Expressiveness of the morphological language
Júnior Barrera, Gerald Jean Francis Banon · Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE · 1992
An important aspect of mathematical morphology is the description of set mappings by the use of a formal language, called here the morphological language (ML), whose vocabulary are erosions, dilations, anti-erosions, anti-dilations, infimum and supremum. Since the sixties, special machines, the Morphological maChines (MC), have been built to efficiently perform this language. These machines have proved to be very useful by solving hundreds of image analysis problems. A natural question that arises is: what class of mappings are phrases of the ML? Now, we can answer this question precisely. In 1991, Banon and Barrera proved that any translation-invariant (TI) mapping can be decomposed as the supremum of sup-generating mappings (the infimum of an erosion and an anti-dilation), with structuring elements that are extremities of closed set intervals contained in the kernel. Adding the hypothesis of upper semi-continuity (USC), they simplified the result by taking a minimal subcollection of sup- generating mappings. Now, we follow the same idea and generalize the concept of kernel, in order to state that any set mapping (non necessarily TI) can be built in a MC. We present decompositions for set mappings in terms of sets of (non-TI) sup-generating mappings, defined from the generalized kernels. Under the USC hypothesis, we also arrive to minimal decompositions. Some examples illustrate the main results.