Track joining

The idea is to implement an algorithm that is able to connect and merge tracks together according to similar criteria as one would connect single spots. The method is based on the same linear optimization problem as the method of Karrenbauer et al. but now for the connection of tracks instead of single spots.

This leads to some subtle amendments. It is not only possible to continue a trajectory by connecting it with another one, but also to really merge overlapping trajectories in the sense that one track is allowed to fill the time gaps of the other one (like shown in Figure4.6 going from panel C to panel D). In the next paragraphs the steps of this procedure are discussed in more detail. The focus of the description is on new features and changes with regard to the technique of Karrenbauer et al. a detailed introduction to the construction of the linear optimization problem is given in Section4.3.

Virtual track endings

In order to take into account the past and future evolution of the tracks for the selection of the most probable joinings, the decision time length𝑡_decis introduced. Virtual starting points and endpoints of the trajectories are retrieved by computing the average (=̂center of mass) of the positions of the first and the last𝑡_decsnapshots. If there are gaps at the endings of the track, they reduce the number of spots that go into the average, so that one always obtains time averaged virtual endings. A choice of𝑡_dec = 1avoids any averaging, then the actual endings of the tracks are used.

Search for connectable non-overlapping tracks

For each virtual endpoint one searches for virtual starting points of other tracks at a distance closer than 𝛿𝑟_max. The time interval between endpoint and starting point has to be smaller than𝛿𝑡_maxand bigger than zero, which only allows non-overlapping tracks. Connectable tracks found this way are added to the list of possible links.

Search for connectable overlapping tracks

For each track one also searches for other tracks that have their starting time during the course of that track and have their virtual starting point closer than 𝑓 ⋅𝛿𝑟_maxto that track’s virtual endpoint. A good choice for all purposes is𝑓 = 3, which avoids the missing of any possible connections. Before adding these links to the list, one has to do more tests. For each of the other tracks found, one needs to check if the other track only fills time gaps, or if there are spots at common times in both tracks. If only gaps are filled, everything is fine, else it depends on whether one wants to allow “joined spots” (see section4.3.3).

Figure 4.8:For overlapping tracks one looks at the spots𝑡_decbefore and after a transition from one track to the other. Here they are marked with lighter colors (𝑡_dec= 3). The average position of all light green spots has to be closer than𝛿𝑟_max to the average position of all light magenta spots. If spot joining is allowed, spots at common times, here at𝑡= 38, must be closer than the join radius𝛿𝑟_join

4.4 Iterative tracking If they shall be allowed, one checks if the distance of spots at common times is always smaller than the join radius𝛿𝑟_join. If this is not the case, this possible link is discarded. Finally, before adding the link to the list, one has to decide if the spatial distance between the two overlapping tracks is small enough. In order to do this one has to look at all transitions from one track to the other, more precisely at the spots 𝑡_decbefore and𝑡_decafter the transitions (see Figure4.8). For each of both tracks the average position of all those spots is computed. Only if the distance between these two averages is smaller than𝛿𝑟_maxthe corresponding link is added to the list of possible links.

Search for conflicts

The connection of overlapping tracks can lead to conflicts. In this context it is not a transitive relation, in the sense that if A↔B and B↔C are allowed it does not follow that A↔C is possible, too. Track A is compatible with track C only if either A↔C is on the list of possible connections or if the time lines of A and C do not overlap. Otherwise their connection leads to wrong and unexpected results, e.g. tracks with two spots at one time. Figure4.9gives examples for such conflicts.

coordinate

Figure 4.9: Conflicts in the connection of overlapping tracks: A↔B or B↔C is allowed, but not both, since A↮C. Another conflict is: A↔B or A↔D is allowed but not both, since B↮D. A third one is:

A↔B or A↔E is allowed but not both, since B↮E. Note that the latter conflict is already avoided by the rule that a track can only have one successor track if no “eating” is involved (the concept of “eating” is explained later). The solution for this case would be A↔D↔E and B↔C.

