Introducing PC n-Manifolds and P-well-composedness in Partially Ordered Sets

Authors

Nicolas Boutry

Journal

Journal of Mathematical Imaging and Vision

Type

article

Date

2023-01-01

Abstract

In discrete topology, discrete surfaces are well-known for their strong topological and regularity properties. Their definition is recursive, and checking if a poset is a discrete surface is tractable. Their applications are numerous: when domain unicoherence is ensured, they lead access to the tree of shapes, and then to filtering in the shape space (shapings); they also lead to Laplacian zero-crossing extraction, to brain tumor segmentation, and many other applications related to mathematical morphology. They have many advantages in digital geometry and digital topology since discrete surfaces do not have any pinches (and then the underlying polyhedron of their geometric realization can be parameterized). However, contrary to topological manifolds known in continuous topology, discrete surfaces do not have any boundary, which is not always realizable in practice (finite hyper-rectangles cannot be discrete surfaces due to their non-empty boundary). For this reasonwe propose the three following contributions: (1) we introduce a new definition of boundary, called border, based on the definition of discrete surfaces, and which allows us to delimit any partially ordered set whenever it is not embedded in a greater ambient space, (2) we introduce $P$-well-com­po­sed­ness similar to well-com­po­sed­ness in the sense of Alexandrov but based on borders, (3) we propose new (possibly geometrical) structures called (smooth) $n$-PCM's which represent almost the same regularity as discrete surfaces and that are tractable thanks to their recursive definition, and (4) we prove several fundamental theorems relative to PCM's and their relations with discrete surfaces. We deeply believe that these new $n$-dimensional structures are promising for the discrete topology and digital geometry fields.

Bibtex (lrde.bib)

@Article{	  boutry.23.jmiv.2,
  author	= {Nicolas Boutry},
  title		= {Introducing PC $n$-Manifolds and $P$-well-composedness in
		  Partially Ordered Sets},
  year		= {2023},
  journal	= {Journal of Mathematical Imaging and Vision},
  abstract	= {In discrete topology, discrete surfaces are well-known for
		  their strong topological and regularity properties. Their
		  definition is recursive, and checking if a poset is a
		  discrete surface is tractable. Their applications are
		  numerous: when domain unicoherence is ensured, they lead
		  access to the tree of shapes, and then to filtering in the
		  shape space (shapings); they also lead to Laplacian
		  zero-crossing extraction, to brain tumor segmentation, and
		  many other applications related to mathematical morphology.
		  They have many advantages in digital geometry and digital
		  topology since discrete surfaces do not have any pinches
		  (and then the underlying polyhedron of their geometric
		  realization can be parameterized). However, contrary to
		  topological manifolds known in continuous topology,
		  discrete surfaces do not have any boundary, which is not
		  always realizable in practice (finite hyper-rectangles
		  cannot be discrete surfaces due to their non-empty
		  boundary). For this reason, we propose the three following
		  contributions: (1) we introduce a new definition of
		  boundary, called border, based on the definition of
		  discrete surfaces, and which allows us to delimit any
		  partially ordered set whenever it is not embedded in a
		  greater ambient space, (2) we introduce
		  $P$-well-com\-po\-sed\-ness similar to
		  well-com\-po\-sed\-ness in the sense of Alexandrov but
		  based on borders, (3) we propose new (possibly geometrical)
		  structures called (smooth) $n$-PCM's which represent almost
		  the same regularity as discrete surfaces and that are
		  tractable thanks to their recursive definition, and (4) we
		  prove several fundamental theorems relative to PCM's and
		  their relations with discrete surfaces. We deeply believe
		  that these new $n$-dimensional structures are promising for
		  the discrete topology and digital geometry fields. }
}