The paper [5] defines a notion of digital deformation and claims to prove that if (X, p) is k-deformable into (A, p), then these two pointed images have isomorphic fundamental groups. We present a simple counterexample to this claim

Boxer, Laurence

English

Università del Salento: ESE - Salento University Publishing

Note di Matematica 28, n. 1, 2008, 69–76.Remarks on digital deformationLaurence BoxeriDepartment of Computer and Information Sciences, Niagara University,Niagara University, NY 14109, USAboxer@niagara.eduReceived: 24/10/2006; accepted: 09/01/2007.Abstract. The paper [5] defines a notion of digital deformation and claims to prove that if(X, p) is k-deformable into (A, p), then these two pointed images have isomorphic fundamentalgroups. We present a simple counterexample to this claim.Keywords: digital image, digitally continuous, deformation, homotopy, fundamental group,digital topologyMSC 2000 classification: primary 55Q99, 62HR35, 68U101 IntroductionIn digital topology, we examine geometric properties of digital images viatools adapted from Euclidean topology. These tools include digital versions ofcontinuous functions, homotopy (continuous deformation), homotopy type, andthe fundamental group. A theme of several recent papers [3, 4, 5] is the rela-tionship between the fundamental groups Πk01 (X,x) and Πk11 (f(X), f(x)), wheref : (X,x) → (f(X), f(x)) is a (k0, k1)-continuous function.Of interest in this paper is the case of f being a pointed deformation. It isclaimed in [5] that for pointed deformations f : (X, p) → (A, p), Πk01 (X, p) andΠk11 (A, p) are isomorphic. In this paper, we present a simple counterexample tothis claim.2 Preliminaries2.1 General propertiesLet N be the set of natural numbers and let Z denote the set of integers.Then Zn is the set of lattice points in Euclidean n-dimensional space.Adjacency relations frequently used for digital images include the follow-ing [8]. Two points p and q in Z2 are 8-adjacent if they are distinct and differiDepartment of Computer and Information Sciences, Niagara University, Niagara Univer-sity, NY 14109, USA; and Department of Computer Science and Engineering, State Universityof New York at Buffalo.Note di MatematicaNote Mat. 1(2008), 69-76ISSN 1123-2536, e-ISSN 1590-0932DOI 10.1285/i15900932v28n1p69http://siba2.unile.it/notemat______________________________________________________________________________________________70 L. Boxerby at most 1 in each coordinate; p and q in Z2 are 4-adjacent if they are 8-adjacent and differ in exactly one coordinate. Two points p and q in Z3 are26-adjacent if they are distinct and differ by at most 1 in each coordinate; theyare 18-adjacent if they are 26-adjacent and differ in at most two coordinates;they are 6-adjacent if they are 18-adjacent and differ in exactly one coordinate.For k ∈ {4, 8, 6, 18, 26}, a k − neighbor of a lattice point p is a point that isk-adjacent to p.Let κ be an adjacency relation defined on Zn. A digital image X ⊂ Zn isκ-connected [6] if and only if for every pair of points {x, y} ⊂ X, x = y, thereexists a set {x0, x1, . . . , xc} ⊂ X such that x = x0, xc = y, and xi and xi+1 areκ-neighbors, i ∈ {0, 1, . . . , c−1}. A κ-component of X is a maximal κ-connectedsubset of X.1 Definition ([2]). Let a, b ∈ Z, a < b. A digital interval is a set of theform[a, b]Z = {z ∈ Z | a ≤ z ≤ b}in which 2-adjacency is assumed.2 Definition ([3]; see also [11]). Let X ⊂ Zn0, Y ⊂ Zn1 . Let f : X → Ybe a function. Let κi be an adjacency relation defined on Zni , i ∈ {0, 1}. Wesay f is (κ0, κ1)-continuous if for every κ0-connected subset A of X, f(A) is aκ1-connected subset of Y .We say a function satisfying Definition 2 is digitally continuous. This defi-nition implies the following.3 Proposition ([3]; see also [11]). Let X and Y be digital images. Then