Shadow sequences

We conclude this section with the introduction of a shadow sequence, of which the importance will become more apparent in coming sections. When the shadows of $S(t_0, t_f)$ for a fixed path $\tau$ are put together, a sequential structure comes up. This structure, which we call a shadow sequence, captures the combinatorial changes of the labeled shadows through component events. A graphical illustration of the shadow sequence for the example in Fig. 1 is given in Fig. 5.

Figure 5: A graphical illustration of the shadow sequence for the example from Fig. 1. The numbers in the circles represent the labels of the shadows. The four events marked on the time line, from top to bottom, are appear, merge, split, and disappear.