We discuss the basic concepts of Mathematical Morphology on the
classical context of binary images and also on the abstract context
of complete lattices. In particular, we present an estrategy to construct
dilation and erosion morphological operators on sup-generated complete
lattices.
Next, we present multi-triangulation, a mesh representation based on
variable resolution meshes. To this representation, we show that we can
associate a sup-generated complete lattice. On this lattice, we see how to
define the dilation and erosion operators that reproduce a given sequence
of local refinements and simplifications in a multi-triangulation.
Finally, we show how to represent a given image using a
multi-triangulation and, by constructing dilation operators
on the sup-generated complete lattice associated to this
multi-triangulation, we obtain sequences of images leading
progressively to the given image.