thefunction f : X → Y is (κ0, κ1)-continuous if and only if for every {x0, x1} ⊂ Xsuch that x0 and x1 are κ0-adjacent, either f(x0) = f(x1) or f(x0) and f(x1)are κ1-adjacent.For example, if κ is an adjacency relation on a digital image Y , then f :[a, b]Z → Y is (2, κ)-continuous if and only if for every {c, c+1} ⊂ [a, b]Z, eitherf(c) = f(c+ 1) or f(c) and f(c+ 1) are κ-adjacent.2.2 Digital homotopyRoughly, a homotopy between continuous functions is a continuous defor-mation of one of the functions into the other over a time period.4 Definition ([3]; see also [7]). Let X and Y be digital images. Let f, g :X → Y be (κ, κ′)-continuous functions. Suppose there is a positive integer mand a function F : X × [0,m]Z → Y such that• for all x ∈ X, F (x, 0) = f(x) and F (x,m) = g(x);________________________________________________________________________________________________Remarks on digital deformation 71• for all x ∈ X, the induced function Fx : [0,m]Z → Y defined byFx(t) = F (x, t) for all t ∈ [0,m]Zis (2, κ′)-continuous.• for all t ∈ [0,m]Z, the induced function Ft : X → Y defined byFt(x) = F (x, t) for all x ∈ Xis (κ, κ′)-continuous.Then F is a digital (κ, κ′)-homotopy between f and g, and f and g are digitally(κ, κ′)-homotopic in Y . If for some x ∈ X we have F (x, t) = F (x, 0) for allt ∈ [0,m]Z, we say F holds x fixed.The notationf 	κ,κ′ gindicates that functions f and g are digitally (κ, κ′)-homotopic in Y .If (X,κ) is a digital image and x0 ∈ X, the triple (X,x0, κ) is a pointeddigital image.For p ∈ Y , we denote by p the constant function p : X → Y defined byp(x) = p for all x ∈ X.5 Definition. A digital image (X,κ) is κ-contractible [7, 2] if its identitymap is (κ, κ)-homotopic to a constant function p for some p ∈ X. If the homo-topy of the contraction holds p fixed, we say (X, p, κ) is pointed κ-contractible.6 Example ([2]). Every digital interval [0,m]Z is pointed contractible.2.3 Digital loops7 Definition (See [7]). A digital κ-path in a digital image X is a (2, κ)-continuous function f : [0,m]Z → X. If, further, f(0) = f(m), we call f adigital κ-loop, and the point f(0) is the basepoint of the loop f . If f is aconstant function, it is called a trivial loop.If f and g are digital κ-paths in X such that g starts where f ends, theproduct (see [7]) of f and g, written f · g, is, intuitively, the κ-path obtainedby following f by g. Formally, if f : [0,m1]Z → X, g : [0,m2]Z → X, andf(m1) = g(0), then (f · g) : [0,m1 +m2]Z → X is defined by(f · g)(t) ={f(t) if t ∈ [0,m1]Z;g(t−m1) if t ∈ [m1,m1 +m2]Z.Unlike its Euclidean model, a digital interval is a finite set, so were we torestrict homotopy classes of loops to loops defined on the same digital interval,________________________________________________________________________________________________72 L. Boxerwe would limit the class of a given loop undesirably. The following notion oftrivial extension permits a loop to “stretch” and remain in the same pointedhomotopy class. Intuitively, f ′ is a trivial extension of f if f ′ follows the samepath as f , but more slowly, with pauses for rest (subintervals of the domain onwhich f ′ is constant).8 Definition ([3]). Let f and f ′ be κ-paths in a digital image X. We sayf ′ is a trivial extension of f if there are sets of κ-paths {f1, f2, . . . , fk} and{F1, F2, . . . , Fp} in X such that(1) k ≤ p;(2) f = f1 · f2 · · · · · fk;(3) f ′ = F1 · F2 · · · · · Fp; and(4) there are indices 1 ≤ i1 < i2 < · · · < ik ≤ p such that• Fij = fj, 1 ≤ j ≤ k, and• i ∈ {i1, i2, . . . , ik} implies Fi is a trivial loop.This notion lets us compare the digital homotopy properties of loops evenif their domains have differing cardinality, since if m1 ≤ m2, we obtain a trivialextension of a loop f : [0,m1]Z → X to f ′ : [0,m2]Z → X viaf ′(t) ={f(t) if 0 ≤ t ≤ m1;f(m1) if m1 ≤ t ≤ m2.The