An item in the two-dimensional track file is described by an eight-component state vector
where the component vectors on the RHS of Equation 18.6 are four-element kinematic state vectors as defined for Equation 18.1, referred to the standard measurement axes:
In principle, each track described by Equation 18.6 has an associated covariance matrix with 36 independent elements. In order to reduce the storage and CPU resource requirements of two-dimensional tracking, a simplifying assumption is made. The measurement error matrix for a two-dimensional datum
is taken to have the simple form
with the same effective value used to describe the measurement variance for each projection, and no correlation of the measurement errors. The assumption in Equations 18.7 and 18.8 is reasonable, provided the effective measurement error is made large enough, and reduces the number of independent components in the covariance matrix from 36 to 10.
The central task of the two-dimensional track extension module is to find all plausible trackhit associations, subject to a set of criteria which define ``plausible.'' The primary association criterion is based on the track association score
where is the variance of the prediced data position along a reference axis and
is the difference between the actual data value and that predicted by Equation 18.6 for the time of the datum. Equation 18.9 is simply a dimensionless measure of the size of the mismatch in Equation 18.10, normalized by the expected prediction error.
The first step in limiting TrackHit associations is a simple cut on the association score of Equation 18.9. For the dense, multitarget environments used in Sim89, this simple cut is not sufficiently restrictive, and a variety of additional heuristic cuts are made. The most important of these are
The actual track scoring cut is a bit more complicated than the preceding paragraph implies. Let denote the nominal extension score of Equation 18.9. In addition, define a cumulative association score which is updated on associations in a fading memory fashion
with (typically) . An extension is accepted only if is below some nominal cutoff (typically 3-4) and is below a more restrictive cut (2-3). This second cut prevents creation of poor tracks with barely acceptable extension scores at each step.
The preceding rules for TrackHit associations define the basic two-dimensional track extension formalism. There are, however, two additional problems which must be addressed:
In regard to the first problem, two entries in the track file are said to be equivalent if they involve the same associated data points over the past four scans. If an equivalent track pair is found in the track file, the track with a higher cumulative score is simply deleted. The natural mechanism for track deletion in a track-splitting model is based on the track's data association history. If no data items give acceptable association scores over some preset number of scans (typically 0-2), the track is simply discarded.
The equivalent-track merging and poor track deletion mechanisms are not sufficient to prevent track file ``explosions'' in truly dense environments. A final track-limiting mechanism is simply a hard cutoff on the number of tracks maintained for any item in the data set (this cut is typically ). If more than tracks give acceptable association scores to a particular datum, only the pairings with the lowest total association scores are kept.
The complexity of the track extension algorithm is nominally for new data items and existing tracks. This computational burden is easily reduced to something closer to by sorting both the incoming data and the predicted data values for existing tracks.