13 Pattern Recognition and Reconstruction
587
if the field is zero, the track model is a straight line. In other cases the track model
may have to be computed numerically (see Sect. 13.1.3.2).
13.2.3.1 Least-Squares Methods
The conventional approach to estimating the vertex position is the minimization
of some quadratic objective function, yielding a least-squares estimate. There are
two main flavors of least-squares estimation in vertex fitting, constrained and
unconstrained minimization. In the first case the vertex constraint is introduced into
the objective function via a Lagrange multiplier, in the second case the constraint is
implicit in the track model.
As an example, consider a vertex fit with n straight tracks. The n straight tracks
originating from the common vertex v = (x v , y v , z v ) T can be represented by n
straight lines with parameters λ i :
x = x v + λ i a i , y = y v + λ i b i , z = z v + λ i , i = 1, . . . , n,
where a i and b i are the direction tangents at the vertex. At the reference surface
z = z ref track i is specified by its parameter vector q i = (x i , y i , a i , b i ) T , consisting
of the intersection point (x i , y i ) and the two direction tangents (a i , b i ). The track
fit delivers estimates ˜
q i and information matrices G i for i = 1, . . . , n. In the
constrained problem the sum of the squared residuals
M(q 1 , . . . , q n ) =
n
i=1
e i
T G i e i , e i = ˜
q i − q i ,
(13.7)
must be minimized under the 2n nonlinear constraints
x v = x i + a i (z − z ref ), y v = y i + b i (z − z ref ), i = 1, . . . , n
There are 4n + 3 unknowns, 4n observations and 2n constraints, giving 4n + 2n −
(4n + 3) = 2n − 3 degrees of freedom. The resulting track parameters ¯
q i fit best,
in the least-squares sense, to the track fit estimates ˜
q i and at the same time have a
common vertex. For the solution of the constrained vertex fit see Sect. 13.2.4.2.
In the example, the constraints can be rewritten as
x i = x v + (z ref − z)a i , y i = y v + (z ref − z)b i , i = 1, . . . , n.
(13.8)
Insertion of Eq. (13.8) into Eq. (13.7) gives the objective function of the unconstrained nonlinear least-squares problem:
M(v, a 1 , b 1 , . . . , a n , b n ) =
n
i=1
e i
T G i e i , e i = ˜
q i − q i .
587
if the field is zero, the track model is a straight line. In other cases the track model
may have to be computed numerically (see Sect. 13.1.3.2).
13.2.3.1 Least-Squares Methods
The conventional approach to estimating the vertex position is the minimization
of some quadratic objective function, yielding a least-squares estimate. There are
two main flavors of least-squares estimation in vertex fitting, constrained and
unconstrained minimization. In the first case the vertex constraint is introduced into
the objective function via a Lagrange multiplier, in the second case the constraint is
implicit in the track model.
As an example, consider a vertex fit with n straight tracks. The n straight tracks
originating from the common vertex v = (x v , y v , z v ) T can be represented by n
straight lines with parameters λ i :
x = x v + λ i a i , y = y v + λ i b i , z = z v + λ i , i = 1, . . . , n,
where a i and b i are the direction tangents at the vertex. At the reference surface
z = z ref track i is specified by its parameter vector q i = (x i , y i , a i , b i ) T , consisting
of the intersection point (x i , y i ) and the two direction tangents (a i , b i ). The track
fit delivers estimates ˜
q i and information matrices G i for i = 1, . . . , n. In the
constrained problem the sum of the squared residuals
M(q 1 , . . . , q n ) =
n
i=1
e i
T G i e i , e i = ˜
q i − q i ,
(13.7)
must be minimized under the 2n nonlinear constraints
x v = x i + a i (z − z ref ), y v = y i + b i (z − z ref ), i = 1, . . . , n
There are 4n + 3 unknowns, 4n observations and 2n constraints, giving 4n + 2n −
(4n + 3) = 2n − 3 degrees of freedom. The resulting track parameters ¯
q i fit best,
in the least-squares sense, to the track fit estimates ˜
q i and at the same time have a
common vertex. For the solution of the constrained vertex fit see Sect. 13.2.4.2.
In the example, the constraints can be rewritten as
x i = x v + (z ref − z)a i , y i = y v + (z ref − z)b i , i = 1, . . . , n.
(13.8)
Insertion of Eq. (13.8) into Eq. (13.7) gives the objective function of the unconstrained nonlinear least-squares problem:
M(v, a 1 , b 1 , . . . , a n , b n ) =
n
i=1
e i
T G i e i , e i = ˜
q i − q i .