1. A method for use in an electronic surveillance apparatus comprising a processor and a memory, the memory storing instructions which when executed cause the processor to perform the method comprising the following steps: a) establishing a plurality of hypotheses associated with a particular point in time, the plurality of hypotheses including at least first and second hypotheses, each of the at least first and second hypotheses comprising a set of objects hypothesized to be in an area under surveillance by the system at the particular point in time and further comprising hypothesized prior trajectories through the area under surveillance for at least ones of the hypothesized objects, b) detecting objects present in the area under surveillance at a subsequent point in time, c) establishing a first plurality of extended hypotheses associated with the subsequent point in time, the first plurality of extended hypotheses including at least a first extended hypothesis and one or more other extended hypotheses, i) the first extended hypothesis in the first plurality comprising 1) at least ones of the trajectories of the first hypothesis, 2) one or more of the detected objects, and 3) a particular set of connections extending at least ones of the trajectories of the first hypothesis to the one or more of the detected objects, the first extended hypothesis in the first plurality thereby comprising a set of objects hypothesized to be in the area under surveillance at the subsequent point in time and further comprising hypothesized prior trajectories through the area under surveillance for at least ones of the set of hypothesized objects, ii) the one or more other extended hypotheses in the first plurality each comprising 1) at least ones of the trajectories of the first hypothesis, 2) one or more of the detected objects, and 3) a particular set of connections extending at least ones of the trajectories of the first hypothesis to the one or more of the detected objects, each of the one or more other extended hypotheses in the first plurality thereby comprising a set of objects hypothesized to be in the area under surveillance at the subsequent point in time and further comprising hypothesized prior trajectories through the area under surveillance for at least ones of the set of hypothesized objects, iii) the first extended hypothesis of the first plurality and at least one of said one or more other extended hypotheses of the first plurality being different from one another, d) establishing a second plurality of extended hypotheses associated with said subsequent point in time, the second plurality of extended hypotheses including at least a first extended hypothesis and one or more other extended hypotheses, i) the first extended hypothesis in the second plurality comprising 1) at least ones of the trajectories of the second hypothesis, 2) one or more of the detected objects, and 3) a particular set of connections extending at least ones of the trajectories of the second hypothesis to the one or more of the detected objects, the first extended hypothesis in the second plurality thereby comprising a set of objects hypothesized to be in the area under surveillance at said subsequent point in time and further comprising hypothesized prior trajectories through the area under surveillance for at least ones of the set of hypothesized objects, ii) the one or more other extended hypotheses in the second plurality each comprising 1) at least ones of the trajectories of the second hypothesis, 2) so one or more of the detected objects, and 3) a particular set of connections extending at least ones of the trajectories of the second hypothesis to the one or more of the detected objects, each of the one or more other extended hypothesis in the second plurality thereby comprising a set of objects hypothesized to be in the area under surveillance at the subsequent point in time and further comprising hypothesized prior trajectories through the area under surveillance for at least ones of the set of hypothesized objects, iii) the first extended hypothesis of the second plurality and at least one of said one or more other extended hypotheses of the second plurality being different from one another, e) identifying at least one of the plurality of extended hypotheses associated with said subsequent point in time as being more likely than others of the extended hypotheses associated with that point in time as comprising the actual trajectories of the actual objects in the area under surveillance at that point in time, and f) utilizing the identified extended hypothesis to determine whether at least one predefined alert condition has occurred within the area under surveillance.