Connected operators and pyramids

This paper deals with the notion of connected operators in the context of mathematical morphology. In the case of gray level functions, the flat zones over a space E are defined as the largest connected components of E on which the function is constant (a flat zone may be reduced to a single point). Hence, the flat zones of every function make a partition of the space. A connected operator acting on a function is a mapping which enlarges the partition of the space created by the flat zones of the functions. In this paper, it is shown that, from any connected operator acting on sets, one can construct a connected operator for functions. Then, the concept of pyramid is introduced and one of the most important results of this study is that, if a pyramid is based on connected operators, the flat zones of the functions increase with the level of the pyramid. In other words, the flat zones are nested. Then, a very important class of connected filter called `filter by reconstruction' is defined and its properties are stated and discussed. Rules to create pyramids relying on filters by reconstruction are proposed.