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 properties such as one-dimensionality of digital curves and (n-1)-dimensionality of digital hypersurfaces that makes them discrete analogs of corresponding notions in continuous topology. The presented approach is fully based on the concept of adjacency relation and complements the concept of dimension as common 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 digital curve is a one-dimensional object and a digital hypersurface is an (n-1)-dimensional object, as it is in the case of curves and hypersurfaces in the Euclidean space. Relying on the obtained properties of digital hypersurfaces, we propose a uniform approach for studying good pairs defined by separations and obtain a classification of good pairs in arbitrary dimension. We also discuss possible applications of the presented definitions and results.

In this paper we investigate the advantages of using hexagonal grids in raster and volume graphics. In 2D, we present a hexagonal graphical model based on a hexagonal grid. In 3D, we introduce two honeycomb graphical models in which the voxels are hexagonal prisms, and we show that these are the only possible models under certain reasonable conditions. In the framework of the proposed models we design the 2D and 3D analytical honeycomb geometry of linear objects as well as of circles and spheres. We demonstrate certain advantages of the honeycomb models and address algorithmic and complexity issues.

In classical combinatorial string matching repetitions and other regularities play a central role. Besides their theoretical importance, repetitions in strings have been found relevant to coding and automata theory, formal languages, data compression, and molecular biology. An important motivation for developing a 2D pattern matching theory is seen in its relation with pattern recognition, image processing, computer vision and multimedia. Repetitions in 2D arrays have been defined and classified recently.5 In this paper we present an optimally fast CRCW-PRAM algorithm for testing whether a given n × n array contains repetitions of certain type. The algorithm takes optimal O( log log n) time with [Formula: see text] processors.