Figure 4.10:Walking the network of possible links of overlapping tracks in the search for conflicts in the connections of the tracks shown in Figure4.9. In the left panel we see the network of possible links, to the right we see how the network is explored, starting individually from each of the five nodes indicated with large capitals. Numbers give the order in which track compatibility is tested. Jagged lines denote incompatible tracks that lead to the registration of a set of conflicting possible links in the list of conflicts.

In this case the sets{A↔B, B↔C}and{A↔B, A↔D}are added to this list.

In order to avoid them, starting from each track individually one needs to walk through the network given by the list of possible links of overlapping tracks and check for a conflict between connectable tracks.

Tracks are the nodes of this network. The search must go forward and backward in time, if the link A→B is possible, the link B→A is possible too, there is no direction in track connectivity. Figure4.10 illustrates how the network is explored. As an example, let us look at the walk starting from D (magenta arrows): From D one can go directly to A. Now the task is to search for other connections starting at D and conflicting with D↔A. First one has to test if D↔E is allowed at the same time and yes, since A and E do not overlap in time they are compatible (marked with the number 1). Therefore, the links D↔A and D↔E are not conflicting. There are no other links starting directly at D or E, therefore one goes further with the direct links starting at A: Is D↔A allowed together with A↔B? No it is not, tracks B and D are not compatible (marked as test number 2). No other links start from A and since the path to B is not allowed it is not necessary to continue the search with B. The search for conflicts starting from D can be stopped here.

Since the network is walked starting individually from each node, it is clear that one finds many of the incompatibilities more than once (see Figure 4.10). Each incompatibility can be described by a set of conflicting links that might not be selected together in the optimization step (in the example above there are only sets with two links, but larger sets are definitely possible). When a set is added to the list of all the sets that have been found before, it makes sense to check if adding this set is necessary and/or if other sets in the list become obsolete after this set has been added. This keeps the list of conflicts as short as possible.

Cost function

Again the aim of the optimization step is to minimize the distances squared with the proper selection of possible links. Therefore, a very similar cost function is used for track joining. It depends on the squared space distance between the virtual endings of the tracks and the squared time interval between their real endings. The cost for a connection of a track A and a track B is defined as:

𝑐_𝐴𝐵 = (𝑥_𝐵,𝑠^∗−𝑥_𝐴,𝑒∗)²+ (𝑦_𝐵,𝑠^∗−𝑦_𝐴,𝑒∗)²+ (𝑧_𝐵,𝑠^∗−𝑧_𝐴,𝑒∗)²+ (𝑡_𝐵,𝑠−𝑡_𝐴,𝑒)² (4.8) Here𝑠^∗and𝑒^∗denote virtual starting points and endpoints, as well as𝑠and𝑒denote real ones. For each track that has no successive track the costs are𝛿𝑟²_max+𝛿𝑡²_max, which is analogue to Karrenbauer’s method (see Equation4.7). The same cost is paid for having no predecessive track, which means one has to pay twice for an unconnected track. Note that this cost function can also be adapted to include other particle features like e.g. particle diameters or detection quality.

Modifications to the optimization step: Conflicting links and “eating” links

The general form of the linear optimization problem given above in Equations4.4and4.5stays the same as before. However, new rules must be included in the linear equations of constraints𝐌 𝐱=𝐛in order to avoid the selection of conflicting links. In the list of conflicts each entry is a set of links that may not be selected all at once. Remember that selecting link number𝑗 means that in the solution vector𝐱we have 𝑥_𝑗 = 1. Let us assume the set𝑐 = {𝑗, 𝑘, 𝑙}represents a set of conflicting links, saying that selecting together all the links with the numbers𝑗, 𝑘and𝑙is not allowed. If the inequality

𝑥_𝑗+𝑥_𝑘+𝑥_𝑙 <3

is fulfilled it is clear that at least one of the three links is not selected in the solution vector.

4.4 Iterative tracking

Figure 4.11:Track A “eats” track B: In this situation one should allow a second track to connect to track A and forbid further connections to track B.

