Problem formulation

In the basic setup, besides the availability of a sequence of component and FOV event observations (e.g. Fig. 7), the following assumptions are made:

1. Component events are observed without error.
2. Targets are indistinguishable. The initial condition is given as a joint probability distribution $P(s_1, \ldots, s_n)$ of targets in the $n$ shadows at $t = t_0$ .
3. When a split component event happens, a probabilistic split rule decides how the targets should redistribute.
4. Observations of FOV events follows distribution given by $P(\mathbf{e}=e\vert\mathbf{y}=y), e\in E_{FOV}, y \in Y_{FOV}$ .

After general algorithms are presented, we discuss extensions relaxing the first two assumptions. The last two assumptions can be satisfied by collecting and analyzing sensor data from the same environment; the necessity of these two assumptions will become self-evident shortly. Given these assumptions, we want to obtain the target distribution in the $m$ shadows, $P(s_{1}', \ldots, s_{m}')$ , at time $t = t_f$ .

The resulting joint probability distribution is useful in solving many decision making problems; for example, in a fire evacuation scenario, knowing the the expected number of people trapped in various parts (shadows) of a building (possibly estimated through observations from infrared beam sensors or security cameras), firefighters can better decide which region of the building should be given priority when they look around. The expected number of people in each shadow is readily available from the joint probability distribution.