following notions are useful for defining the class of a pointed loop.9 Definition. Let f, g : [0,m]Z → (X,x0) be digital loops with basepointx0. If H : [0,m]Z × [0,M ]Z → X is a digital homotopy between f and g suchthat for all t ∈ [0,M ]Z we haveH(0, t) = H(m, t),we say H is loop-preserving. If, further, for all t ∈ [0,M ]Z we haveH(0, t) = H(m, t) = x0,we say H holds the endpoints fixed.The notion of H holding the endpoints fixed was introduced in [4]. Theterm “loop-preserving” suggests that every (time) stage of the homotopy yieldsa digital loop.Digital κ-loops f and g in X with the same basepoint p belong to the sameκ-loop class in X if there are trivial extensions f ′ and g′ of f and g, respectively,________________________________________________________________________________________________Remarks on digital deformation 73with domains of the same cardinality, and a loop-preserving homotopy betweenf ′ and g′ that holds the endpoints fixed [3].Membership in the same loop class in (X,x0) is an equivalence relationamong digital κ-loops [3].Let [f ] be the loop class of a loop f in X. We have the following.10 Proposition ([3, 7]). Suppose f1, f2, g1, g2 are digital loops in a pointeddigital image (X,x0), with f2 ∈ [f1] and g2 ∈ [g1]. Then f2 · g2 ∈ [f1 · g1].2.4 Digital fundamental groupThe digital fundamental group is derived from the classical fundamentalgroup of algebraic topology (see [10]).Let (X, p, κ) be a pointed digital image. Consider the set Πκ1(X, p) of κ-loopclasses [f ] in X with basepoint p. By Proposition 10, the product operation[f ] ∗ [g] = [f · g]is well-defined on Πκ1(X, p).The operation ∗ is associative on Πκ1(X, p) [7].11 Lemma ([3]). Let (X, p) be a pointed digital image. Let p : [0,m]Z → Xbe the constant function p(t) = p. Then [p] is an identity element for Πκ1(X, p).12 Lemma ([3]). If f : [0,m]Z → X represents an element of Π1(X, p),then the function g : [0,m]Z → X defined byg(t) = f(m− t) for t ∈ [0,m]Zis an element of [f ]−1 in Πκ1(X, p).13 Theorem ([3]). Πκ1(X, p) is a group under the ∗ product operation, theκ-fundamental group of (X, p).We may interpret the following result to say that in a connected digital imageX, the digital fundamental group is independent of the choice of basepoint.14 Theorem ([3]). Let X be a digital image with adjacency relation κ. Ifp and q belong to the same κ-component of X, then Πκ1(X, p) and Πκ1(X, q) areisomorphic groups.15 Proposition ([4]). Let X be a pointed κ-contractible digital image andlet p ∈ X. Then Πκ1(X, p) is a trivial group.2.5 Deformation and deformation retractionWe have the following.________________________________________________________________________________________________74 L. Boxer16 Definition ([5]). Let (X,κ) be a digital image and let A be a nonemptysubset of X. Then X is κ-deformable into A if there is a κ-homotopy D :X × [0,m]Z → X such that D(x, 0) = x and D(x,m) ∈ A for all x ∈ X.D is called a κ-deformation. If for some x0 ∈ A we have D(x0, t) = x0 forall t ∈ [0,m]Z, we say X is pointed κ-deformable into A, and D is a pointedκ-deformation.Classical notions of topology [1] yielded the concepts of digital retractionand deformation retraction in [2]. Let (X,κ) be a digital image and let A be anonempty subset of X. A retraction of X onto A is a (κ, κ)-continuous functionr : X → A such that r(a) = a for all a ∈ A. A κ-deformation retraction of X toA is a κ-homotopy H : X × [0,m]Z → X such that the induced map H( , 0) isthe identity map 1X , and the induced map H( ,m) is a retraction of X onto A.3 Deformations, deformation retractions, and funda-mental groupsNotice that a deformation retraction is a pointed deformation. We have thefollowing.17 Theorem ([4]). Let A be a nonempty subset of a digital image X andlet H : X× [0,m]Z → X be a κ-deformation retraction of X onto