Planar Visibility Counting
Abstract
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.
Cite
@article{arxiv.0810.0052,
title = {Planar Visibility Counting},
author = {Matthias Fischer and Matthias Hilbig and Claudius Jähn and Friedhelm Meyer auf der Heide and Martin Ziegler},
journal= {arXiv preprint arXiv:0810.0052},
year = {2009}
}
Comments
added Section 4: Implementation and Empirical Evaluation