13 Pattern Recognition and Reconstruction
591
A further simplification was proposed in [126]. In the vicinity of the vertex
the track is approximated by a straight line. The estimated track parameters are
transformed to a coordinate system the x-axis of which is parallel to the track.
The vertex is then estimated by minimizing the sum of the weighted transverse
distances of the tracks to the vertex. The resulting objective function has the same
form as in Eq. (13.12), again with weight matrices of rank 2. The estimate is exact
for straight tracks.
A different type of a fast vertex fitting algorithm is described in [127]. It is
based on approximating the tracks by straight lines both in the x–y plane and in
the x–z plane. In either projection, the lines representing the tracks are Houghtransformed to points in the dual plane of line parameters. The vertex coordinates
are then obtained by a weighted linear least-squares fit in the dual plane.
Adding Prior Information
If the vertex to be fitted is the primary vertex, there may be prior information about
the vertex position from the beam profile in a collider experiment or the target
location in a fixed target experiment. The prior information usually comes in the
form of a position v 0 plus a covariance matrix C 0 . The objective function is then
augmented by an additional term
(v 0 − v)
T C
−1
0 (v 0 − v).
For instance, the Gauss–Newton estimate Eq. (13.10) is modified in the following
way:
˜
v n = C n
C
−1
0 v 0 +
n
i=1
A i
T G i
B ( ˜
q i − c i )
, Var(˜ v n ) = C n =
C
−1
0 +
n
i=1
A i
T G i
B A i
−1
.
Similar modifications apply to the Newton–Raphson estimate and the fast vertex
fits.
13.2.3.2 Vertex Quality and Outlier Removal
Some tracks used in the vertex fit may be outliers in the sense that they do not
actually belong to the vertex. Also, the estimated track parameters may be distorted
by outliers or distorted hits in the track fit. Both types of outliers distort the vertex
estimate and need to be identified.
In the case of Gaussian errors and a linear model the contribution of each track
to the minimum value of the objective function is distributed according to a χ 2 -
distribution with two degrees of freedom. The contribution χ 2
i of track i has to
be computed relative to the vertex estimated without track i. For instance, in the
591
A further simplification was proposed in [126]. In the vicinity of the vertex
the track is approximated by a straight line. The estimated track parameters are
transformed to a coordinate system the x-axis of which is parallel to the track.
The vertex is then estimated by minimizing the sum of the weighted transverse
distances of the tracks to the vertex. The resulting objective function has the same
form as in Eq. (13.12), again with weight matrices of rank 2. The estimate is exact
for straight tracks.
A different type of a fast vertex fitting algorithm is described in [127]. It is
based on approximating the tracks by straight lines both in the x–y plane and in
the x–z plane. In either projection, the lines representing the tracks are Houghtransformed to points in the dual plane of line parameters. The vertex coordinates
are then obtained by a weighted linear least-squares fit in the dual plane.
Adding Prior Information
If the vertex to be fitted is the primary vertex, there may be prior information about
the vertex position from the beam profile in a collider experiment or the target
location in a fixed target experiment. The prior information usually comes in the
form of a position v 0 plus a covariance matrix C 0 . The objective function is then
augmented by an additional term
(v 0 − v)
T C
−1
0 (v 0 − v).
For instance, the Gauss–Newton estimate Eq. (13.10) is modified in the following
way:
˜
v n = C n
C
−1
0 v 0 +
n
i=1
A i
T G i
B ( ˜
q i − c i )
, Var(˜ v n ) = C n =
C
−1
0 +
n
i=1
A i
T G i
B A i
−1
.
Similar modifications apply to the Newton–Raphson estimate and the fast vertex
fits.
13.2.3.2 Vertex Quality and Outlier Removal
Some tracks used in the vertex fit may be outliers in the sense that they do not
actually belong to the vertex. Also, the estimated track parameters may be distorted
by outliers or distorted hits in the track fit. Both types of outliers distort the vertex
estimate and need to be identified.
In the case of Gaussian errors and a linear model the contribution of each track
to the minimum value of the objective function is distributed according to a χ 2 -
distribution with two degrees of freedom. The contribution χ 2
i of track i has to
be computed relative to the vertex estimated without track i. For instance, in the