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 tradeoff 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 Graphwhich 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.
 Publication:

arXiv eprints
 Pub Date:
 September 2008
 arXiv:
 arXiv:0810.0052
 Bibcode:
 2008arXiv0810.0052F
 Keywords:

 Computer Science  Computational Geometry;
 Computer Science  Data Structures and Algorithms;
 I.3.5;
 F.2.2
 EPrint:
 added Section 4: Implementation and Empirical Evaluation