Mathematical Morphology Based on Linear Combined MetricSpaces on Z2 (Part I): Fast Distance Transforms

• Authors: J. L. Díaz De León S.;J. H. Sossa-Azuela
• Venue: Journal of Mathematical Imaging and Vision
• Year: 2000

Mathematical Morphology (MM) is a general method forimage processing based on set theory. The two basic morphologicaloperators are dilation and erosion. From these, several non linearfilters have been developed usually with polynomial complexity, andthis because the two basic operators depend strongly on thedefinition of the structural element. Most efforts to improve thealgorithm's speed for each operator are based on structural elementdecomposition and/or efficient codification.A new framework and a theoretical basis toward the construction of fastmorphological operators (of zero complexity) for an infinite (countable)family of regular metric spaces are presented in work. The framework iscompletely defined by the three axioms of metric. The theoretical basis heredeveloped points out properties of some metric spaces and relationshipsbetween metric spaces in the same family, just in terms of the properties ofthe four basic metrics stated in this work. Concepts such as bounds,neighborhoods and contours are also related by the same framework.The presented results, are general in the sense that they cover the mostcommonly used metrics such as the chamfer, the city block and the chessboard metrics. Generalizations and new results related with distances anddistance transforms, which in turn are used to develop the morphologicoperations in constant time, in contrast with the polynomial time algorithmsare also given.