Minimal representations for translation-invariant set mappings by mathematical morphology
SIAM Journal on Applied Mathematics
Minimal representation of t -openings via pattern bases
Pattern Recognition Letters
Automatic programming of morphological machines by PAC learning
Fundamenta Informaticae - Special issue on mathematical morphology
A Combinatorial Optimization Technique for the Sequential Decomposition of Erosions and Dilations
Journal of Mathematical Imaging and Vision
From the Sup-Decomposition to Sequential Decompositions
Journal of Mathematical Imaging and Vision
Image Analysis and Mathematical Morphology
Image Analysis and Mathematical Morphology
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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-. 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).