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 Chi-Square Fitter (GX2F) [wip] Even though all of them are least-squares 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 backwards-propagation and one does not.
Note
This chapter will be extended in the future.
Gaussian Sum Filter¶
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 Kalman-Filter that allows to handle non-gaussian errors by modelling the track state as a gaussian mixture:
A common use case of this is electron fitting. The energy-loss of Bremsstrahlung for electrons in matter are highly non-gaussian, and thus are not modeled accurately by the default material interactions in the Kalman Filter. Instead, the Bremsstrahlung is modeled as a Bethe-Heitler distribution, which is approximated as a gaussian mixture.
Implementation¶
To implement the GSF, a special stepper is needed, that can handle a multi-component state internally: The Acts::MultiEigenStepperLoop. On a surface with material, the Bethe-Heitler energy-loss distribution is approximated with a fixed number of gaussian distributions for each component. Since 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. The Kalman update mechanism is based on the code for the Acts::KalmanFitter.
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 traj_t, typename bethe_heitler_approx_t = detail::BetheHeitlerApprox<6, 5>>
struct GaussianSumFitter Public Functions
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 a Acts::KalmanFitterResult object, the same as the Acts::KalmanFitter. This allows to use the same analysis tools for both fitters. Currently, the states of the individual components are not returned by the fitter.
A GSF example can be found in the Acts Examples Framework here.
Customizing the Bethe-Heitler approximation¶
To be able to evaluate the approximation of the Bethe-Heitler distribution for different materials and thicknesses, the individual gaussian components (weight, mean, variance of the ratio \(E_f/E_i\)) are parameterized as polynomials in \(x/x_0\). The default parametrization uses 6 components and 5th order polynomials.
This approximation of the Bethe-Heitler distribution is described in Acts::detail::BetheHeitlerApprox. The class is templated on the number of components and the degree of the polynomial, and is designed to be used with the parameterization files from ATLAS. However, in principle the GSF could be constructed with custom classes with the same interface as Acts::detail::BetheHeitlerApprox.
For small \(x/x_0\) the Acts::detail::BetheHeitlerApprox only returns a one-component mixture or no change at all. When loading a custom parametrization, it is possible to specify different parameterizations for high and for low \(x/x_0\). The thresholds are currently not configurable.