Lefschetz fixed point theorem for digital images
 Ozgur Ege^{1}Email author and
 Ismet Karaca^{2}
https://doi.org/10.1186/168718122013253
© Ege and Karaca; licensee Springer. 2013
Received: 5 July 2013
Accepted: 30 September 2013
Published: 7 November 2013
Abstract
In this article we study the fixed point properties of digital images. Moreover, we prove the Lefschetz fixed point theorem for a digital image. We then give some examples about the fixed point property. We conclude that spherelike digital images have the fixed point property.
MSC:55N35, 68R10, 68U05, 68U10.
Keywords
1 Introduction
Digital topology is important for computer vision, image processing and computer graphics. In this area many researchers, such as Rosenfeld, Kong, Kopperman, Kovalevsky, Boxer, Karaca, Han and others, have characterized the properties of digital images with tools from topology, especially algebraic topology.
The homology theory is the major subject of algebraic topology. It is used in the classification problems of topological spaces. The simplicial homology is one of the homology theories. Homology groups are topological invariants which are related to the different ndimensional holes, connected components of a geometric object. For example, the torus contains one 0dimensional hole, two 1dimensional holes and one 2dimensional hole.
The Lefschetz fixed point theorem is a formula that counts fixed points of a continuous mapping from a compact topological space X to itself by means of traces of the induced mappings on the homology groups of X. In 1926, Lefschetz introduced the Lefschetz number of a map and proved that if the number is nonzero, then the map has a fixed point. Since the Lefschetz number is defined in terms of the homology homomorphism induced by the map, it is a homotopy invariant. As a result, a nonzero Lefschetz number implies that each of the maps in the given homotopy class has at least one fixed point. The Lefschetz fixed point theorem generalizes a collection of fixed point theorems for different topological spaces. We will work entirely with digital images that are digital simplicial complexes or retracts of digital simplicial complexes.
Arslan et al. [1] introduced the simplicial homology groups of ndimensional digital images from algebraic topology. They also computed simplicial homology groups of $MS{S}_{18}$. Boxer et al. [2] expanded the knowledge of simplicial homology groups of digital images. They studied the simplicial homology groups of certain minimal simple closed surfaces, extended an earlier definition of the Euler characteristics of a digital image, and computed the Euler characteristic of several digital surfaces. Demir and Karaca [3] computed the simplicial homology groups of some digital surfaces such as $MS{S}_{18}\mathrm{\u266f}MS{S}_{18}$, $MS{S}_{6}$ and $MS{S}_{6}\mathrm{\u266f}MS{S}_{6}$.
Karaca and Ege [4] studied some results related to the simplicial homology groups of 2D digital images. They showed that if a bounded digital image $X\subset \mathbb{Z}$ is nonempty and κconnected, then its homology groups at the first dimension are a trivial group. They also proved that the homology groups of the operands of a wedge of digital images need not be additive. Ege and Karaca [5] gave characteristic properties of the simplicial homology groups of digital images and then investigated the EilenbergSteenrod axioms for the simplicial homology groups of digital images.
This paper is organized as follows. The second section provides the general notions of digital images with κadjacency relations, digital homotopy groups and digital homology groups. In Section 3 we present the Lefschetz fixed point theorem for digital images and provide some of its important applications. In the last section we make some conclusions about this topic.
