.. _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.