  For the emerging Internet of Things (IoT), one of the most critical problems
is the real-time reconstruction of signals from a set of aged measurements.
During the reconstruction, distortion occurs between the observed signal and
the reconstructed signal due to sampling and transmission. In this paper, we
focus on minimizing the average distortion defined as the 1-norm of the
difference of the two signals under the scenario that a Poisson counting
process is reconstructed in real-time on a remote monitor. Especially, we
consider the reconstruction under uniform sampling policy and two non-uniform
sampling policies, i.e., the threshold-based policy and the zero-wait policy.
For each of the policy, we derive the closed-form expression of the average
distortion by dividing the overall distortion area into polygons and analyzing
their structures. It turns out that the polygons are built up by sub-polygons
that account for distortions caused by sampling and transmission. The
closed-form expressions of the average distortion help us find the optimal
sampling parameters that achieve the minimum distortion. Simulation results are
provided to validate our conclusion.
