Issue No. 04 - April (2008 vol. 30)
In this paper we define and study digital manifolds of arbitrary dimension, and provide (in particular)a general theoretical basis for curve or surface tracing in picture analysis. The studies involve propertiessuch as one-dimensionality of digital curves and (n-1)-dimensionality of digital hypersurfaces thatmakes them discrete analogs of corresponding notions in continuous topology. The presented approachis fully based on the concept of adjacency relation and complements the concept of dimension ascommon in combinatorial topology. This work appears to be the first one on digital manifolds based ona graph-theoretical definition of dimension. In particular, in the n-dimensional digital space, a digitalcurve is a one-dimensional object and a digital hypersurface is an (n-1)-dimensional object, as it isin the case of curves and hypersurfaces in the Euclidean space. Relying on the obtained properties ofdigital hypersurfaces, we propose a uniform approach for studying good pairs defined by separationsand obtain a classification of good pairs in arbitrary dimension. We also discuss possible applicationsof the presented definitions and results.
digital geometry, digital topology, discrete dimension, digital manifold, digital curve, digital hypersurface, good pair
Valentin Brimkov, Reinhard Klette, "Border and SurfaceTracing - Theoretical Foundations", IEEE Transactions on Pattern Analysis & Machine Intelligence, vol. 30, no. , pp. 577-590, April 2008, doi:10.1109/TPAMI.2007.70725