted in ascending order for the | |

``pose_id``. | |

The executable :member:`pose_graph_3d` expects the first argument to be the | |

path to the problem definition. The executable can be run via | |

.. code-block:: bash | |

/path/to/bin/pose_graph_3d /path/to/dataset/dataset.g2o | |

A script is provided to visualize the resulting output files. There is also | |

an option to enable equal axes using ``--axes_equal`` | |

.. code-block:: bash | |

/path/to/repo/examples/slam/pose_graph_3d/plot_results.py --optimized_poses ./poses_optimized.txt --initial_poses ./poses_original.txt | |

As an example, a standard synthetic benchmark dataset [#f9]_ where the robot is | |

traveling on the surface of a sphere which has 2500 nodes with a total of | |

4949 edges was solved. Visualizing the results with the provided script | |

produces: | |

.. figure:: pose_graph_3d_ex.png | |

:figwidth: 600px | |

:height: 300px | |

:align: center |