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-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.
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