2 Preliminaries
Let ℤ be the set of integers. A (binary) digital image is a pair $(X,\kappa )$, where $X\subset {\mathbb{Z}}^{n}$ for some positive integer n and κ represents certain adjacency relation for the members of X. A variety of adjacency relations are used in the study of digital images. We give one of them. Let l, n be positive integers, $1\le l\le n$ and two distinct points $p=({p}_{1},{p}_{2},\dots ,{p}_{n})$, $q=({q}_{1},{q}_{2},\dots ,{q}_{n})$ in ${\mathbb{Z}}^{n}$, p and q are ${k}_{l}$adjacent [6] if there are at most l distinct coordinates j, for which ${p}_{j}{q}_{j}=1$ and for all other coordinates j, ${p}_{j}={q}_{j}$. The number of points $q\in {\mathbb{Z}}^{n}$ that are adjacent to a given point $p\in {\mathbb{Z}}^{n}$ is represented by a ${k}_{l}$adjacency relation. From this viewpoint, the ${k}_{1}$adjacency on ℤ is denoted by the number 2 and ${k}_{1}$adjacent points are called 2adjacent. In a similar way, we call 4adjacent and 8adjacent for ${k}_{1}$ and ${k}_{2}$adjacent points of ${\mathbb{Z}}^{2}$; and in ${\mathbb{Z}}^{3}$, 6adjacent, 18adjacent and 26adjacent for ${k}_{1}$, ${k}_{2}$ and ${k}_{3}$adjacent points, respectively.
is said to be a digital interval [8], where $a,b\in \mathbb{Z}$ with $a<b$.
Let $(X,{\kappa}_{0})\subset {\mathbb{Z}}^{{n}_{0}}$ and $(Y,{\kappa}_{1})\subset {\mathbb{Z}}^{{n}_{1}}$ be digital images. A function $f:X\u27f6Y$ is called $({\kappa}_{0},{\kappa}_{1})$continuous [9] if for every ${\kappa}_{0}$connected subset U of X, $f(U)$ is a ${\kappa}_{1}$connected subset of Y.
In a digital image X, if there is a $(2,\kappa )$continuous function $f:{[0,m]}_{\mathbb{Z}}\u27f6X$ such that $f(0)=x$ and $f(m)=y$, then we say that there exists a digital κpath [10] from x to y. If $f(0)=f(m)$, then f is called digital κloop and the point $f(0)$ is the base point of the loop f. When a digital loop f is a constant function, it is said to be a trivial loop. A simple closed κcurve of $m\ge 4$ points in a digital image X is a sequence $\{f(0),f(1),\dots ,f(m1)\}$ of images of the κpath $f:{[0,m1]}_{\mathbb{Z}}\u27f6X$ such that $f(i)$ and $f(j)$ are κadjacent if and only if $j=i\pm modm$.
Definition 2.1 Let $(X,{\kappa}_{0})\subset {\mathbb{Z}}^{{n}_{0}}$ and $(Y,{\kappa}_{1})\subset {\mathbb{Z}}^{{n}_{1}}$ be digital images. A function $f:X\u27f6Y$ is a $({\kappa}_{0},{\kappa}_{1})$isomorphism [1] if f is $({\kappa}_{0},{\kappa}_{1})$continuous and bijective and ${f}^{1}:Y\u27f6X$ is $({\kappa}_{1},{\kappa}_{0})$continuous and it is denoted by $X{\approx}_{({\kappa}_{0},{\kappa}_{1})}Y$.
For a digital image $(X,\kappa )$ and its subset $(A,\kappa )$, we call $(X,A)$ a digital image pair with κadjacency. If A is a singleton set $\{{x}_{0}\}$, then $(X,{x}_{0})$ is called a pointed digital image.
Definition 2.2 [9]
Let $(X,{\kappa}_{0})\subset {\mathbb{Z}}^{{n}_{0}}$ and $(Y,{\kappa}_{1})\subset {\mathbb{Z}}^{{n}_{1}}$ be digital images. Two $({\kappa}_{0},{\kappa}_{1})$continuous functions $f,g:X\u27f6Y$ are said to be digitally $({\kappa}_{0},{\kappa}_{1})$homotopic in Y if there is a positive integer m and a function $H:X\times {[0,m]}_{\mathbb{Z}}\u27f6Y$ such that

for all $x\in X$, $H(x,0)=f(x)$ and $H(x,m)=g(x)$;

for all $x\in X$, the induced function ${H}_{x}:{[0,m]}_{\mathbb{Z}}\u27f6Y$ defined by${H}_{x}(t)=H(x,t)\phantom{\rule{1em}{0ex}}\text{for all}t\in {[0,m]}_{\mathbb{Z}}$
is $(2,{\kappa}_{1})$continuous; and

for all $t\in {[0,m]}_{\mathbb{Z}}$, the induced function ${H}_{t}:X\u27f6Y$ defined by${H}_{t}(x)=H(x,t)\phantom{\rule{1em}{0ex}}\text{for all}x\in X$
is $({\kappa}_{0},{\kappa}_{1})$continuous.
