Map Preserving Local Properties of a Digital Image

Abstract

In order to study digital topological properties of a k-surface in Z n , we generalize the topological number in Bertrand (Pattern Recogn. Lett. 15:1003-1011, 1994. Furthermore, we show that a local (k 0 , k 1 )-isomorphism preserves some digital-topological properties, such as a generalized topological number and a simple k 0 -point, and prove that a local (k 0 , k 1 )-isomorphism takes a simple k 0 -surface in Z n 0 into a simple k 1 -surface in Z n 1 .Keywords Digital k-surface · Digital k-fundamental group · Simple k-curve point · Generalized topological number · Generalized geodesic neighborhood · Local (k 0 , k 1 )-isomorphism · Discrete topology Mathematics Subject Classification (2000) 52xx · 55Q70 · 52Cxx · 55P10 · 55P15 · 68U05

Ultra regular covering space and its automorphism group

“…This local (k 0 , k 1 )-isomorphism has often been used in studying the preservation of local k 0 -properties of a digital space (X, k 0 ) into its corresponding k 1 -ones in digital geometry (Han, 2008c).…”

“…Useful tools from algebraic topology and geometric topology for studying digital topological properties of a (binary) digital space include a digital covering space, a (digital) k-fundamental group, a digital k-surface and so forth. These have been studied in numerous papers (Boxer, 1999;Boxer and Karaca, 2008;Han, 2005b;2005c;2005d;2006a;2006b;2006c;2006d;2007a;2007b;2008a;2008b;2008c;2008d;2009a;2009b;2009c;2010a;2010b;2010c;Malgouyres and Lenoir, 2000;Khalimsky, 1987;Rosenfeld and Klette, 2003).…”

Regular Covering Space in Digital Covering Theory and Its Applications

“…This local (k 0 , k 1 )-isomorphism can be strongly used in studying the investigation of the preservation of local k 0 -properties of a digital space (X, k 0 ) into its corresponding k 1 -ones in digital geometry [17]. Since a (k 0 , k 1 )-isomorphism is equivalent to locally (k 0 , k 1 )-isomorphic bijection and a restriction map of a (k 0 , k 1 )-isomorphism is also a (k 0 , k 1 )-isomorphism [15], we obtain the following property.…”

“…Useful tools from algebraic topology and geometric topology for studying digital topological properties of a (binary) digital space include a digital covering space, the (digital) k-fundamental group, a digital k-surface, and so forth. These have been studied in papers including [1,2,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,25,26,31]. Recently, motivated by both a regular covering space and a Deck's transformation group in algebraic topology [32], their digital versions in digital covering theory have been developed [10,12,14,16], which play important roles in classifying digital covering spaces.…”

Properties of a Generalized Universal Covering Space Over a Digital Wedge

“…This has been studied in many papers including [2,4,5,7,8,9,10,11,12,13,14,15,16,17,18]. Motivated by the study of a covering space over a figure eight in algebraic topology [28], the recent papers [6] (see also [3,11,16,17,18,19]) studied its digital version, which plays an important role in classifying digital spaces.…”