Robust signal decompositions on the circle

Imagine an agent moving along a circular path in the plane with some stationary landmarks, whose number and exact locations are unknown to the agent. Suppose that each landmark transmits an omnidirectional signal with a finite range, which we can model as a function that equals 1 inside a circular disk centered at the landmark and 0 outside. The boundaries of these disks, whose radii are in general different, may intersect the agent's path at one or two points or not at all. As the agent moves along its path, it can perceive these signals and so it knows, at each point, the number of landmarks that are within range. It cannot, however, identify different landmarks by their signals, and neither can it discern anything about each signal's strength other than its presence or absence. The agent's knowledge of its position on the circle may also not be precise, and the signal transmissions or measurements may occur with some sampling frequency rather than continuously in time. For these reasons, all that the agent can reliably reconstruct is a sequence of nonnegative integers corresponding to local landmark counts around the circle, and it may not be sure of the precise count at the exact points where this count changes. In this scenario, we want to pose the following questions: Can the agent figure out the total number of landmarks (excluding, of course, those whose signals do not reach any points on the circle)?