Inside and Outside Within Combinatorial Pyramids

Abstract

Irregular pyramids are made of a stack of successively reduced graphs embedded in the plane. Such pyramids are often used within the segmentation and the connected component analysis frameworks to detect meaningful objects together with their spatial and topological relationships. The graphs reduced in the pyramid may be region adjacency graphs, dual graphs or combinatorial maps. Using any of these graphs each vertex of a reduced graph encodes a region of the image. Using simple graphs one edge between two vertices encodes the existence of a common boundary between two regions. Using dual graphs and combinatorial maps, each connected boundary segment between two regions is associated to one edge. Moreover, special edges called loops may be used to differentiate a special type of adjacency where one region surrounds the other. We show in this article that the loop information does not allow to distinguish inside and outside of the loop by local computations. We provide a method based on the combinatorial pyramid framework which uses the orientation explicitly encoded by combinatorial maps to determine inside and outside with local calculus.