293 lines
12 KiB
ReStructuredText
293 lines
12 KiB
ReStructuredText
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.. _chapter-tricks:
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===================
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FAQS, Tips & Tricks
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===================
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Answers to frequently asked questions, tricks of the trade and general
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wisdom.
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Building
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========
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#. Use `google-glog <http://code.google.com/p/google-glog>`_.
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Ceres has extensive support for logging detailed information about
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memory allocations and time consumed in various parts of the solve,
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internal error conditions etc. This is done logging using the
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`google-glog <http://code.google.com/p/google-glog>`_ library. We
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use it extensively to observe and analyze Ceres's
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performance. `google-glog <http://code.google.com/p/google-glog>`_
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allows you to control its behaviour from the command line `flags
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<http://google-glog.googlecode.com/svn/trunk/doc/glog.html>`_. Starting
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with ``-logtostderr`` you can add ``-v=N`` for increasing values
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of ``N`` to get more and more verbose and detailed information
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about Ceres internals.
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In an attempt to reduce dependencies, it is tempting to use
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`miniglog` - a minimal implementation of the ``glog`` interface
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that ships with Ceres. This is a bad idea. ``miniglog`` was written
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primarily for building and using Ceres on Android because the
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current version of `google-glog
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<http://code.google.com/p/google-glog>`_ does not build using the
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NDK. It has worse performance than the full fledged glog library
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and is much harder to control and use.
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Modeling
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========
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#. Use analytical/automatic derivatives.
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This is the single most important piece of advice we can give to
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you. It is tempting to take the easy way out and use numeric
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differentiation. This is a bad idea. Numeric differentiation is
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slow, ill-behaved, hard to get right, and results in poor
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convergence behaviour.
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Ceres allows the user to define templated functors which will
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be automatically differentiated. For most situations this is enough
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and we recommend using this facility. In some cases the derivatives
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are simple enough or the performance considerations are such that
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the overhead of automatic differentiation is too much. In such
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cases, analytic derivatives are recommended.
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The use of numerical derivatives should be a measure of last
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resort, where it is simply not possible to write a templated
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implementation of the cost function.
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In many cases it is not possible to do analytic or automatic
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differentiation of the entire cost function, but it is generally
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the case that it is possible to decompose the cost function into
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parts that need to be numerically differentiated and parts that can
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be automatically or analytically differentiated.
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To this end, Ceres has extensive support for mixing analytic,
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automatic and numeric differentiation. See
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:class:`CostFunctionToFunctor`.
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#. Putting `Inverse Function Theorem
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<http://en.wikipedia.org/wiki/Inverse_function_theorem>`_ to use.
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Every now and then we have to deal with functions which cannot be
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evaluated analytically. Computing the Jacobian in such cases is
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tricky. A particularly interesting case is where the inverse of the
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function is easy to compute analytically. An example of such a
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function is the Coordinate transformation between the `ECEF
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<http://en.wikipedia.org/wiki/ECEF>`_ and the `WGS84
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<http://en.wikipedia.org/wiki/World_Geodetic_System>`_ where the
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conversion from WGS84 to ECEF is analytic, but the conversion
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back to WGS84 uses an iterative algorithm. So how do you compute the
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derivative of the ECEF to WGS84 transformation?
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One obvious approach would be to numerically
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differentiate the conversion function. This is not a good idea. For
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one, it will be slow, but it will also be numerically quite
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bad.
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Turns out you can use the `Inverse Function Theorem
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<http://en.wikipedia.org/wiki/Inverse_function_theorem>`_ in this
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case to compute the derivatives more or less analytically.
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The key result here is. If :math:`x = f^{-1}(y)`, and :math:`Df(x)`
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is the invertible Jacobian of :math:`f` at :math:`x`. Then the
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Jacobian :math:`Df^{-1}(y) = [Df(x)]^{-1}`, i.e., the Jacobian of
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the :math:`f^{-1}` is the inverse of the Jacobian of :math:`f`.
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Algorithmically this means that given :math:`y`, compute :math:`x =
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f^{-1}(y)` by whatever means you can. Evaluate the Jacobian of
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:math:`f` at :math:`x`. If the Jacobian matrix is invertible, then
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its inverse is the Jacobian of :math:`f^{-1}(y)` at :math:`y`.
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One can put this into practice with the following code fragment.
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.. code-block:: c++
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Eigen::Vector3d ecef; // Fill some values
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// Iterative computation.
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Eigen::Vector3d lla = ECEFToLLA(ecef);
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// Analytic derivatives
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Eigen::Matrix3d lla_to_ecef_jacobian = LLAToECEFJacobian(lla);
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bool invertible;
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Eigen::Matrix3d ecef_to_lla_jacobian;
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lla_to_ecef_jacobian.computeInverseWithCheck(ecef_to_lla_jacobian, invertible);
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#. When using Quaternions, use :class:`QuaternionParameterization`.
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TBD
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#. How to choose a parameter block size?
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TBD
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Solving
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=======
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#. Choosing a linear solver.
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When using the ``TRUST_REGION`` minimizer, the choice of linear
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solver is an important decision. It affects solution quality and
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runtime. Here is a simple way to reason about it.
