Encoding Long-Range E ffects Locally

The MAP solution of the chain graph model is obtained from all time slices at once allowing the model to consider track properties over several time slices. This may sound counterintuitive since the factors in the chain graph model are only defined over variables from at most two consecutive time slices and do not consider tracks as a whole. For instance, in contrast to actual cell nuclei, false positive detections tend to suddenly appear and only exist for a comparatively short time span since they are mostly caused by noise clusters in the raw data. To mitigate this issue we could add higher order factors to the model to encode our belief of the expected lifetime of a track. We now show that the model in its current form already considers the above situation without the need to add any higher order factors.

Consider a highly idealized situation with only one detection per time slice, a small number of time slices n, and a fixed energy cost for all tracking events as defined in the following table:

7. Discussion Space Slice Space Slice s s + 1 s + 2

vs.

(a) Two explanations for the same detec- tions: either all true positives (in blue) or all false positives (in red).

Energy

Track Lifetime

(b) Schematic of the energy costs for the two explanations (same color coding as on the left).

Figure 7.1.: Local costs determine wide-range decisions. A series of detections is explained either as all true positives or all false positives (left). For short lifetimes the latter explanation is more likely in the model. To be considered a temporal sequence of true objects the track has to reach a certain minimal lifetime to exhibit less energy costs than the other explanation. The costs for both explanations are linear functions of the lifetime but with different slope and intercept (right).

Event Energy

Appearance 25

Disappearance 25

Move 5

True Positive Detection 7 False Positive Detection 30

Opportunity 0

In case of n = 1 there is only one decision to be made: is the one detection a true positive or a false positive? The former has to be explained in terms of an appearance (25 energy units), a disappearance (25 energy units), and a true positive detection event (7 energy units) and has a total energy cost of 57. The latter can be explained in terms of just one false positive detection event and we only need to expend 30 energy units. We therefore explain the one detection as being a false positive. Now compare that with n = 4. The two most likely explanations are that all four detections are either the same object existing during four time slices or are all false positives. For the former explanation we would now need to expend in total 93 energy units (one appearance, four true positive detections, three moves, and one disappearance) and for the later 120 energy units (four false positive

7.1. A Holistic Model Over All Time Slices Helps Tracking detections). In this case we would pick the other explanation that the detections constitute a true track. The two possible decisions are illustrated in Fig. 7.1a on the preceding page.

In general, the energy costs for the false positive explanation are a linear function of n with a slope equal to the false positive detection energy and an intercept of zero. The true positive explanation costs are also a linear function of n with a less steep slope equal to the sum of true positive detection and move energy but a positive intercept equal to the sum of appearance and disappearance energy. That is, a object or track needs a minimal lifespan to overcome the track initialization and termination costs and the costs for the alternative explanation of being all false positives. Fig. 7.1b on the facing page illustrates the idea. To conclude, this effect is exploited in the chain graph model to reason about minimal track lengths given evidence encoded in the energy values.

Another more obvious way to consider wide-range effects with local factors is the minimal cell cycle length extension. It controls the minimal temporal distance between divisions—not directly with higher order factors but indirectly using the counting trick (see Sec. 4.4.2 on page 36)—and is another example of how the chain graph model can reason globally using local factors..

We argue that local factors are the better choice compared to global factors because they allow to build up a model from the same basic elements for any data input. In particular, time slices can be easily added to the chain graph by just repeating the same construction used for all other slices. Higher order factors over time would involve newly added variables for the new slice and already added variables from older slices and some factors could only be added after the model has reached a certain size. This would complicate both the theoretical formulation of the model and its implementation in software, increasing the chance of errors. However, when using a linear programming based inference method global factors can nevertheless be useful. There might be some constraints (i.e. zero probability configurations) that are only rarely violated in a typical solution. Such cases can be handled efficiently with an iterative cutting planes (Kelley, 1960) approach for which a few global constraints make more sense than many local ones (again, for simplicity reasons but also because of possible performance gains).