| .. _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 ``-logtostderr`` 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:`CostFunctionToFunctor`. |
| |
| #. When using Quaternions, consider using :class:`QuaternionParameterization`. |
| |
| `Quaternions <https://en.wikipedia.org/wiki/Quaternion>`_ are a |
| four dimensional parameterization of the space of three dimensional |
| rotations :math:`SO(3)`. However, the :math:`SO(3)` is a three |
| dimensional set, and so is the tangent space of a |
| Quaternion. Therefore, it is sometimes (not always) benefecial to |
| associate a local parameterization with parameter blocks |
| representing a Quaternion. Assuming that the order of entries in |
| your parameter block is :math:`w,x,y,z`, you can use |
| :class:`QuaternionParameterization`. |
| |
| If however, you are using `Eigen's Quaternion |
| <http://eigen.tuxfamily.org/dox/classEigen_1_1Quaternion.html>`_ |
| object, whose layout is :math:`x,y,z,w`, then we recommend you use |
| Lloyd Hughes's `Ceres Extensions |
| <https://github.com/system123/ceres_extensions>`_. |
| |
| #. How do I solve problems with general linear & non-linear |
| **inequality** constraints with Ceres Solver? |
| |
| Currently, Ceres Solver only supports upper and lower bounds |
| constraints on the parameter blocks. |
| |
| A crude way of dealing with inequality constraints is have one or |
| more of your cost functions check if the inequalities you are |
| interested in are satisfied, and if not return false instead of |
| true. This will prevent the solver from ever stepping into an |
| infeasible region. |
| |
| This requires that the starting point for the optimization be a |
| feasible point. You also risk pre-mature convergence using this |
| method. |
| |
| #. How do I solve problems with general linear & non-linear **equality** |
| constraints with Ceres Solver? |
| |
| There is no built in support in ceres for solving problems with |
| equality constraints. Currently, Ceres Solver only supports upper |
| and lower bounds constraints on the parameter blocks. |
| |
| The trick described above for dealing with inequality |
| constraints will **not** work for equality constraints. |
| |
| #. How do I set one or more components of a parameter block constant? |
| |
| Using :class:`SubsetParameterization`. |
| |
| #. 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 to ECEF is analytic, but the conversion |
| back to WGS84 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 |
| its inverse is the Jacobian of :math:`f^{-1}(y)` 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); |
| |
| |
| Solving |
| ======= |
| |
| #. How do I evaluate the Jacobian for a solver problem? |
| |
| Using :func:`Problem::Evaluate`. |
| |
| #. 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 build Ceres with support for ``SuiteSparse``, |
| ``CXSparse`` or Eigen's sparse linear algebra libraries. |
| |
| If you do not have access to these libraries for whatever |
| reason, ``ITERATIVE_SCHUR`` with ``SCHUR_JACOBI`` is an |
| excellent alternative. |
| |
| 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. |
| |
| .. NOTE:: |
| |
| If you are solving small to medium sized problems, consider |
| setting ``Solver::Options::use_explicit_schur_complement`` to |
| ``true``, it can result in a substantial performance boost. |
| |
| 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 :func:`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 v1.12.0 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. |
