| .. _chapter-tricks: | |
| =================== | |
| FAQS, Tips & Tricks | |
| =================== | |
| Answers to frequently asked questions, tricks of the trade and general | |
| wisdom. | |
| Building | |
| ======== | |
| #. Use `google-glog <http://code.google.com/p/google-glog>`_. | |
| Ceres has extensive support for logging detailed information about | |
| memory allocations and time consumed in various parts of the solve, | |
| internal error conditions etc. This is done logging using the | |
| `google-glog <http://code.google.com/p/google-glog>`_ library. We | |
| use it extensively to observe and analyze Ceres's | |
| performance. `google-glog <http://code.google.com/p/google-glog>`_ | |
| allows you to control its behaviour from the command line `flags | |
| <http://google-glog.googlecode.com/svn/trunk/doc/glog.html>`_. Starting | |
| with ``-logtostdterr`` you can add ``-v=N`` for increasing values | |
| of ``N`` to get more and more verbose and detailed information | |
| about Ceres internals. | |
| In an attempt to reduce dependencies, it is tempting to use | |
| `miniglog` - a minimal implementation of the ``glog`` interface | |
| that ships with Ceres. This is a bad idea. ``miniglog`` was written | |
| primarily for building and using Ceres on Android because the | |
| current version of `google-glog | |
| <http://code.google.com/p/google-glog>`_ does not build using the | |
| NDK. It has worse performance than the full fledged glog library | |
| and is much harder to control and use. | |
| Modeling | |
| ======== | |
| #. Use analytical/automatic derivatives. | |
| This is the single most important piece of advice we can give to | |
| you. It is tempting to take the easy way out and use numeric | |
| differentiation. This is a bad idea. Numeric differentiation is | |
| slow, ill-behaved, hard to get right, and results in poor | |
| convergence behaviour. | |
| Ceres allows the user to define templated functors which will | |
| be automatically differentiated. For most situations this is enough | |
| and we recommend using this facility. In some cases the derivatives | |
| are simple enough or the performance considerations are such that | |
| the overhead of automatic differentiation is too much. In such | |
| cases, analytic derivatives are recommended. | |
| The use of numerical derivatives should be a measure of last | |
| resort, where it is simply not possible to write a templated | |
| implementation of the cost function. | |
| In many cases it is not possible to do analytic or automatic | |
| differentiation of the entire cost function, but it is generally | |
| the case that it is possible to decompose the cost function into | |
| parts that need to be numerically differentiated and parts that can | |
| be automatically or analytically differentiated. | |
| To this end, Ceres has extensive support for mixing analytic, | |
| automatic and numeric differentiation. See | |
| :class:`NumericDiffFunctor` and :class:`CostFunctionToFunctor`. | |
| #. Putting `Inverse Function Theorem | |
| <http://en.wikipedia.org/wiki/Inverse_function_theorem>`_ to use. | |
| Every now and then we have to deal with functions which cannot be | |
| evaluated analytically. Computing the Jacobian in such cases is | |
| tricky. A particularly interesting case is where the inverse of the | |
| function is easy to compute analytically. An example of such a | |
| function is the Coordinate transformation between the `ECEF | |
| <http://en.wikipedia.org/wiki/ECEF>`_ and the `WGS84 | |
| <http://en.wikipedia.org/wiki/World_Geodetic_System>`_ where the | |
