Mathematical morphology based on linear combined metric spaces on Z² (Part I): Fast distance transforms

Mathematical Morphology (MM) is a general method for image processing based on set theory. The two basic morphological operators are dilation and erosion. From these, several non linear filters have been developed usually with polynomial complexity, and this because the two basic operators depend strongly on the definition of the structural element. Most efforts to improve the algorithm's speed for each operator are based on structural element decomposition and/or efficient codification. A new framework and a theoretical basis toward the construction of fast morphological operators (of zero complexity) for an infinite (countable) family of regular metric spaces are presented in work. The framework is completely defined by the three axioms of metric. The theoretical basis here developed points out properties of some metric spaces and relationships between metric spaces in the same family, just in terms of the properties of the 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 most commonly used metrics such as the chamfer, the city block and the chess board metrics. Generalizations and new results related with distances and distance transforms, which in turn are used to develop the morphologic operations in constant time, in contrast with the polynomial time algorithms are also given.