A. For a ∈ A,the inclusion map i : (A, a) ↪→ (X, a) induces an isomorphism of Πκ1(A, a) andΠκ1(X, a).However, a pointed deformation from a digital image X into its nonemptysubset A need not yield an isomorphism of the fundamental groups of X and A,despite the claim of Han (presented as Theorem 3 of [5]) to the contrary. Indeed,Han’s claim is false even if the deformation is required to be onto A at the endof the homotopy. Consider the pair (X,A) defined as follows. X = ([0, 2]Z ×[0, 2]Z) ∪ {(j, 0)}7j=3. Let A ⊂ X be the set A = ([0, 2]Z × [0, 2]Z) \ {(1, 1)}, asimple closed 4-curve; hence Π41(A, a) is isomorphic to Z [3, 8, 9]. It is easilyseen that X is 4-contractible via the function h : X × [0, 9]Z → X defined byh(x, y, t) ={(x, max{0, y − t}) for 0 ≤ t ≤ 2;(max{0, x + 2− t}, 0) for 3 ≤ t ≤ 9.Hence X has a trivial 4-fundamental group.We show there is a 4-deformation of X onto A as follows. Consider the________________________________________________________________________________________________Remarks on digital deformation 75function H : X × [0, 8]Z → X given as follows.H(x, y, t) =⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩(max{0, x − t}, y) if t ∈ [0, 2]Z;(max{2, x − t}, 0) if (x, y) ∈ {(j, 0)}7j=5, 3 ≤ t ≤ 5;H(x, y, 2) if (x, y) ∈ {(j, 0)}7j=5, 3 ≤ t ≤ 5;(2, 1) if (x, y) ∈ {(j, 0)}7j=5, t = 6;H(x, y, 5) if (x, y) ∈ {(j, 0)}7j=5, t = 6;(2, 2) if (x, y) ∈ {(6, 0), (7, 0)}, t = 7;H(x, y, 6) if (x, y) ∈ {(6, 0), (7, 0)}, t = 7;(1, 2) if (x, y) = (7, 0), t = 8;H(x, y, 7) if (x, y) = (7, 0), t = 8.It is easily seen that this function is a 4-homotopy between 1X and the functionH8 : X → X, defined by H8(x, y) = H(x, y, 8), that is onto A.What is valid from Han’s paper is the following.18 Theorem. Let X be a digital image and let A be a non-empty subsetof X. Let D : X × [0,m]Z → X be a pointed κ-deformation of X into A, withD(p, t) = p for some p ∈ A and all t ∈ [0,m]Z . Let r : X → A be the mapdefined by r(x) = D(x,m) for all x ∈ X. Then the induced homomorphismr∗ : Πκ1(X, p) → Πκ1(A, p) is one to one.Proof. [5] Let i : A → X be the inclusion map. Then D is a homotopybetween 1X and i◦r. Therefore, 1Πκ1 (X,p) = i∗◦r∗. The assertion follows. QED4 SummaryWe have shown that the claim of [5], that a pointed digital deformationinduces an isomorphism between fundamental groups, is false.Acknowledgements. We gratefully acknowledge a suggestion of an anon-ymous referee.References[1] K. Borsuk: Theory of Retracts, Polish Scientific Publishers, Warsaw 1967.[2] L. Boxer: Digitally continuous functions, Pattern Recognition Letters, 15 (1994), 833–839.[3] L. Boxer: A classical construction for the digital fundamental group, Journal of Mathe-matical Imaging and Vision, 10 (1999), 51–62.[4] L. Boxer: Properties of digital homotopy, Journal of Mathematical Imaging and Vision,22 (2005), 19–26.________________________________________________________________________________________________76 L. Boxer[5] S.-E. Han: Generalized digital (k0, k1)-homeomorphism, Note di Matematica, 22 (2003),157–166.[6] G. T. Herman: Oriented surfaces in digital spaces, CVGIP: Graphical Models and ImageProcessing, 55 (1993), 381–396.[7] E. Khalimsky: Motion, deformation, and homotopy in finite spaces, in Proceedings IEEEIntl. Conf. on Systems, Man, and Cybernetics, (1987) 227–234.[8] T. Y. Kong: A digital fundamental group, Computers and Graphics, 13 (1989), 159–166.[9] T. Y. Kong, A. W. Roscoe, A. Rosenfeld: Concepts of digital topology, Topology andits Applications, 46 (1992), 219–262.[10] W. S. Massey: Algebraic Topology: An Introduction, Harcourt, Brace, and World, NewYork 1967.[11] A. Rosenfeld: ‘Continuous’ functions on digital pictures, Pattern Recognition Letters,4 (1986), 177–184.________________________________________________________________________________________________

