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Ceres Solver
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1.13
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Why?
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Installation
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Tutorial
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On Derivatives
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Modeling Non-linear Least Squares
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Solving Non-linear Least Squares
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Covariance Estimation
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General Unconstrained Minimization
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FAQS, Tips & Tricks
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Modeling
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Solving
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Further Reading
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Users
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Ceres Solver
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Modeling
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ModelingΒΆ
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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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CostFunctionToFunctor.
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When using Quaternions, consider using QuaternionParameterization.
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Quaternions are a
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four dimensional parameterization of the space of three dimensional
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rotations \(SO(3)\). However, the \(SO(3)\) is a three
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dimensional set, and so is the tangent space of a
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Quaternion. Therefore, it is sometimes (not always) benefecial to
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associate a local parameterization with parameter blocks
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representing a Quaternion. Assuming that the order of entries in
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your parameter block is \(w,x,y,z\), you can use
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QuaternionParameterization.
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Note
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If you are using Eigen’s Quaternion
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object, whose layout is \(x,y,z,w\), then you should use
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EigenQuaternionParameterization.
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How do I solve problems with general linear & non-linear
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inequality constraints with Ceres Solver?
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Currently, Ceres Solver only supports upper and lower bounds
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constraints on the parameter blocks.
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A crude way of dealing with inequality constraints is have one or
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more of your cost functions check if the inequalities you are
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interested in are satisfied, and if not return false instead of
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true. This will prevent the solver from ever stepping into an
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infeasible region.
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This requires that the starting point for the optimization be a
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feasible point. You also risk pre-mature convergence using this
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method.
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How do I solve problems with general linear & non-linear equality
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constraints with Ceres Solver?
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There is no built in support in ceres for solving problems with
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equality constraints. Currently, Ceres Solver only supports upper
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and lower bounds constraints on the parameter blocks.
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The trick described above for dealing with inequality
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constraints will not work for equality constraints.
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How do I set one or more components of a parameter block constant?
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Using SubsetParameterization.
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Putting 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 and the WGS84 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 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 \(x = f^{-1}(y)\), and \(Df(x)\)
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is the invertible Jacobian of \(f\) at \(x\). Then the
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Jacobian \(Df^{-1}(y) = [Df(x)]^{-1}\), i.e., the Jacobian of
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the \(f^{-1}\) is the inverse of the Jacobian of \(f\).
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Algorithmically this means that given \(y\), compute \(x =
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f^{-1}(y)\) by whatever means you can. Evaluate the Jacobian of
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\(f\) at \(x\). If the Jacobian matrix is invertible, then
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its inverse is the Jacobian of \(f^{-1}(y)\) at \(y\).
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One can put this into practice with the following code fragment.
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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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