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Rename tricks.rst to faq.rst.
Reorganize into sections and add some advice on linear solvers. Change-Id: Ia2d8665720c64b17da67f466f2fc154efb2b6c50
This commit is contained in:
Sameer Agarwal
@@ -1,13 +1,43 @@
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.. _chapter-tricks:
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===================
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Tips, Tricks & FAQs
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FAQS, Tips & Tricks
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===================
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A collection of miscellanous tips, tricks and answers to frequently
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asked questions.
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Answers to frequently asked questions, tricks of the trade and general
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wisdom.
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1. Use analytical/automatic derivatives when possible.
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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 ``-logtostdterr`` 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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@@ -36,32 +66,100 @@ asked questions.
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automatic and numeric differentiation. See
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:class:`NumericDiffFunctor` and :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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2. Use `google-glog <http://code.google.com/p/google-glog>`_.
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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 back
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to ECEF 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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Ceres has extensive support for logging various stages of the
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solve. This includes detailed information about memory allocations
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and time consumed in various parts of the solve, internal error
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conditions etc. This logging structure is built on top of the
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`google-glog <http://code.google.com/p/google-glog>`_ library and
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can easily be controlled from the command line.
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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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We use it extensively to observe and analyze Ceres's
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performance. Starting with ``-logtostdterr`` you can add ``-v=N``
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for increasing values of N to get more and more verbose and
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detailed information about Ceres internals.
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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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Building Ceres like this introduces an external dependency, and it
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is tempting instead to use the `miniglog` implementation that ships
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inside Ceres instead. This is a bad idea.
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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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``miniglog`` was written primarily for building and using Ceres on
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Android because the 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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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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the inverse is the Jacobian of the inverse at :math:`y`.
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3. `Solver::Summary::FullReport` is your friend.
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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 have ``SuiteSparse`` or ``CXSparse``
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installed.
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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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preconditioners 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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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 `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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@@ -169,50 +267,3 @@ asked questions.
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Total 0.998
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The preprocessor time has gone down by more than 4x!.
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4. 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 back
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to ECEF 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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the inverse is the Jacobian of the inverse 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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@@ -43,7 +43,7 @@ squares problems.
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tutorial
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modeling
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solving
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tricks
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faqs
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reading
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contributing
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acknowledgements
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