# FAQS, Tips & Tricks¶

Answers to frequently asked questions, tricks of the trade and general wisdom.

## Building¶

Use 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 library. We use it extensively to observe and analyze Ceres’s performance. google-glog allows you to control its behaviour from the command line flags. 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 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

`CostFunctionToFunctor`.Putting 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 and the WGS84 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 in this case to compute the derivatives more or less analytically.

The key result here is. If \(x = f^{-1}(y)\), and \(Df(x)\) is the invertible Jacobian of \(f\) at \(x\). Then the Jacobian \(Df^{-1}(y) = [Df(x)]^{-1}\), i.e., the Jacobian of the \(f^{-1}\) is the inverse of the Jacobian of \(f\).

Algorithmically this means that given \(y\), compute \(x = f^{-1}(y)\) by whatever means you can. Evaluate the Jacobian of \(f\) at \(x\). If the Jacobian matrix is invertible, then its inverse is the Jacobian of \(f^{-1}(y)\) at \(y\).

One can put this into practice with the following code fragment.

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

`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.For small (a few hundred parameters) or dense problems use

`DENSE_QR`.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.For bundle adjustment problems with up to a hundred or so cameras, use

`DENSE_SCHUR`.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.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

`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./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

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 1.220Which 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

Linear solver ordering AUTOMATIC 22106, 16Which indicates that we are using automatic ordering for the

SPARSE_SCHURsolver. This can be expensive at times. A straight forward way to deal with this is to give the ordering manually. Forbundle_adjusterthis can be done by passing the flag-ordering=user. Doing so and looking at the timing block of the full report gives usTime (in seconds): Preprocessor 0.051 Residual evaluation 0.053 Jacobian evaluation 0.344 Linear solver 0.372 Minimizer 0.854 Postprocessor 0.002 Total 0.935The 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.