On Making nD Images Well-Composed by a Self-Dual Local Interpolation

From LRDE

Abstract

Natural and synthetic discrete images are generally not well-composed, leading to many topological issues: connectivities in binary images are not equivalent, the Jordan Separation theorem is not true anymore, and so on. Conversely, making images well-composed solves those problems and then gives access to many powerful tools already known in mathematical morphology as the Tree of Shapes which is of our principal interest. In this paperwe present two main results: a characterization of 3D well-composed gray-valued images; and a counter-example showing that no local self-dual interpolation with a classical set of properties makes well-composed images with one subdivision in 3D, as soon as we choose the mean operator to interpolate in 1D. Then, we briefly discuss various constraints that could be interesting to change to make the problem solvable in nD.

Documents

Bibtex (lrde.bib)

@InProceedings{	  boutry.14.dgci,
  author	= {Nicolas Boutry and Thierry G\'eraud and Laurent Najman},
  title		= {On Making {$n$D} Images Well-Composed by a Self-Dual Local
		  Interpolation},
  booktitle	= {Proceedings of the 18th International Conference on
		  Discrete Geometry for Computer Imagery (DGCI)},
  year		= 2014,
  month		= sep,
  pages		= {320--331},
  address	= {Siena, Italy},
  series	= {Lecture Notes in Computer Science},
  volume	= {8668},
  publisher	= {Springer},
  editor	= {E. Barcucci and A. Frosini and S. Rinaldi},
  abstract	= {Natural and synthetic discrete images are generally not
		  well-composed, leading to many topological issues:
		  connectivities in binary images are not equivalent, the
		  Jordan Separation theorem is not true anymore, and so on.
		  Conversely, making images well-composed solves those
		  problems and then gives access to many powerful tools
		  already known in mathematical morphology as the Tree of
		  Shapes which is of our principal interest. In this paper,
		  we present two main results: a characterization of 3D
		  well-composed gray-valued images; and a counter-example
		  showing that no local self-dual interpolation with a
		  classical set of properties makes well-composed images with
		  one subdivision in 3D, as soon as we choose the mean
		  operator to interpolate in 1D. Then, we briefly discuss
		  various constraints that could be interesting to change to
		  make the problem solvable in nD.}
}