  We introduce a new distortion measure for point processes called
functional-covering distortion. It is inspired by intensity theory and is
related to both the covering of point processes and logarithmic loss
distortion. We obtain the distortion-rate function with feedforward under this
distortion measure for a large class of point processes. For Poisson processes,
the rate-distortion function is obtained under a general condition called
constrained functional-covering distortion, of which both covering and
functional-covering are special cases. Also for Poisson processes, we
characterize the rate-distortion region for a two-encoder CEO problem and show
that feedforward does not enlarge this region.
