Track Fitting¶
The track fitting algorithms estimate the track parameters. It is part of the pattern recognition/track reconstruction/tracking. We can run the track fitting algorithms, after we allocated all hits to single tracks with the help of a track finding algorithm. It is not necessary, that all points of a track are present.
Currently we have implementations for three different fitters:
Kalman Filter
GSF
Global ChiSquare Fitter (GX2F) [wip] Even though all of them are leastsquares fits, the concepts are quite different. Therefore, we should not expect identical results from all of them.
Kalman Filter (KF) [wip]¶
The Kalman Filter is an iterative fitter. It successively combines measurements to obtain an estimate of the track parameters. The KF needs an estimate as a starting point. The procedure alternates between two methods:
Extrapolate the current state to the next surface.
Update the extrapolation using the measurement of the new surface.1 The meaning of “this surface” and “the next surface” changes with the context. There are three different interpretations for this. The KF can give us those three interpretations as sets of track parameters:
predicted: Uses “older” data (i.e. from the last surfaces) to make the prediction. This prediction is an extrapolation from the old data onto the current surface.
filtered: Uses the “current” data (i.e. the predicted data updated with the measurement on the current surface). It is some kind of weighted mean.
smoothed: Uses the “future” data to predict the current parameters. This can only be evaluated if the whole propagation is finished once. This can be done in to ways: one uses backwardspropagation and one does not.
Note
This chapter will be extended in the future.
Gaussian Sum Filter (GSF)¶
Note
The GSF is not considered as production ready yet, therefore it is located in the namespace Acts::Experimental
.
The GSF is an extension of the KalmanFilter that allows to handle nongaussian errors by modelling the track state as a gaussian mixture:
A common use case of this is electron fitting. The energyloss of Bremsstrahlung for electrons in matter are highly nonGaussian, and thus cannot be modelled accurately by the default material interactions in the Kalman Filter. Instead, the Bremsstrahlung is modelled as a BetheHeitler distribution, where \(z\) is the fraction of the energy remaining after the interaction (\(E_f/E_i\)), and \(t\) is the material thickness in terms of the radiation length:
To be able to handle this with the Kalman filter mechanics, this distribution is approximated by a gaussian mixture as well (see Fig. 30). The GSF Algorithm works then as follows (see also Fig. 31)
On a surface with material, the BetheHeitler energyloss distribution is approximated with a fixed number of gaussian components for each component. Since this way the number of components would grow exponentially with each material interaction, components that are close in terms of their Kullback–Leibler divergence are merged to limit the computational cost.
On a measurement surface, for each component a Kalman update is performed. Afterwards, the component weights are corrected according to each components compatibility with the measurement.
The MultiStepper¶
To implement the GSF, a special stepper is needed, that can handle a multicomponent state internally: The Acts::MultiEigenStepperLoop
, which is based on the Acts::EigenStepper
and thus shares a lot of code with it. It interfaces to the navigation as one aggregate state to limit the navigation overhead, but internally processes a multicomponent state. How this aggregation is performed can be configured via a template parameter, by default weighted average is used (WeightedComponentReducerLoop
).
At the end of the fit the multicomponent state must be reduced to a single set of parameters with a corresponding covariance matrix. This is supported by the Acts::MultiEigenStepperLoop
in two different ways currently: The mean method computes the mean and the covariance matrix of the multicomponent state, whereas the maximum weight method just returns the component with the maximum weight. This can be configured in the constructor of the Acts::MultiEigenStepperLoop
with the Acts::FinalReductionMethod
. In the future there is planned to add a mode finding method as well.
Note
In practice it turned out that the maximum weight method leads to better results so far.
Even though the multistepper interface exposes only one aggregate state and thus is compatible with most standard tools, there is a special aborter is required to stop the navigation when the surface is reached, the Acts::MultiStepperSurfaceReached
. It checks if all components have reached the target surface already and updates their state accordingly. Optionally, it also can stop the propagation when the aggregate state reaches the surface.
Using the GSF¶
The GSF is implemented in the class Acts::Experimental::GaussianSumFitter
. The interface of its fit(...)
functions is very similar to the one of the Acts::KalmanFitter
(one for the standard Acts::Navigator
and one for the Acts::DirectNavigator
that takes an additional std::vector<const Acts::Surface *>
as an argument):

template<typename propagator_t, typename bethe_heitler_approx_t, typename traj_t>
struct GaussianSumFitter Public Functions

template<typename source_link_it_t, typename start_parameters_t, typename track_container_t, template<typename> class holder_t>
inline auto fit(source_link_it_t begin, source_link_it_t end, const start_parameters_t &sParameters, const GsfOptions<traj_t> &options, const std::vector<const Surface*> &sSequence, TrackContainer<track_container_t, traj_t, holder_t> &trackContainer) const

template<typename source_link_it_t, typename start_parameters_t, typename track_container_t, template<typename> class holder_t>
inline auto fit(source_link_it_t begin, source_link_it_t end, const start_parameters_t &sParameters, const GsfOptions<traj_t> &options, TrackContainer<track_container_t, traj_t, holder_t> &trackContainer) const

template<typename source_link_it_t, typename start_parameters_t, typename track_container_t, template<typename> class holder_t>
The fit can be customized with several options, e.g., the maximum number of components. All options can be found in the Acts::GsfOptions
.
To simplify integration, the GSF returns an Acts::KalmanFitterResult
object, the same as the Acts::KalmanFitter
. This allows to use the same analysis tools for both fitters.
If the GSF finds the column with the string identifier “gsffinalmulticomponentstate” (defined in Acts::Experimental::GsfConstants::kFinalMultiComponentStateColumn
) in the track container, it adds the final multicomponent state to the track as a std::optional<Acts::MultiComponentBoundTrackParameters<SinglyCharged>>
object.
A GSF example can be found in the Acts Examples Framework here.
Customising the BetheHeitler approximation¶
The GSF needs an approximation of the BetheHeitler distribution as a Gaussian mixture on each material interaction (see above). This task is delegated to a separate class, that can be provided by a template parameter to Acts::Experimental::GaussianSumFitter
, so in principle it can be implemented in different ways.
However, ACTS ships with the class Acts::Experimental::AtlasBetheHeitlerApprox
that implements the ATLAS strategy for this task: To be able to evaluate the approximation of the BetheHeitler distribution for different materials and thicknesses, the individual Gaussian components (weight, mean, variance of the ratio \(E_f/E_i\)) are parametrised as polynomials in \(x/x_0\). This class can load files in the ATLAS format that can be found here. A default parameterization can be created with Acts::Experimental::makeDefaultBetheHeitlerApprox()
.
The Acts::Experimental::AtlasBetheHeitlerApprox
is constructed with two parameterizations, allowing to use different parameterizations for different \(x/x_0\). In particular, it has this behaviour:
\(x/x_0 < 0.0001\): Return no change
\(x/x_0 < 0.002\): Return a single gaussian approximation
\(x/x_0 < 0.1\): Return the approximation for low \(x/x_0\).
\(x/x_0 \geq 0.1\): Return the approximation for high \(x/x_0\). The maximum possible value is \(x/x_0 = 0.2\), for higher values it is clipped to 0.2 and the GSF emits a warning.