| conversion from WGS84 from ECEF is analytic, but the conversion | |
| back to ECEF uses an iterative algorithm. So how do you compute the | |
| derivative of the ECEF to WGS84 transformation? | |
| One obvious approach would be to numerically | |
| differentiate the conversion function. This is not a good idea. For | |
| one, it will be slow, but it will also be numerically quite | |
| bad. | |
| Turns out you can use the `Inverse Function Theorem | |
| <http://en.wikipedia.org/wiki/Inverse_function_theorem>`_ in this | |
| case to compute the derivatives more or less analytically. | |
| The key result here is. If :math:`x = f^{-1}(y)`, and :math:`Df(x)` | |
| is the invertible Jacobian of :math:`f` at :math:`x`. Then the | |
| Jacobian :math:`Df^{-1}(y) = [Df(x)]^{-1}`, i.e., the Jacobian of | |
| the :math:`f^{-1}` is the inverse of the Jacobian of :math:`f`. | |
| Algorithmically this means that given :math:`y`, compute :math:`x = | |
| f^{-1}(y)` by whatever means you can. Evaluate the Jacobian of | |
| :math:`f` at :math:`x`. If the Jacobian matrix is invertible, then | |
| the inverse is the Jacobian of the inverse at :math:`y`. | |
| One can put this into practice with the following code fragment. | |
| .. code-block:: c++ | |
| Eigen::Vector3d ecef; // Fill some values | |
| // Iterative computation. | |
| Eigen::Vector3d lla = ECEFToLLA(ecef); | |
| // Analytic derivatives | |
| Eigen::Matrix3d lla_to_ecef_jacobian = LLAToECEFJacobian(lla); | |
| bool invertible; | |
| Eigen::Matrix3d ecef_to_lla_jacobian; | |
| lla_to_ecef_jacobian.computeInverseWithCheck(ecef_to_lla_jacobian, invertible); | |
| #. When using Quaternions, use :class:`QuaternionParameterization`. | |
| TBD | |
| #. How to choose a parameter block size? | |
| TBD | |
| Solving | |
| ======= | |
| #. Choosing a linear solver. | |
| When using the ``TRUST_REGION`` minimizer, the choice of linear | |
| solver is an important decision. It affects solution quality and | |
| runtime. Here is a simple way to reason about it. | |
| 1. For small (a few hundred parameters) or dense problems use | |
| ``DENSE_QR``. | |
| 2. For general sparse problems (i.e., the Jacobian matrix has a | |
| substantial number of zeros) use | |
| ``SPARSE_NORMAL_CHOLESKY``. This requires that you have | |
| ``SuiteSparse`` or ``CXSparse`` installed. | |
| 3. For bundle adjustment problems with up to a hundred or so | |
| cameras, use ``DENSE_SCHUR``. | |
| 4. For larger bundle adjustment problems with sparse Schur | |
| Complement/Reduced camera matrices use ``SPARSE_SCHUR``. This | |
| requires that you have ``SuiteSparse`` or ``CXSparse`` | |
| installed. | |
| 5. For large bundle adjustment problems (a few thousand cameras or | |
| more) use the ``ITERATIVE_SCHUR`` solver. There are a number of | |
| preconditioner choices here. ``SCHUR_JACOBI`` offers an | |
| excellent balance of speed and accuracy. This is also the | |
| recommended option if you are solving medium sized problems for | |
| which ``DENSE_SCHUR`` is too slow but ``SuiteSparse`` is not | |
| available. | |
| If you are not satisfied with ``SCHUR_JACOBI``'s performance try | |
| ``CLUSTER_JACOBI`` and ``CLUSTER_TRIDIAGONAL`` in that | |
| order. They require that you have ``SuiteSparse`` | |
| installed. Both of these preconditioners use a clustering | |
| algorithm. Use ``SINGLE_LINKAGE`` before ``CANONICAL_VIEWS``. | |
| #. Use `Solver::Summary::FullReport` to diagnose performance problems. | |
| When diagnosing Ceres performance issues - runtime and convergence, | |
