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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. Furthermorewell-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, and so on. 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 flavours, the different methods existing to produce well-composed signals, and the various topics that are related to well-composedness.

Documents

Bibtex (lrde.bib)

@Article{	  boutry.17.jmiv,
  author	= {Nicolas Boutry and Thierry G\'eraud and Laurent Najman},
  title		= {A Tutorial on Well-Composedness},
  journal	= {Journal of Mathematical Imaging and Vision},
  year		= {2017},
  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, and so on. 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 flavours, the
		  different methods existing to produce well-composed
		  signals, and the various topics that are related to
		  well-composedness.},
  note		= {To appear}
}