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Inf-structuring Functions: A Unifying Theory of Connections and Connected Operators

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Abstract

During the last decade, several theories have been proposed in order to extend the notion of set connections in mathematical morphology. These new theories were obtained by generalizing the definition to wider spaces (namely complete lattices) and/or by relaxing some hypothesis. Nevertheless, the links among those different theories are not always well understood, and this work aims at defining a unifying theoretical framework. The adopted approach relies on the notion of inf-structuring function which is simply a mapping that associates a set of sub-elements to each element of the space. The developed theory focuses on the properties of the decompositions given by an inf-structuring function rather than in trying to characterize the properties of the set of connected elements as a whole. We establish several sets of inf-structuring function properties that enable to recover the existing notions of connections, hyperconnections, and attribute space connections. Moreover, we also study the case of grey-scale connected operators that are obtained by stacking set connected operators and we show that they can be obtained using specific inf-structuring functions. This work allows us to better understand the existing theories, it facilitates the reuse of existing results among the different theories and it gives a better view on the unexplored areas of the connection theories.

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Notes

  1. There is a missing condition in Proposition 3 of [49] (Proposition 4 of [47]) as the property P3 is indeed only equivalent to P2 and P1 if the lattice is co-prime. We thank Pr. Ch. Ronse for pointing out to us this mistake.

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Acknowledgments

I warmly thank the coordinating editor, Pr. Ch. Ronse, whose meticulous work has greatly contributed to the correctness of this article.

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Appendix: Max-sup Coherence and Chain-sup Completness

Appendix: Max-sup Coherence and Chain-sup Completness

The following example shows a max-sup coherent set that is not chain-sup complete. We consider the complete lattice of functions from \([0,1]\) to \([0,1]\) with the classical pointwise partial order, infimum, and supremum. The set \(E\) is composed of two subsets : an infinite chain of flat functions that is not chain sup-complete and an infinite anti-chain of functions whose supremum is equal to the supremum of the family of flat functions: \(E = \left\{ g_k\right\} _{k\in [0,1[} \cup \left\{ f_k\right\} _{k\in [0,1[}\) with \(\left\{ g_k\right\} _{k\in [0,1[}\) the chain of flat functions defined by \(\forall z\in [0,1],\ g_k(z) = k \) and \(\left\{ f_k\right\} _{k\in [0,1[}\) the anti-chain of functions defined by: \(\forall z\in [0,1],\ f_k(z)\) equals \(1\) if \(k=z\) and \(k\) otherwise. Note that we have: \(\max \left\{ g_k\right\} _{k\in [0,1[}=\emptyset \), \(\max \left\{ f_k\right\} _{k\in [0,1[}=\left\{ f_k\right\} _{k\in [0,1[}\), and \(\max (E)=\left\{ f_k\right\} _{k\in [0,1[}\).

\(E\) is not chain-sup-complete as the family \(\left\{ g_k\right\} _{k\in [0,1[}\) forms a chain but \(\bigvee \left\{ g_k\right\} _{k\in [0,1[} = g_1\) the constant function of value 1 does not belong to \(E\). In fact, we can easily see that all the chains that are not sup-complete in \(E\) contains \(\left\{ g_k\right\} _{k\in ]l,1[}\) with some \(l\in [0,1[\) and so we can restrict our study of the max-sup coherence to the critical intervals \([g_k,g_1]\) of \(E\) with \(k\in ]0,1[\), i.e., \(E\cap {{\mathrm{\uparrow }}}(g_k)\cap {{\mathrm{\downarrow }}}(g_1)\). We have \(\max (E\cap {{\mathrm{\uparrow }}}(g_k)\cap {{\mathrm{\downarrow }}}(g_1))=\left\{ f_{\ell }\right\} _{{\ell }\in [k,1[}\) which is not empty and we have \(g_1=\bigvee \max (E\cap {{\mathrm{\uparrow }}}(g_k)\cap {{\mathrm{\downarrow }}}(g_1))\). Therefore, \(E\) is max-sup coherent and max-sup coherence is not equivalent to chain-sup completness.

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Perret, B. Inf-structuring Functions: A Unifying Theory of Connections and Connected Operators. J Math Imaging Vis 51, 171–194 (2015). https://doi.org/10.1007/s10851-014-0515-2

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