13 Pattern Recognition and Reconstruction
577
On the other hand, unbiased algorithms usually do not require iterations, with the
possible exception of problems like non-linearities or rejection of outliers.
Besides the differences between various algorithms it should be noted that the
final result of any track-based alignment is always limited by the tracks used. Basic
quality cuts, like the selection of high momentum tracks to minimize the influence of
multiple scattering or cuts on the minimum number of hits, have a strong influence
on the convergence. More subtle is the effect of an unbalanced mixture of tracks
or the complete absence of some types of tracks, such as tracks from collisions
and cosmic events or tracks taken with and without a magnetic field. This is due
to the fact that any kind of tracks has several unconstrained degrees of freedom,
usually referred to as weak modes, weakly defined modes or χ 2 -invariant modes.
As an example, typical weak modes for straight tracks are shears but not bends, and
vice versa for curved tracks. Combining the information of both kinds of tracks is
therefore a reasonable strategy to avoid these deformations in the final result. The
most obvious weak mode is a translation or rotation of the entire tracking device,
which can be only fixed with some kind of reference frame, be it an external system
or by definition. This, however, is less severe and sometimes even not considered at
all, as it does not affect the internal alignment of the tracking device.
Once a set of alignment parameters is calculated, it should always be validated [83, chapter 11]. Apart from checking the improvement of the residuals,
several physics measurements can be utilized, especially to probe for remaining
weak modes. Known charge, forward-backward or ϕ-symmetries of distinct physics
processes can be used. Distributions of the signed curvature or the signed transverse
impact parameter are also sensitive observables.
13.1.4.2 Examples of Alignment Algorithms
Some modern experiments deploy large tracking devices that require a large number
of alignment parameters, of the order of 10 5 . In such a case the computation
of parameters by using straightforward recipes might become unreasonably slow
or cause numerical problems. The two algorithms presented in this section are
examples of how to cope with such challenging circumstances.
The HIP Algorithm
The HIP algorithm [101] is a straightforward and easy-to-implement biased alignment algorithm. It computes the alignment parameters for each alignable object
separately. Only when iterating on the track sample a certain kind of indirect
feedback between the alignable objects is established due to the track refit.
Since only individual alignable objects are regarded, Eq. (13.5) can be partitioned. This is simply done by evaluating the corresponding expressions for each
alignable object i together with its associated parameters p i :
r i (p i ) = m i − d i ( ˘
q, p i ) ≈ m i − ˘
d i − D pi p i
577
On the other hand, unbiased algorithms usually do not require iterations, with the
possible exception of problems like non-linearities or rejection of outliers.
Besides the differences between various algorithms it should be noted that the
final result of any track-based alignment is always limited by the tracks used. Basic
quality cuts, like the selection of high momentum tracks to minimize the influence of
multiple scattering or cuts on the minimum number of hits, have a strong influence
on the convergence. More subtle is the effect of an unbalanced mixture of tracks
or the complete absence of some types of tracks, such as tracks from collisions
and cosmic events or tracks taken with and without a magnetic field. This is due
to the fact that any kind of tracks has several unconstrained degrees of freedom,
usually referred to as weak modes, weakly defined modes or χ 2 -invariant modes.
As an example, typical weak modes for straight tracks are shears but not bends, and
vice versa for curved tracks. Combining the information of both kinds of tracks is
therefore a reasonable strategy to avoid these deformations in the final result. The
most obvious weak mode is a translation or rotation of the entire tracking device,
which can be only fixed with some kind of reference frame, be it an external system
or by definition. This, however, is less severe and sometimes even not considered at
all, as it does not affect the internal alignment of the tracking device.
Once a set of alignment parameters is calculated, it should always be validated [83, chapter 11]. Apart from checking the improvement of the residuals,
several physics measurements can be utilized, especially to probe for remaining
weak modes. Known charge, forward-backward or ϕ-symmetries of distinct physics
processes can be used. Distributions of the signed curvature or the signed transverse
impact parameter are also sensitive observables.
13.1.4.2 Examples of Alignment Algorithms
Some modern experiments deploy large tracking devices that require a large number
of alignment parameters, of the order of 10 5 . In such a case the computation
of parameters by using straightforward recipes might become unreasonably slow
or cause numerical problems. The two algorithms presented in this section are
examples of how to cope with such challenging circumstances.
The HIP Algorithm
The HIP algorithm [101] is a straightforward and easy-to-implement biased alignment algorithm. It computes the alignment parameters for each alignable object
separately. Only when iterating on the track sample a certain kind of indirect
feedback between the alignable objects is established due to the track refit.
Since only individual alignable objects are regarded, Eq. (13.5) can be partitioned. This is simply done by evaluating the corresponding expressions for each
alignable object i together with its associated parameters p i :
r i (p i ) = m i − d i ( ˘
q, p i ) ≈ m i − ˘
d i − D pi p i