13 Pattern Recognition and Reconstruction
573
An outlier rejection procedure can make use of the chi-squares of the measurements with respect to the smoothed predictions, i.e., a weighted mean of the
predicted states of a forward and a backward Kalman filter. The measurement with
the largest value of the chi-square is removed, and the total chi-square is again
calculated. This procedure is repeated until the value of the total chi-square falls
below a defined threshold.
In the presence of a potentially large fraction of outliers in a track candidate,
the sequential outlier rejection procedure outlined above might become unstable,
because the smoothed predictions may themselves be biased by outliers. An
alternative approach is the Gaussian-sum filter [62]. This algorithm is based on
the assumption that the probability distribution of the measurement error can
be modeled as a two-component Gaussian mixture, where a narrow component
represents the hypothesis that the measurement is real and a wider component
represents the hypothesis that the measurement is an outlier. It takes the form of
a set of Kalman filters running in parallel, each Kalman filter representing a specific
hypothesis of a subset of the measurements that should be classified as outliers.
A weight attached to each Kalman filter can be interpreted as the probability of
correctness of the hypothesis. In the end, the Kalman filter with the largest weight
or a weighted mean of the different filters can be taken as the final estimate.
The Gaussian-sum filter can also be used to deal with a mixture model of the process noise, i.e., the stochastic disturbance of the track because of interactions with
the detector material [63]. In the case of bremsstrahlung, a successful application to
the reconstruction of electrons is described in [50].
For the treatment of outliers, the Gaussian-sum filter has two disadvantages.
First, it may create a large number of Kalman filters running in parallel because
of poor knowledge of the track parameters in the early stages of the filter, making
the approach expensive in terms of computing time. Second, an explicit outlier
model is required. A faster and even more robust alternative is the Deterministic
Annealing Filter [64]. This filter is an iterated Kalman filter with annealing, which
assigns small weights to measurements far away from the track. A temperature
parameter is introduced, facilitating convergence to the globally optimal solution.
The iterations start at a high temperature, continue with a gradual lowering of the
temperature and converge at the nominal value of the temperature. The procedure is
easily generalized to the situation of several measurements being present in the same
detector layer. In this case the measurements compete for inclusion in the track. As
opposed to a standard outlier rejection approach, the assignment of measurements
is soft. This means that several measurements in the same detector layer might
contribute to the final estimate of the track parameters, each with a weight equal to
the assignment probability. A further generalization is the multi-track filter, where
several tracks are allowed to compete for compatible hits in all detector layers [65].
For an experimental application, see [66].
573
An outlier rejection procedure can make use of the chi-squares of the measurements with respect to the smoothed predictions, i.e., a weighted mean of the
predicted states of a forward and a backward Kalman filter. The measurement with
the largest value of the chi-square is removed, and the total chi-square is again
calculated. This procedure is repeated until the value of the total chi-square falls
below a defined threshold.
In the presence of a potentially large fraction of outliers in a track candidate,
the sequential outlier rejection procedure outlined above might become unstable,
because the smoothed predictions may themselves be biased by outliers. An
alternative approach is the Gaussian-sum filter [62]. This algorithm is based on
the assumption that the probability distribution of the measurement error can
be modeled as a two-component Gaussian mixture, where a narrow component
represents the hypothesis that the measurement is real and a wider component
represents the hypothesis that the measurement is an outlier. It takes the form of
a set of Kalman filters running in parallel, each Kalman filter representing a specific
hypothesis of a subset of the measurements that should be classified as outliers.
A weight attached to each Kalman filter can be interpreted as the probability of
correctness of the hypothesis. In the end, the Kalman filter with the largest weight
or a weighted mean of the different filters can be taken as the final estimate.
The Gaussian-sum filter can also be used to deal with a mixture model of the process noise, i.e., the stochastic disturbance of the track because of interactions with
the detector material [63]. In the case of bremsstrahlung, a successful application to
the reconstruction of electrons is described in [50].
For the treatment of outliers, the Gaussian-sum filter has two disadvantages.
First, it may create a large number of Kalman filters running in parallel because
of poor knowledge of the track parameters in the early stages of the filter, making
the approach expensive in terms of computing time. Second, an explicit outlier
model is required. A faster and even more robust alternative is the Deterministic
Annealing Filter [64]. This filter is an iterated Kalman filter with annealing, which
assigns small weights to measurements far away from the track. A temperature
parameter is introduced, facilitating convergence to the globally optimal solution.
The iterations start at a high temperature, continue with a gradual lowering of the
temperature and converge at the nominal value of the temperature. The procedure is
easily generalized to the situation of several measurements being present in the same
detector layer. In this case the measurements compete for inclusion in the track. As
opposed to a standard outlier rejection approach, the assignment of measurements
is soft. This means that several measurements in the same detector layer might
contribute to the final estimate of the track parameters, each with a weight equal to
the assignment probability. A further generalization is the multi-track filter, where
several tracks are allowed to compete for compatible hits in all detector layers [65].
For an experimental application, see [66].