Remarks on digital deformation

ESE - Salento University Publishing

Note di Matematica 28, n. 1, 2008, 69–76.Remarks on digital deformationLaurence BoxeriDepartment of Computer and Information Sciences, Niagara University,Niagara University, NY 14109, USAboxer@niagara.eduReceived: 24/10/2006; accepted: 09/01/2007.Abstract. The paper [5] deﬁnes a notion of digital deformation and claims to prove that if(X, p) is k-deformable into (A, p), then these two pointed images have isomorphic fundamentalgroups. We present a simple counterexample to this claim.Keywords: digital image, digitally continuous, deformation, homotopy, fundamental group,digital topologyMSC 2000 classiﬁcation: primary 55Q99, 62HR35, 68U101 IntroductionIn digital topology, we examine geometric properties of digital images viatools adapted from Euclidean topology. These tools include digital versions ofcontinuous functions, homotopy (continuous deformation), homotopy type, andthe fundamental group. A theme of several recent papers [3, 4, 5] is the rela-tionship between the fundamental groups Πk01 (X,x) and Πk11 (f(X), f(x)), wheref : (X,x) → (f(X), f(x)) is a (k0, k1)-continuous function.Of interest in this paper is the case of f being a pointed deformation. It isclaimed in [5] that for pointed deformations f : (X, p) → (A, p), Πk01 (X, p) andΠk11 (A, p) are isomorphic. In this paper, we present a simple counterexample tothis claim.2 Preliminaries2.1 General propertiesLet N be the set of natural numbers and let Z denote the set of integers.Then Zn is the set of lattice points in Euclidean n-dimensional space.Adjacency relations frequently used for digital images include the follow-ing [8]. Two points p and q in Z2 are 8-adjacent if they are distinct and diﬀeriDepartment of Computer and Information Sciences, Niagara University, Niagara Univer-sity, NY 14109, USA; and Department of Computer Science and Engineering, State Universityof New York at Buﬀalo.Note di MatematicaNote Mat. 1(2008), 69-76ISSN 1123-2536, e-ISSN 1590-0932DOI 10.1285/i15900932v28n1p69http://siba2.unile.it/notemat______________________________________________________________________________________________70 L. Boxerby at most 1 in each coordinate; p and q in Z2 are 4-adjacent if they are 8-adjacent and diﬀer in exactly one coordinate. Two points p and q in Z3 are26-adjacent if they are distinct and diﬀer by at most 1 in each coordinate; theyare 18-adjacent if they are 26-adjacent and diﬀer in at most two coordinates;they are 6-adjacent if they are 18-adjacent and diﬀer in exactly one coordinate.For k ∈ {4, 8, 6, 18, 26}, a k − neighbor of a lattice point p is a point that isk-adjacent to p.Let κ be an adjacency relation deﬁned on Zn. A digital image X ⊂ Zn isκ-connected [6] if and only if for every pair of points {x, y} ⊂ X, x = y, thereexists a set {x0, x1, . . . , xc} ⊂ X such that x = x0, xc = y, and xi and xi+1 areκ-neighbors, i ∈ {0, 1, . . . , c−1}. A κ-component of X is a maximal κ-connectedsubset of X.1 Deﬁnition ([2]). Let a, b ∈ Z, a < b. A digital interval is a set of theform[a, b]Z = {z ∈ Z | a ≤ z ≤ b}in which 2-adjacency is assumed.2 Deﬁnition ([3]; see also [11]). Let X ⊂ Zn0, Y ⊂ Zn1 . Let f : X → Ybe a function. Let κi be an adjacency relation deﬁned on Zni , i ∈ {0, 1}. Wesay f is (κ0, κ1)-continuous if for every κ0-connected subset A of X, f(A) is aκ1-connected subset of Y .We say a function satisfying Deﬁnition 2 is digitally continuous. This deﬁ-nition implies the following.3 Proposition ([3]; see also [11]). Let X and Y be digital images. Then thefunction f : X → Y is (κ0, κ1)-continuous if and only if for every {x0, x1} ⊂ Xsuch that x0 and x1 are κ0-adjacent, either f(x0) = f(x1) or f(x0) and f(x1)are κ1-adjacent.For example, if κ is an adjacency relation on a digital image Y , then f :[a, b]Z → Y is (2, κ)-continuous if and only if for every {c, c+1} ⊂ [a, b]Z, eitherf(c) = f(c+ 1) or f(c) and f(c+ 1) are κ-adjacent.2.2 Digital homotopyRoughly, a homotopy between continuous functions is a continuous defor-mation of one of the functions into the other over a time period.4 Deﬁnition ([3]; see also [7]). Let X and Y be digital images. Let f, g :X → Y be (κ, κ′)-continuous functions. Suppose there is a positive integer mand a function F : X × [0,m]Z → Y such that• for all x ∈ X, F (x, 0) = f(x) and F (x,m) = g(x);________________________________________________________________________________________________Remarks on digital deformation 71• for all x ∈ X, the induced function Fx : [0,m]Z → Y deﬁned byFx(t) = F (x, t) for all t ∈ [0,m]Zis (2, κ′)-continuous.• for all t ∈ [0,m]Z, the induced function Ft : X → Y deﬁned byFt(x) = F (x, t) for all x ∈ Xis (κ, κ′)-continuous.Then F is a digital (κ, κ′)-homotopy between f and g, and f and g are digitally(κ, κ′)-homotopic in Y . If for some x ∈ X we have F (x, t) = F (x, 0) for allt ∈ [0,m]Z, we say F holds x ﬁxed.The notationf 	κ,κ′ gindicates that functions f and g are digitally (κ, κ′)-homotopic in Y .If (X,κ) is a digital image and x0 ∈ X, the triple (X,x0, κ) is a pointeddigital image.For p ∈ Y , we denote by p the constant function p : X → Y deﬁned byp(x) = p for all x ∈ X.5 Deﬁnition. A digital image (X,κ) is κ-contractible [7, 2] if its identitymap is (κ, κ)-homotopic to a constant function p for some p ∈ X. If the homo-topy of the contraction holds p ﬁxed, we say (X, p, κ) is pointed κ-contractible.6 Example ([2]). Every digital interval [0,m]Z is pointed contractible.2.3 Digital loops7 Deﬁnition (See [7]). A digital κ-path in a digital image X is a (2, κ)-continuous function f : [0,m]Z → X. If, further, f(0) = f(m), we call f adigital κ-loop, and the point f(0) is the basepoint of the loop f . If f is aconstant function, it is called a trivial loop.If f and g are digital κ-paths in X such that g starts where f ends, theproduct (see [7]) of f and g, written f · g, is, intuitively, the κ-path obtainedby following f by g. Formally, if f : [0,m1]Z → X, g : [0,m2]Z → X, andf(m1) = g(0), then (f · g) : [0,m1 +m2]Z → X is deﬁned by(f · g)(t) ={f(t) if t ∈ [0,m1]Z;g(t−m1) if t ∈ [m1,m1 +m2]Z.Unlike its Euclidean model, a digital interval is a ﬁnite set, so were we torestrict homotopy classes of loops to loops deﬁned on the same digital interval,________________________________________________________________________________________________72 L. Boxerwe would limit the class of a given loop undesirably. The following notion oftrivial extension permits a loop to “stretch” and remain in the same pointedhomotopy class. Intuitively, f ′ is a trivial extension of f if f ′ follows the samepath as f , but more slowly, with pauses for rest (subintervals of the domain onwhich f ′ is constant).8 Deﬁnition ([3]). Let f and f ′ be κ-paths in a digital image X. We sayf ′ is a trivial extension of f if there are sets of κ-paths {f1, f2, . . . , fk} and{F1, F2, . . . , Fp} in X such that(1) k ≤ p;(2) f = f1 · f2 · · · · · fk;(3) f ′ = F1 · F2 · · · · · Fp; and(4) there are indices 1 ≤ i1 < i2 < · · · < ik ≤ p such that• Fij = fj, 1 ≤ j ≤ k, and• i ∈ {i1, i2, . . . , ik} implies Fi is a trivial loop.This notion lets us compare the digital homotopy properties of loops evenif their domains have diﬀering cardinality, since if m1 ≤ m2, we obtain a trivialextension of a loop f : [0,m1]Z → X to f ′ : [0,m2]Z → X viaf ′(t) ={f(t) if 0 ≤ t ≤ m1;f(m1) if m1 ≤ t ≤ m2.The following notions are useful for deﬁning the class of a pointed loop.9 Deﬁnition. Let f, g : [0,m]Z → (X,x0) be digital loops with basepointx0. If H : [0,m]Z × [0,M ]Z → X is a digital homotopy between f and g suchthat for all t ∈ [0,M ]Z we haveH(0, t) = H(m, t),we