Targets

Our interest in shadows lies with obtaining information that is not available to the robots' sensors. To effectively investigate how to track such information, it is necessary to formally characterize what we mean by information. We assume that there is a non-negative integer number of targets in $F$ , which are point entities that move arbitrarily fast, following some continuous, unpredictable trajectories. The robots' sensors can detect certain attributes of the targets. We are interested in two types of attributes, each with several levels of granularities:
1) Location. When the targets move in/out the sensors' FOV, their appearance/disappearance may be detected. Depending on the sensors' capabilities, at least two levels of precision are possible:

1. The sensors can tell whether the FOV contains no target or at least one. In other words, one or more targets in FOV will appear the same to the sensors, making the sensors' output binary. One such binary location sensor is a motion detector, which is surprisingly reliable.
2. Each target inside the FOV can be precisely located and counted. Cars equipped with high resolution GPS tracking units fall into this category.

2) Identity. When multiple targets are being detected by sensors, it may be possible to tell them apart. That is, the sensors may be able to distinguish the targets in the FOV. Roughly speaking, the targets may be:

1. Fully distinguishable. When targets possess unique IDs recognizable by the sensors, they are fully distinguishable. This is like a set of merchandise with pairwise different UPCs: A label scanner can easily identify each one with no ambiguity.
2. Indistinguishable. Although it appears that full distinguishability is the most powerful, it is not always available due to cheap sensors or even desirable due to concerns such as privacy. It is not hard to make targets indistinguishable: In the sensor output, erase any attributes that can be used to distinguish among the targets.
3. Partially distinguishable. Everything between the previous two notions of distinguishability belongs to this class, which may again be of various fineness. For instance, targets may form teams that are distinguishable by color.

Location and identity are related - full distinguishability implies that the sensors should be able to locate targets in the FOV. On the other hand, tracking locations over time can be used to distinguish targets. However, these two attributes are not identical and it benefits to treat them orthogonally. For example, when colored teams of targets are present, a low resolution overhead camera can easily tell whether a team is present in the FOV via a color scan, acting as a combination of binary location sensor and identity sensor. Given sensors that can detect some subsets of the above mentioned attributes of targets, each labeled shadow can be assigned one or more variables that describe these attributes of the targets residing in the shadow. Note that although we deal mostly with binary and integer variables in this paper, variables of other forms, such as real numbers, can also be incorporated over the structure of shadows and component events introduced here. When we consider targets in the shadows, a type of invariance arises:

Observation 1   In an environment with only component events, the number of targets hidden in a workspace-time shadow is invariant along its span over time; furthermore, a workspace-time shadow is a maximal set in which such invariance holds.

By the assumption that a hidden target moves continuously, its trajectory is contained in the same workspace-time shadow when no component events happen. Two workspace shadows, as different time slices of the same workspace-time shadow, must intersect the same number of such trajectories since no target enters or exits the component in the time being. This yields the invariance. The second claim follows the definition of workspace-time shadow as a maximal union of all such workspace shadows.