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A Tutorial on Well-Composedness

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Abstract

Due to digitization, usual discrete signals generally present topological paradoxes, such as the connectivity paradoxes of Rosenfeld. To get rid of those paradoxes, and to restore some topological properties to the objects contained in the image, like manifoldness, Latecki proposed a new class of images, called well-composed images, with no topological issues. Furthermore, well-composed images have some other interesting properties: for example, the Euler number is locally computable, boundaries of objects separate background from foreground, the tree of shapes is well defined. Last, but not the least, some recent works in mathematical morphology have shown that very nice practical results can be obtained thanks to well-composed images. Believing in its prime importance in digital topology, we then propose this state of the art of well-composedness, summarizing its different flavors, the different methods existing to produce well-composed signals, and the various topics that are related to well-composedness.

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Notes

  1. A topological space is said to be unicoherent iff it is connected and for any two closed connected sets such that their union equals the whole space, their intersection is also connected.

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  1. EPITA Research and Development Laboratory (LRDE), 14-16 rue Voltaire, 94276, Le Kremlin-Bicêtre, France

    Nicolas Boutry & Thierry Géraud

  2. Université Paris-Est, LIGM, Équipe A3SI, ESIEE, Marne-la-Vallée, France

    Laurent Najman

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  1. Nicolas Boutry
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Boutry, N., Géraud, T. & Najman, L. A Tutorial on Well-Composedness. J Math Imaging Vis 60, 443–478 (2018). https://doi.org/10.1007/s10851-017-0769-6

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