The function H is called a digital $({\kappa}_{0},{\kappa}_{1})$homotopy between f and g. If these functions are digitally $({\kappa}_{0},{\kappa}_{1})$homotopic, this is denoted by $f{\simeq}_{{\kappa}_{0},{\kappa}_{1}}g$. The digital $({\kappa}_{0},{\kappa}_{1})$homotopy relation [9] is equivalence among digitally continuous functions $f:(X,{\kappa}_{0})\u27f6(Y,{\kappa}_{1})$.
 (i)
A digital image $(X,\kappa )$ is said to be κcontractible if its identity map is $(\kappa ,\kappa )$homotopic to a constant function $\overline{c}$ for some $c\in X$, where the constant function $\overline{c}:X\u27f6X$ is defined by $\overline{c}(x)=c$ for all $x\in X$.
 (ii)
Let $(X,A)$ be a digital image pair with κadjacency. Let $i:A\u27f6X$ be the inclusion function. A is called a κretract of X if and only if there is a κcontinuous function $r:X\u27f6A$ such that $r(a)=a$ for all $a\in A$. Then the function r is called a κretraction of X onto A.
 (iii)
A digital homotopy $H:X\times {[0,m]}_{\mathbb{Z}}\u27f6X$ is a deformation retract [11] if the induced map $H(,0)$ is the identity map ${1}_{X}$ and the induced map $H(,m)$ is retraction of X onto $H(X\times \{m\})\subset X$. The set $H(X\times \{m\})$ is called a deformation retract of X.
Definition 2.4 [12]
 (a)
If p and q are distinct points of $s\in S$, then p and q are κadjacent.
 (b)
If $s\in S$ and $\mathrm{\varnothing}\ne t\subset s$, then $t\in S$.
An msimplex is a simplex S such that $S=m+1$.
Let P be a digital msimplex. If ${P}^{\prime}$ is a nonempty proper subset of P, then ${P}^{\prime}$ is called a face of P. We write $Vert(P)$ to denote the vertex set of P, namely the set of all digital 0simplexes in P. A digital subcomplex A of a digital simplicial complex X with κadjacency is a digital simplicial complex [12] contained in X with $Vert(A)\subset Vert(X)$.
 (1)
If P belongs to X, then every face of P also belongs to X.
 (2)
If $P,Q\in X$, then $P\cap Q$ is either empty or a common face of P and Q.
The dimension of a digital simplicial complex X is the largest integer m such that X has an msimplex. ${C}_{q}^{\kappa}(X)$ is a free abelian group [1] with basis of all digital $(\kappa ,q)$simplexes in X.
Corollary 2.6 [1]
Let $(X,\kappa )\subset {\mathbb{Z}}^{n}$ be a digital simplicial complex of dimension m. Then, for all $q>m$, ${C}_{q}^{\kappa}(X)$ is a trivial group.
Definition 2.7 [2]
 (1)
${Z}_{q}^{\kappa}(X)=Ker{\partial}_{q}$ is called the group of digital simplicial qcycles.
 (2)
${B}_{q}^{\kappa}(X)=Im{\partial}_{q+1}$ is called the group of digital simplicial qboundaries.
 (3)
${H}_{q}^{\kappa}(X)={Z}_{q}^{\kappa}(X)/{B}_{q}^{\kappa}(X)$ is called the q th digital simplicial homology group.
We recall some important examples about digital homology groups of certain digital images.