More generally written, we demand 𝑀_{2𝑛+𝑐,∗}⋅𝐱=∑

𝑘

𝑀_{2𝑛+𝑐,𝑘}𝑥_𝑘 < 𝑛^! _𝑐 =𝑏_2𝑛+𝑐 with 𝑀_{2𝑛+𝑐,𝑙} = 1 for every link𝑙∈𝑐, (4.9) where𝑀_{2𝑛+𝑐,∗}is the(2𝑛+𝑐)th row of𝐌,𝑏_2𝑛+𝑐is the(2𝑛+𝑐)th element of the constraint vector𝐛and 𝑛_𝑐is the number of links in set number𝑐from the list of conflicts. The constraint matrix is thus extended with𝐶new rows, where𝐶is the number of sets in the list of conflicts. Analogously, the constraint vector 𝐛is also extended with𝐶new elements, each of them being the number of links𝑛_𝑐of the corresponding set of conflicting links. So for each set in the list of conflicting links one linear inequality in the form of Equation4.9is appended to the constraints of the linear optimization problem.

Another subtle problem arises if a possible link means that one track “eats” another one. A “eats” B means that track B only fills gaps in the timeline of track A but does not extend the track in time (see Figure4.11). In this case there should be the possibility to connect a second track to track A and further connections to track B should be forbidden. In order to allow for a second connection starting at A one has to change the equation of constraints, because one of the key rules is that one spot may only have one successor spot. A second change in the constraints is required to prohibit further connections starting from B.

To this end, the solution vector𝐱is extended with𝐸 more entries (before it had2𝑛elements),𝐸being the total number of possible eating links. Consequently, also the matrix of constraints𝐌 is extended with𝐸more columns (entries of𝐌not mentioned later are all set to “0”). By adding another row to the constraints for each eating link, the solution vector𝐱is forced to be “1” at position𝑚+ 2𝑛+𝑒if an eating connection𝑗 is selected:

𝑀_{2𝑛+𝐶+𝑒,∗}⋅𝐱=∑

𝑘

𝑀_{2𝑛+𝐶+𝑒}𝑥_𝑘 = 0^! with 𝑀_{2𝑛+𝐶+𝑒,𝑗} = −1and𝑀2𝑛+𝐶+𝑒,𝑚+2𝑛+𝑒 = 1. (4.10) Here𝑗 is the position of the eating link in the list of all possible links, 𝑒 is its position in the list of eating links only; 𝐶 is the number of rows that have been added before to deal with conflicting links.

Let’s say the eating link𝑒connects track number𝑙(A in Fig.4.11) with track number𝑟(B in Fig.4.11).

Equation4.10enforces𝑥_{𝑚+2𝑛+𝑒} = 1if the “eating” link𝑗 is selected. If𝑗is selected, the permission for track𝑙to have a successor must be re-enabled, which is done by setting𝑀_{𝑛+𝑙,𝑚+2𝑛+𝑒}= −1. Furthermore, track𝑟may not have a successor if𝑗is selected, which is enforced by setting𝑀_{𝑛+𝑟,𝑚+2𝑛+𝑒}= 1.

Implementation

For tests of Karrenbauer and Wöll’s algorithm on experimental data, their code was already translated to a Python/NumPy version for 3D data points. Their MATLAB script published onarXiv.org[78]

therefore serves as the basis for the implementation of the new track joining algorithm. However, since

so many details and subtleties have been added, there is not so much left of their code in the end. Some key parts like the (adapted) matrix of constraints and the solution of the linear optimization problem with the IBM ILOG CPLEX optimizer are still used.

The computational performance of the implementation depends on the number and complexity of the tracks to be connected: With many small tracks a big part of the computation time is consumed by the CPLEX optimizer. With fewer and longer tracks at complex constellations the search for possible links takes most of the time. As an example, the reduction of∼ 40000small tracks to∼ 20000longer tracks is done in about 30 seconds on a3GHz machine, at a RAM usage of∼ 1GB. In the evaluation of the experimental data discussed here, this is comparable to an average step in an iterative tracking sequence, so the code is fast enough to do about 20iterations in about ten minutes, and does not need excessive amounts of RAM. Compared to Karrenbauer and Wöll’s code that was not applicable due to its large memory requirement, this is a great improvement.