Field-of-view events

If a location sensor also has memory, it will be able to detect changes to the number of targets in the FOV during a short time interval. We call such a change a field-of-view event (FOV event for short), which is a second type of critical events of our interest. Furthermore, if the sensors know where a FOV event happens, these events can be associated with corresponding shadows. For a shadow $s_i$ , three FOV events are possible: 1) A target enters $s_i$ from the FOV, 2) A target exits $s_i$ into the FOV, and 3) Nothing happens at the boundaries between $s_i$ and the FOV (for a period of time), or null event. Denoting these events $e_e, e_x, e_n$ , respectively, the collection of possible field-of-view events for a shadow $s_i$ is the set

$$E_{FOV} = \{e_e, e_x, e_n \}.$$

Some sensors may only detect the enter and exit events explicitly, such as a sensing node in a sensor network that only senses targets passing through the boundary of its sensing range. For detection beams, the FOV is a line segment, which causes two FOV events to happen consecutively (see Figure 6).

Figure 6: Illustration of FOV events for an environment with obstructed visibility (left) and for an environment with detection beams (right). 1) A target is about to exit a shadow into the FOV of the sensor (yellow disc). 2) A target is about to enter a shadow from the sensor's FOV. 3) A target is about to enter and exit the FOV of a beam sensor.

Certain systems may not have FOV events at all; an instance is a pursuit evasion game in which the evader always avoids appearing in the pursuer's FOV. The game ends when an evader is found or when it is confirmed that no evader is in the environment.

Figure 7: A typical sequence of critical events. The circles with numbers represent the shadows; the labeled arrows associate field-of-view events to shadows.

Since component events and FOV events both happen as robots move along some path $\tau$ in the free space $F$ , it makes sense to treat them as a whole. It does not take much to represent them together: We can simply augment the shadow sequence to include the FOV events. A typical combined sequence of critical events is shown in Fig. 7. With the introduction of FOV events, the invariance from Observation 1 needs to be updated.

Observation 2   In an environment with component and FOV events, the number of targets hidden in a workspace-time shadow is invariant between FOV events (excluding null events) associated with the shadow; the time span of such invariance is again maximal.

To see why the above statement is true, note that an enter FOV event can be viewed as an appear component event immediately followed by a merge component event. Same breakdown holds for exit FOV events. The case is then reduced to Observation 1. Observation 2 establishes that for the task of tracking hidden targets that move nondeterministically, nearly all information (sensor data) can be discarded.