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1. For small (a few hundred parameters) or dense problems use
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``DENSE_QR``.
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2. For general sparse problems (i.e., the Jacobian matrix has a
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substantial number of zeros) use
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``SPARSE_NORMAL_CHOLESKY``. This requires that you have
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``SuiteSparse`` or ``CXSparse`` installed.
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3. For bundle adjustment problems with up to a hundred or so
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cameras, use ``DENSE_SCHUR``.
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4. For larger bundle adjustment problems with sparse Schur
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Complement/Reduced camera matrices use ``SPARSE_SCHUR``. This
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requires that you build Ceres with support for ``SuiteSparse``,
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``CXSparse`` or Eigen's sparse linear algebra libraries.
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If you do not have access to these libraries for whatever
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reason, ``ITERATIVE_SCHUR`` with ``SCHUR_JACOBI`` is an
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excellent alternative.
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5. For large bundle adjustment problems (a few thousand cameras or
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more) use the ``ITERATIVE_SCHUR`` solver. There are a number of
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preconditioner choices here. ``SCHUR_JACOBI`` offers an
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excellent balance of speed and accuracy. This is also the
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recommended option if you are solving medium sized problems for
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which ``DENSE_SCHUR`` is too slow but ``SuiteSparse`` is not
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available.
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.. NOTE::
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If you are solving small to medium sized problems, consider
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setting ``Solver::Options::use_explicit_schur_complement`` to
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``true``, it can result in a substantial performance boost.
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If you are not satisfied with ``SCHUR_JACOBI``'s performance try
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``CLUSTER_JACOBI`` and ``CLUSTER_TRIDIAGONAL`` in that
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order. They require that you have ``SuiteSparse``
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installed. Both of these preconditioners use a clustering
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algorithm. Use ``SINGLE_LINKAGE`` before ``CANONICAL_VIEWS``.
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#. Use :func:`Solver::Summary::FullReport` to diagnose performance problems.
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When diagnosing Ceres performance issues - runtime and convergence,
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the first place to start is by looking at the output of
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``Solver::Summary::FullReport``. Here is an example
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.. code-block:: bash
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./bin/bundle_adjuster --input ../data/problem-16-22106-pre.txt
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iter cost cost_change |gradient| |step| tr_ratio tr_radius ls_iter iter_time total_time
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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
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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
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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
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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
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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
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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
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Ceres Solver v1.11.0 Solve Report
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----------------------------------
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Original Reduced
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Parameter blocks 22122 22122
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Parameters 66462 66462
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Residual blocks 83718 83718
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Residual 167436 167436
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Minimizer TRUST_REGION
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Sparse linear algebra library SUITE_SPARSE
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Trust region strategy LEVENBERG_MARQUARDT
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Given Used
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Linear solver SPARSE_SCHUR SPARSE_SCHUR
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Threads 1 1
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Linear solver threads 1 1
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Linear solver ordering AUTOMATIC 22106, 16
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Cost:
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Initial 4.185660e+06
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Final 1.803391e+04
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Change 4.167626e+06
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Minimizer iterations 5
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Successful steps 5
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Unsuccessful steps 0
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Time (in seconds):
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Preprocessor 0.283
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Residual evaluation 0.061
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Jacobian evaluation 0.361
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Linear solver 0.382
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Minimizer 0.895
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Postprocessor 0.002
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Total 1.220
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Termination: NO_CONVERGENCE (Maximum number of iterations reached.)
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Let us focus on run-time performance. The relevant lines to look at
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are
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.. code-block:: bash
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Time (in seconds):
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Preprocessor 0.283
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Residual evaluation 0.061
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Jacobian evaluation 0.361
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Linear solver 0.382
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Minimizer 0.895
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Postprocessor 0.002
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Total 1.220
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Which tell us that of the total 1.2 seconds, about .3 seconds was
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spent in the linear solver and the rest was mostly spent in
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preprocessing and jacobian evaluation.
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The preprocessing seems particularly expensive. Looking back at the
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report, we observe
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.. code-block:: bash
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Linear solver ordering AUTOMATIC 22106, 16
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Which indicates that we are using automatic ordering for the
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``SPARSE_SCHUR`` solver. This can be expensive at times. A straight
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forward way to deal with this is to give the ordering manually. For
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``bundle_adjuster`` this can be done by passing the flag
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``-ordering=user``. Doing so and looking at the timing block of the
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full report gives us
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.. code-block:: bash
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Time (in seconds):
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Preprocessor 0.051
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Residual evaluation 0.053
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Jacobian evaluation 0.344
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Linear solver 0.372
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Minimizer 0.854
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Postprocessor 0.002
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Total 0.935
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The preprocessor time has gone down by more than 5.5x!.
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Further Reading
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===============
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For a short but informative introduction to the subject we recommend
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the booklet by [Madsen]_ . For a general introduction to non-linear
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optimization we recommend [NocedalWright]_. [Bjorck]_ remains the
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seminal reference on least squares problems. [TrefethenBau]_ book is
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our favorite text on introductory numerical linear algebra. [Triggs]_
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provides a thorough coverage of the bundle adjustment problem.
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