From the Sup-Decomposition to Sequential Decompositions

W-operators are discrete set operators that are both translation invariant and locally defined within a finite window W. A particularly interesting property of W-operators is that they have a sup-decomposition in terms of a family sup-generating operators, that are parameterized by the operator basis. The sup-decomposition has a parallel structure that usually is not efficient for computation in conventional sequential machines. In this paper, we formalize the problem of transforming sup-decompositions into purely sequential decompositions (when they exist). The techniques were developed for general W-operators, specialized for increasing W-operators and applied on operators built by compositions of dilations and erosions.