say H is loop-preserving. If, further, for all t ∈ [0,M ]Z we haveH(0, t) = H(m, t) = x0,we say H holds the endpoints ﬁxed.The notion of H holding the endpoints ﬁxed was introduced in [4]. Theterm “loop-preserving” suggests that every (time) stage of the homotopy yieldsa digital loop.Digital κ-loops f and g in X with the same basepoint p belong to the sameκ-loop class in X if there are trivial extensions f ′ and g′ of f and g, respectively,________________________________________________________________________________________________Remarks on digital deformation 73with domains of the same cardinality, and a loop-preserving homotopy betweenf ′ and g′ that holds the endpoints ﬁxed [3].Membership in the same loop class in (X,x0) is an equivalence relationamong digital κ-loops [3].Let [f ] be the loop class of a loop f in X. We have the following.10 Proposition ([3, 7]). Suppose f1, f2, g1, g2 are digital loops in a pointeddigital image (X,x0), with f2 ∈ [f1] and g2 ∈ [g1]. Then f2 · g2 ∈ [f1 · g1].2.4 Digital fundamental groupThe digital fundamental group is derived from the classical fundamentalgroup of algebraic topology (see [10]).Let (X, p, κ) be a pointed digital image. Consider the set Πκ1(X, p) of κ-loopclasses [f ] in X with basepoint p. By Proposition 10, the product operation[f ] ∗ [g] = [f · g]is well-deﬁned on Πκ1(X, p).The operation ∗ is associative on Πκ1(X, p) [7].11 Lemma ([3]). Let (X, p) be a pointed digital image. Let p : [0,m]Z → Xbe the constant function p(t) = p. Then [p] is an identity element for Πκ1(X, p).12 Lemma ([3]). If f : [0,m]Z → X represents an element of Π1(X, p),then the function g : [0,m]Z → X deﬁned byg(t) = f(m− t) for t ∈ [0,m]Zis an element of [f ]−1 in Πκ1(X, p).13 Theorem ([3]). Πκ1(X, p) is a group under the ∗ product operation, theκ-fundamental group of (X, p).We may interpret the following result to say that in a connected digital imageX, the digital fundamental group is independent of the choice of basepoint.14 Theorem ([3]). Let X be a digital image with adjacency relation κ. Ifp and q belong to the same κ-component of X, then Πκ1(X, p) and Πκ1(X, q) areisomorphic groups.15 Proposition ([4]). Let X be a pointed κ-contractible digital image andlet p ∈ X. Then Πκ1(X, p) is a trivial group.2.5 Deformation and deformation retractionWe have the following.________________________________________________________________________________________________74 L. Boxer16 Deﬁnition ([5]). Let (X,κ) be a digital image and let A be a nonemptysubset of X. Then X is κ-deformable into A if there is a κ-homotopy D :X × [0,m]Z → X such that D(x, 0) = x and D(x,m) ∈ A for all x ∈ X.D is called a κ-deformation. If for some x0 ∈ A we have D(x0, t) = x0 forall t ∈ [0,m]Z, we say X is pointed κ-deformable into A, and D is a pointedκ-deformation.Classical notions of topology [1] yielded the concepts of digital retractionand deformation retraction in [2]. Let (X,κ) be a digital image and let A be anonempty subset of X. A retraction of X onto A is a (κ, κ)-continuous functionr : X → A such that r(a) = a for all a ∈ A. A κ-deformation retraction of X toA is a κ-homotopy H : X × [0,m]Z → X such that the induced map H( , 0) isthe identity map 1X , and the induced map H( ,m) is a retraction of X onto A.3 Deformations, deformation retractions, and funda-mental groupsNotice that a deformation retraction is a pointed deformation. We have thefollowing.17 Theorem ([4]). Let A be a nonempty subset of a digital image X andlet H : X× [0,m]Z → X be a κ-deformation retraction of X onto A. For a ∈ A,the inclusion map i : (A, a) ↪→ (X, a) induces an isomorphism of Πκ1(A, a) andΠκ1(X, a).However, a pointed deformation from a digital image X into its nonemptysubset A need not yield an isomorphism of the fundamental groups of X and A,despite