 
Summary: On the number of views of polyhedral scenes
Boris Aronov 1 , Herv’e Br˜onnimann 1 , Dan Halperin 2 , and Robert Schi#enbauer 1
1 Polytechnic University, Brooklyn NY 11201, USA
{aronov,hbr,rschiff}@photon.poly.edu 2 TelAviv University, TelAviv, Israel
halperin@math.tau.ac.il
Abstract. It is known that a scene consisting of k convex polyhedra of
total complexity n has at most O(n 4 k 2 ) distinct orthographic views,
and that the number of such views
is# ((nk 2 + n 2 ) 2 ) in the worst
case. The corresponding bounds for perspective views are O(n 6 k 3 ) and
# ((nk 2 +n 2 ) 3 ), respectively. In this paper, we close these gaps by improv
ing the lower bounds. We construct an example of a scene with #(n 4 k 2 )
orthographic views, and another with #(n 6 k 3 ) perspective views. Our
construction can also be used to improve the known lower bounds for
the number of silhouette views and for the number of distinct views
from a viewpoint moving along a straight line.
1 Introduction
Aspect graphs have been studied in image analysis as a way to encode all topo
logically distinct views of a scene [2]. In this paper, we concentrate on simply
bounding the number of such views, in the case where the scene consists of k