| the first place to start is by looking at the output of | |
| ``Solver::Summary::FullReport``. Here is an example | |
| .. code-block:: bash | |
| ./bin/bundle_adjuster --input ../data/problem-16-22106-pre.txt | |
| iter cost cost_change |gradient| |step| tr_ratio tr_radius ls_iter iter_time total_time | |
| 0 4.185660e+06 0.00e+00 2.16e+07 0.00e+00 0.00e+00 1.00e+04 0 7.50e-02 3.58e-01 | |
| 1 1.980525e+05 3.99e+06 5.34e+06 2.40e+03 9.60e-01 3.00e+04 1 1.84e-01 5.42e-01 | |
| 2 5.086543e+04 1.47e+05 2.11e+06 1.01e+03 8.22e-01 4.09e+04 1 1.53e-01 6.95e-01 | |
| 3 1.859667e+04 3.23e+04 2.87e+05 2.64e+02 9.85e-01 1.23e+05 1 1.71e-01 8.66e-01 | |
| 4 1.803857e+04 5.58e+02 2.69e+04 8.66e+01 9.93e-01 3.69e+05 1 1.61e-01 1.03e+00 | |
| 5 1.803391e+04 4.66e+00 3.11e+02 1.02e+01 1.00e+00 1.11e+06 1 1.49e-01 1.18e+00 | |
| Ceres Solver v1.10.0 Solve Report | |
| ---------------------------------- | |
| Original Reduced | |
| Parameter blocks 22122 22122 | |
| Parameters 66462 66462 | |
| Residual blocks 83718 83718 | |
| Residual 167436 167436 | |
| Minimizer TRUST_REGION | |
| Sparse linear algebra library SUITE_SPARSE | |
| Trust region strategy LEVENBERG_MARQUARDT | |
| Given Used | |
| Linear solver SPARSE_SCHUR SPARSE_SCHUR | |
| Threads 1 1 | |
| Linear solver threads 1 1 | |
| Linear solver ordering AUTOMATIC 22106, 16 | |
| Cost: | |
| Initial 4.185660e+06 | |
| Final 1.803391e+04 | |
| Change 4.167626e+06 | |
| Minimizer iterations 5 | |
| Successful steps 5 | |
| Unsuccessful steps 0 | |
| Time (in seconds): | |
| Preprocessor 0.283 | |
| Residual evaluation 0.061 | |
| Jacobian evaluation 0.361 | |
| Linear solver 0.382 | |
| Minimizer 0.895 | |
| Postprocessor 0.002 | |
| Total 1.220 | |
| Termination: NO_CONVERGENCE (Maximum number of iterations reached.) | |
| Let us focus on run-time performance. The relevant lines to look at | |
| are | |
| .. code-block:: bash | |
| Time (in seconds): | |
| Preprocessor 0.283 | |
| Residual evaluation 0.061 | |
| Jacobian evaluation 0.361 | |
| Linear solver 0.382 | |
| Minimizer 0.895 | |
| Postprocessor 0.002 | |
| Total 1.220 | |
| Which tell us that of the total 1.2 seconds, about .3 seconds was | |
| spent in the linear solver and the rest was mostly spent in | |
| preprocessing and jacobian evaluation. | |
| The preprocessing seems particularly expensive. Looking back at the | |
| report, we observe | |
| .. code-block:: bash | |
| Linear solver ordering AUTOMATIC 22106, 16 | |
| Which indicates that we are using automatic ordering for the | |
| ``SPARSE_SCHUR`` solver. This can be expensive at times. A straight | |
| forward way to deal with this is to give the ordering manually. For | |
| ``bundle_adjuster`` this can be done by passing the flag | |
| ``-ordering=user``. Doing so and looking at the timing block of the | |
| full report gives us | |
| .. code-block:: bash | |
| Time (in seconds): | |
| Preprocessor 0.051 | |
| Residual evaluation 0.053 | |
| Jacobian evaluation 0.344 | |
| Linear solver 0.372 | |
| Minimizer 0.854 | |
| Postprocessor 0.002 | |
| Total 0.935 | |
| The preprocessor time has gone down by more than 5.5x!. | |
| Further Reading | |
| =============== | |
| For a short but informative introduction to the subject we recommend | |
| the booklet by [Madsen]_ . For a general introduction to non-linear | |
| optimization we recommend [NocedalWright]_. [Bjorck]_ remains the | |
| seminal reference on least squares problems. [TrefethenBau]_ book is | |
| our favorite text on introductory numerical linear algebra. [Triggs]_ | |
| provides a thorough coverage of the bundle adjustment problem. |