the claim of Han (presented as Theorem 3 of [5]) to the contrary. Indeed,Han’s claim is false even if the deformation is required to be onto A at the endof the homotopy. Consider the pair (X,A) deﬁned as follows. X = ([0, 2]Z ×[0, 2]Z) ∪ {(j, 0)}7j=3. Let A ⊂ X be the set A = ([0, 2]Z × [0, 2]Z) \ {(1, 1)}, asimple closed 4-curve; hence Π41(A, a) is isomorphic to Z [3, 8, 9]. It is easilyseen that X is 4-contractible via the function h : X × [0, 9]Z → X deﬁned byh(x, y, t) ={(x, max{0, y − t}) for 0 ≤ t ≤ 2;(max{0, x + 2− t}, 0) for 3 ≤ t ≤ 9.Hence X has a trivial 4-fundamental group.We show there is a 4-deformation of X onto A as follows. Consider the________________________________________________________________________________________________Remarks on digital deformation 75function H : X × [0, 8]Z → X given as follows.H(x, y, t) =⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩(max{0, x − t}, y) if t ∈ [0, 2]Z;(max{2, x − t}, 0) if (x, y) ∈ {(j, 0)}7j=5, 3 ≤ t ≤ 5;H(x, y, 2) if (x, y) ∈ {(j, 0)}7j=5, 3 ≤ t ≤ 5;(2, 1) if (x, y) ∈ {(j, 0)}7j=5, t = 6;H(x, y, 5) if (x, y) ∈ {(j, 0)}7j=5, t = 6;(2, 2) if (x, y) ∈ {(6, 0), (7, 0)}, t = 7;H(x, y, 6) if (x, y) ∈ {(6, 0), (7, 0)}, t = 7;(1, 2) if (x, y) = (7, 0), t = 8;H(x, y, 7) if (x, y) = (7, 0), t = 8.It is easily seen that this function is a 4-homotopy between 1X and the functionH8 : X → X, deﬁned by H8(x, y) = H(x, y, 8), that is onto A.What is valid from Han’s paper is the following.18 Theorem. Let X be a digital image and let A be a non-empty subsetof X. Let D : X × [0,m]Z → X be a pointed κ-deformation of X into A, withD(p, t) = p for some p ∈ A and all t ∈ [0,m]Z . Let r : X → A be the mapdeﬁned by r(x) = D(x,m) for all x ∈ X. Then the induced homomorphismr∗ : Πκ1(X, p) → Πκ1(A, p) is one to one.Proof. [5] Let i : A → X be the inclusion map. Then D is a homotopybetween 1X and i◦r. Therefore, 1Πκ1 (X,p) = i∗◦r∗. The assertion follows. QED4 SummaryWe have shown that the claim of [5], that a pointed digital deformationinduces an isomorphism between fundamental groups, is false.Acknowledgements. We gratefully acknowledge a suggestion of an anon-ymous referee.References[1] K. Borsuk: Theory of Retracts, Polish Scientiﬁc Publishers, Warsaw 1967.[2] L. Boxer: Digitally continuous functions, Pattern Recognition Letters, 15 (1994), 833–839.[3] L. Boxer: A classical construction for the digital fundamental group, Journal of Mathe-matical Imaging and Vision, 10 (1999), 51–62.[4] L. Boxer: Properties of digital homotopy, Journal of Mathematical Imaging and Vision,22 (2005), 19–26.________________________________________________________________________________________________76 L. Boxer[5] S.-E. Han: Generalized digital (k0, k1)-homeomorphism, Note di Matematica, 22 (2003),157–166.[6] G. T. Herman: Oriented surfaces in digital spaces, CVGIP: Graphical Models and ImageProcessing, 55 (1993), 381–396.[7] E. Khalimsky: Motion, deformation, and homotopy in ﬁnite spaces, in Proceedings IEEEIntl. Conf. on Systems, Man, and Cybernetics, (1987) 227–234.[8] T. Y. Kong: A digital fundamental group, Computers and Graphics, 13 (1989), 159–166.[9] T. Y. Kong, A. W. Roscoe, A. Rosenfeld: Concepts of digital topology, Topology andits Applications, 46 (1992), 219–262.[10] W. S. Massey: Algebraic Topology: An Introduction, Harcourt, Brace, and World, NewYork 1967.[11] A. Rosenfeld: ‘Continuous’ functions on digital pictures, Pattern Recognition Letters,4 (1986), 177–184.________________________________________________________________________________________________

http://siba-ese.unile.it/index.php/notemat/article/download/634/479

Remarks on digital deformation

Abstract

Similar works

Full text

Available Versions

Università del Salento: ESE - Salento University Publishing

ESE - Salento University Publishing