How to Make n-D Plain Maps Alexandrov-Well-Composed in a Self-dual Way

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Abstract

In 2013, Najman and Géraud proved that by working on a well-composed discrete representation of a gray-level image, we can compute what is called its tree of shapes, a hierarchical representation of the shapes in this image. This way, we can proceed to morphological filtering and to image segmentation. However, the authors did not provide such a representation for the non-cubical case. We propose in this paper a way to compute a well-composed representation of any gray-level image defined on a discrete surface, which is a more general framework than the usual cubical grid. Furthermore, the proposed representation is self-dual in the sense that it treats bright and dark components in the image the same way. This paper can be seen as an extension to gray-level images of the works of Daragon et al. on discrete surfaces.

Documents

Bibtex (lrde.bib)

@Article{	  boutry.19.jmiv,
  author	= {Nicolas Boutry and Thierry G\'eraud and Laurent Najman},
  title		= {How to Make {$n$-D} Plain Maps {A}lexandrov-Well-Composed
		  in a Self-dual Way},
  journal	= {Journal of Mathematical Imaging and Vision},
  volume	= {61},
  number	= {6},
  pages		= {849--873},
  year		= {2019},
  abstract	= {In 2013, Najman and G\'eraud proved that by working on a
		  well-composed discrete representation of a gray-level
		  image, we can compute what is called its tree of shapes, a
		  hierarchical representation of the shapes in this image.
		  This way, we can proceed to morphological filtering and to
		  image segmentation. However, the authors did not provide
		  such a representation for the non-cubical case. We propose
		  in this paper a way to compute a well-composed
		  representation of any gray-level image defined on a
		  discrete surface, which is a more general framework than
		  the usual cubical grid. Furthermore, the proposed
		  representation is self-dual in the sense that it treats
		  bright and dark components in the image the same way. This
		  paper can be seen as an extension to gray-level images of
		  the works of Daragon et al. on discrete surfaces.}
}