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Continuous Maps on Digital Simple Closed Curves

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Continuous Maps on Digital Simple Closed Curves Applied Mathematics, 2010, 1, 377-386 doi:10.4236/am.2010.15050 Published Online November 2010 (http://www.SciRP.org/journal/am) Continuous Maps on Digital Simple Closed Curves Laurence Boxer1,2 1Department of Computer and Information Sciences, Niagara University, New York, USA 2Department of Computer Science and Engineering, State University of New York at Buffalo, Buffalo, USA E-mail: [email protected] Received August 2, 2010; revised September 14, 2010; accepted September 16, 2010 Abstract We give digital analogues of classical theorems of topology for continuous functions defined on spheres, for digital simple closed curves. In particular, we show the following: 1) A digital simple closed curve S of more than 4 points is not contractible, i.e., its identity map is not nullhomotopic in S ; 2) Let X and Y be digital simple closed curves, each symmetric with respect to the origin, such that | Y |> 5 (where | Y | is the number of points in Y ). Let f : X Y be a digitally continuous antipodal map. Then f is not nullho- motopic in Y ; 3) Let S be a digital simple closed curve that is symmetric with respect to the origin. Let f : S Z be a digitally continuous map. Then there is a pair of antipodes {x,x} S such that | f (x) f (x) | 1. Keywords: Digital Image, Digital Topology, Homotopy, Antipodal Point 1. Introduction Euclidean space; from this viewpoint, it is of interest to study conditions under which this digitization process A digital image is a set X of lattice points that model a preserves topological (or other geometric) properties” “continuous object” Y , where Y is a subset of a Eucli- [3]. dean space. Digital topology is concerned with deve- In this spirit, we obtain in this paper several properties loping a mathematical theory of such discrete objects so of continuous maps on digital simple closed curves, that, as much as possible, digital images have topological inspired by analogs for Euclidean simple closed curves. properties that mirror those of the Euclidean objects they In particular, we show that digital simple closed curves model; however, in digital topology we view a digital of more than 4 points are not contractible, and we obtain image as a graph, rather than, e.g., a metric space, as the several results for continuous maps and antipodal points latter would, for a finite digital image, result in a discrete on digital simple closed curves. topological space. Therefore, the reader is reminded that in digital topology, our “nearness” notion is the graphical 2. Preliminaries notion of adjacency, rather than a neighborhood system as in classical topology; usually, we use one of the na- Let Z be the set of integers. Then Z d is the set of tural cl -adjacencies (see Section 2). Early papers in the lattice points in d -dimensional Euclidean space. Let field, e.g., [1-8], noted that this notion of nearness allows X Z d and let be some adjacency relation for the us to express notions borrowed from classical topology, members of X . Then the pair (X ,) is said to be a e.g., connectedness, continuous function, homotopy, and (binary) digital image. A variety of adjacency relations fundamental group, such that these often mirror their are used in the study of digital images. Well known analogs with respect to Euclidean objects modeled by adjacencies include the following. respective digital images. Applications of digital topo- For a positive integer l with 1 ld and two distinct d logy have been found shape description and in image points ppppqqqq=(12 , , ,dd ), =( 12 , , , ) Z , p processing operations such as thinning and skeletoni- and q are cl -adjacent [10] if zation [9]. • there are at most l indices i such that A. Rosenfeld wrote the following: “The discrete grid | pi qi |= 1 , and (of pixels or voxels) used in digital topology can be • for all other indices j such that | p j q j | 1 , regarded as a ‘digitization’ of (two or three-dimensional) p j =q j . See Figure 1. Copyright © 2010 SciRes. AM 378 L. BOXER morphism). By a digital -path from x to y in a digital image X , we mean a (2,) -continuous function f :[0,m]Z X such that