We show how algebraic methods can be used to provide a mathematical framework suitable for the definition of multidimensional hypersurfaces in digital space, and for proofs of separation theorems. Our work is motivated by the need for a mathematical basis to provide a strong foundation for the creation of image processing algorithms in multidimensions; multidimensional images have been shown to arise naturally in areas as diverse as medical diagnosis and agricultural imaging. Whereas previous work in the area has been either combinatorial or has used the tools of point-set topology, we show how homology and cohomology groups can be defined in digital space. Our definitions are of a broad nature encompassing many of the standard adjacencies used to define digital objects. Given that in Euclidean space these groups satisfy conditions which provide for very neat proofs of separation theorems, we conjecture that an analogous theorem is true in digital space. We further show that the concept of orientability can be given a meaning in digital space more closely analogous to its classical meaning than definitions given previously in the image processing literature.
Charles F. Osborne,
"Algebraic methods for multidimensional digital topology", Proc. SPIE 2060, Vision Geometry II, (1 December 1993); doi: 10.1117/12.164997; https://doi.org/10.1117/12.164997