Planar Visibility Counting
For a fixed virtual scene (=collection of simplices) S and given observer
position p, how many elements of S are weakly visible (i.e. not fully occluded
by others) from p? The present work explores the trade-off between query time
and preprocessing space for these quantities in 2D: exactly, in the approximate
deterministic, and in the probabilistic sense. We deduce the EXISTENCE of an
O(m^2/n^2) space data structure for S that, given p and time O(log n), allows
to approximate the ratio of occluded segments up to arbitrary constant absolute
error; here m denotes the size of the Visibility Graph--which may be quadratic,
but typically is just linear in the size n of the scene S. On the other hand,
we present a data structure CONSTRUCTIBLE in O(n*log(n)+m^2*polylog(n)/k)
preprocessing time and space with similar approximation properties and query
time O(k*polylog n), where k<n is an arbitrary parameter. We describe an
implementation of this approach and demonstrate the practical benefit of the
parameter k to trade memory for query time in an empirical evaluation on three
classes of benchmark scenes.