f (0) = x and f (m) = y . We say m is the length of this path. A simple closed -curve of m 4 points (for some adjacencies, the minimal value of m may be greater than 4; see below) in a digital image X is a sequence {(0),(1),,ff 2 Figure 1. In Z , each of the points p1 = (1,0) and fm(1)} of images of the -path fm:[0, 1]Z p2 = (0,1) is both c1 -adjacent and c2 -adjacent to X such that f (i) and f ( j) are -adjacent if m1 p0 = (0,0) , since both p1 and p2 differ from p0 by 1 in and only if j = (i 1) mod m . If S = {xi }i=0 where exactly one coordinate and coincide in the other coordinate; xi = f (i) for all i [0,m 1]Z , we say the points of S p3 =(-1,1) is c2 -adjacent, but not c1 -adjacent, to p0 , are circularly ordered. Digital simple closed curves are often examples of since p0 and p3 differ by 1 in both coordinates. digital images X Z d for which it is desirable to consider Z d \ X as a digital image with some adja- The notation cl is sometimes also understood as the d cency (not necessarily the same adjacency as used by number of points q Z that are cl -adjacent to a d X ). For example, by analogy with Euclidean topology, given point p Z . Thus, in Z we have c1 = 2 ; in 2 2 3 it is desirable that a digital simple closed curve X Z Z we have c = 4 and c = 8 ; in Z we have 1 2 satisfy the “Jordan curve property” of separating Z 2 c = 6, c = 18, and c = 26 . 1 2 3 into two connected components (one “inside” and the More general adjacency relations are studied in [11]. d other “outside” X ). An example that fails to satisfy this Let be an adjacency relation defined on Z . A - property [10] if we allow | X |= 4 is given by (X ,c ) , neighbor of a lattice point p is -adjacent to p . A 1 d where digital image X Z is -connected [11] if and only if for every pair of different points x, y X , there is a X = {(0,0),(1,0),(1,1),(0,1)} set {x , x ,, x } of points of a digital image X such 2 0 1 r is circularly ordered, and (Z \ X ,c ) is c -connected. that x = x , y = x and x and x are -neighbors 2 2 0 r i i1 However, this anomaly is essentially due to the “small- where i {0,1,,r 1}. ness” of X as a simple closed curve; it is known Let a,b Z with a < b . A digital interval [7] is a [1,15,16] that for m = 8 and m = 4 , if X Z 2 is a set of the form 1 2 digital simple closed ci -curve such that | X | mi , then 2 [a,b]Z = {z Z|a z b}. Z \ X has exactly 2 c3i -connected components, d d i {1,2}. Thus, it is customary to require that a digital Let X Z 0 and Y Z 1 be digital images with simple closed curve X Z 2 satisfy | X | 8 when c - -adjacency and -adjacency respectively. A 1 0 1 adjacency is used; | X | 4 when c -adjacency is used. function f : X Y is said to be ( , ) -continuous 2 0 1 Let X Z d0 and Y Z d1 be digital images with [2,8], if for every -connected subset U of X , 0 -adjacency and -adjacency respectively. Two f (U ) is a -connected subset of Y . We say that 0 1 1 ( , ) -continuous functions f , g : X Y are said to such a function is digitally continuous. 0 1 d d be digitally ( , ) -homotopic in Y [8] if there is a Proposition 2.1 [2,8] Let X Z 0 and Y Z 1 be 0 1 positive integer m and a function H : X [0,m] Y digital images with -adjacency and -adjacency Z 0 1 such that respectively. Then the function f : X Y is ( , ) - 0 1 • for all x X , H (x,0) = f (x) and H (x,m) = g(x) ; continuous if and only if for every -adjacent points 0 • for all x X , the induced function H :[0,m ] {x , x } of X , either f (x ) = f (x ) or f (x ) and x Z 0 1 0 1 0 Y defined by f (x1 ) are 1 -adjacent in Y . This characterization of continuity is what is called an H x (t) = H (x,t) for all t [0,m]Z , immersion, gradually varied operator, or gradually is (2,1 ) -continuous; and varied mapping in [12,13]. • for all t [0,m] , the induced function H : X Y Given digital images (X , ) , i {0,1} , suppose Z t i i defined by there is a ( 0 ,1 ) -continuous bijection f : X 0 X 1 1 H t (x) = H (x,t) for all x X , such that f : X 1 X 0 is (1 , 0 ) -continuous. We say X 0 and X 1 are ( 0 ,1 ) -isomorphic [14] (this is ( 0 ,1 ) -continuous. was called ( 0 ,1 ) -homeomorphic in [7]) and f is a We say that the function H is a digital ( 0 ,1 ) - ( 0 ,1 ) -isomorphism (respectively, a ( 0 ,1 )-homeo- homotopy between f and g .
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