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Sup-compact and inf-compact representations of W-operators

Barrera J., Hashimoto R.F.

Fundamenta Informaticae

2001

Vol. 45, nr 4

283-294

It is well known that any W-operator can be represented as the supremum (respectively, infimum) of sup-generating and (respectively, inf-generating) operators, that is, the families of sup-generating and inf-generating operators constitute the building blocks for representing W-operators. Here, we present two new families of building blocks to represent W-operators: compositions of sup-generating operators with dilations and compositions of inf-generating operators with erosions. The representations based on these new families of operators are called, respectively, sup-compact and inf-compact representations, since they may use less building blocks than the classical sup-generating and inf-generating representations. Considering the W-operators that are both anti-extensive and idempotent -in a strict sense-, we have also gotten a simplification of the sup-compact representation. We have also shown how the inf-compact representation can be simplified for any W-operator such that it is extensive and its dual operator is idempotent -in a strict sense-ź Furthermore, if the W-operators are openings (respectively, closings), we have shown that this simplified sup-compact (respectively, inf-compact) representation reduces to a minimal realization of the classical Matheron's representations for translation invariant openings (respectively, closings).

Automatic programming of Morphological Machines by PAC Learning

Barrera J., Terada R., Hirata R. Jr., Hirata N.S.T.

Fundamenta Informaticae

2000

Vol. 41, Nr 1,2

229-258

An important aspect of mathematical morphology is the description of complete lattice operators by a formal language, the Morphological Language (ML), whose vocabulary is composed of infimum, supremum, dilations, erosions, anti-dilations and anti-erosions. This language is complete (i.e., it can represent any complete lattice operator) and expressive (i.e., many useful operators can be represented as phrases with relatively few words). Since the sixties special machines, the Morphological Machines (MMachs), have been built to implement the ML restricted to the lattices of binary and gray-scale images. However, designing useful MMach programs is not an elementary task. Recently, much research effort has been addressed to automate the programming of MMachs. The goal of the different approaches for this problem is to find suitable knowledge representation formalisms to describe transformations over geometric structures and to translate them automatically into MMach programs by computational systems. We present here the central ideas of an approach based on the representation of transformations by collections of observed-ideal pairs of images and the estimation of suitable operators from these data. In this approach, the estimation of operators is based on statistical optimization or, equivalently, on a branch of Machine Learning Theory known as PAC Learning. These operators are generated as standard form morphological operators that may be simplified (i.e., transformed into equivalent morphological operators that use fewer vocabulary words) by syntactical transformations.