Track Fitting

Gaussian Sum Filter

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:

\[ p(\vec{x}) = \sum_i w_i \varphi(\vec{x}; \mu_i, \Sigma_i), \quad \sum_i w_i = 1 \]

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.


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::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::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::BetheHeitlerApprox.

For small \(x/x_0\) the Acts::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.

Further reading

  • Thomas Atkinson, Electron reconstruction with the ATLAS inner detector, 2006, see here

  • R Frühwirth, Track fitting with non-Gaussian noise, 1997, see here

Kalman Filter


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