Simple pairs of points in digital spaces. Topology-preserving transformations of digital spaces by contracting simple pairs of points

Methods of digital topology are widely used in various image processing operations including topology-preserving thinning, skeletonization, simplification, border and surface tracing and region filling and growing. Usually, transformations of digital objects preserve topological properties. One of the ways to do this is to use simple points, edges and cliques: loosely speaking, a point or an edge of a digital object is called simple if it can be deleted from this object without altering topology. The detection of simple points, edges and cliques is extremely important in image thinning, where a digital image of an object gets reduced to its skeleton with the same topological features. The notion of a simple point was introduced by Rosenfeld [14]. Since then due to its importance, characterizations of simple points in two, three, and four dimensions and algorithms for their detection have been studied in the framework of digital topology by many researchers (see, e.g., [1, 5, 12-13]). Local characterizations of simple points in three dimensions and efficient detection algorithms are particularly essential in such areas as medical image processing [2, 7-8, 15], where the shape correctness is required on the one hand and the image acquisition process is sensitive to the errors produced by the image noise, geometric distortions in the images, subject motion, etc, on the other hand. It has to be noticed that in this paper we use an approach that was developed in [3]. Digital spaces and manifolds are defined axiomatically as specialization graphs with the structure defined by the construction of the nearest neighborhood of every point of the graph. In this approach, the notions of a digital space, a simple point or a simple set are different from those usually to be found in papers on digital topology including papers mentioned above. This paper presents the notion of a simple pair of points based on digital contractible spaces and contractible transformations of digital spaces. Some new properties of digital n-manifolds which are digital models of continuous n-dimensional manifolds are investigated in section 3. In particular, it is shown that M is a digital n-sphere if for any contractible subspace G, the subspace M-G is contractible. Section 4 introduces the notions of the simple splitting of a point and the contraction simple pair of points. It is shown that these transformations convert a given digital space to a homotopy equivalent digital space. Based on the simple contraction, we prove …