Theorem 2.13 [2]
 (1)
${H}_{q}^{\kappa}(X)$ is a finitely generated abelian group for every $q\ge 0$.
 (2)
${H}_{q}^{\kappa}(X)$ is a trivial group for all $q>m$.
 (3)
${H}_{m}^{\kappa}(X)$ is a free abelian group, possible zero.
In [2], it is proven that for each $q\ge 0$, ${H}_{q}^{\kappa}$ is a covariant functor from the category of digital simplicial complexes and simplicial maps to the category of abelian groups.
Definition 2.14 [2]
Let $f:(X,{\kappa}_{0})\u27f6(Y,{\kappa}_{1})$ be a function between two digital images. If for every digital $({\kappa}_{0},m)$simplex P determined by ${\kappa}_{0}$ in X, $f(P)$ is a $({\kappa}_{1},n)$simplex in Y for some $n\le m$, then f is called a digital simplicial map.
where $z\in {Z}_{q}^{\kappa}(X)$, respectively.
Definition 2.15 [10]
Let X and Y be digital images in ${\mathbb{Z}}^{n}$, using the same adjacency notion, denoted by κ, such that $X\cap Y=\{{x}_{0}\}$, where ${x}_{0}$ is the only point of X adjacent to any point of Y and ${x}_{0}$ is the only point of Y adjacent to any point of X. Then the wedge of X and Y, denoted by $X\wedge Y$, is the image $X\wedge Y=X\cup Y$, with κadjacency.
3 The Lefschetz fixed point theorem for digital images
A digital image $(X,\kappa )$ is said to have the fixed point property if for any $(\kappa ,\kappa )$continuous function $f:(X,\kappa )\u27f6(X,\kappa )$, there exists $x\in X$ such that $f(x)=x$. The fixed point property is a topological invariant, i.e., is preserved by any digital isomorphism. The fixed point property is also preserved by any retraction. The Lefschetz fixed point theorem determines when there exist fixed points of a map on a finite digital simplicial complex using a characteristic of the map known as the Lefschetz number.
where ${f}_{\ast}:{H}_{i}^{\kappa}(X)\u27f6{H}_{i}^{\kappa}(X)$.
Theorem 3.2 Let G be a free abelian group, and let ${1}_{G}:G\to G$ be the identity homomorphism. Then $tr({1}_{G})=rank(G)$, where $tr({1}_{G})$ is the trace of ${1}_{G}$.
Proof Since G is a free abelian group, the rank of G is 1; i.e., we have $rank(G)=1$. Moreover, the trace of ${1}_{G}$ is 1 because ${1}_{G}:G\u27f6G$ is the identity map and has an identity $1\times 1$matrix [1]. So, we have $rank(G)=tr({1}_{G})$. □
Theorem 3.3 If $(X,\kappa )$ is a finite digital simplicial complex, or the retract of some finite digital simplicial complex, and $f:(X,\kappa )\u27f6(X,\kappa )$ is a map with $\lambda (f)\ne 0$, then f has a fixed point.
□
Theorem 3.4 (Onedimensional Brouwer fixed point theorem)
Every $(2,2)$continuous function $f:{[0,1]}_{\mathbb{Z}}\u27f6{[0,1]}_{\mathbb{Z}}$ has a fixed point.
Proof Let $f:{[0,1]}_{\mathbb{Z}}\u27f6{[0,1]}_{\mathbb{Z}}$ be $(2,2)$continuous. Since $f(0)=0$ and $f(1)=1$, f has a fixed point. □
Theorem 3.5 (Twodimensional Brouwer fixed point theorem)
Let $X=\{(0,0),(1,0),(0,1),(1,1)\}\subset {\mathbb{Z}}^{2}$ be a digital image with 4adjacency. Every $(4,4)$continuous function $f:(X,4)\u27f6(X,4)$ has a fixed point.
Since $\lambda (f)=1\ne 0$, Theorem 3.3 implies that f has a fixed point. As a result, twodimensional Brouwer fixed point theorem holds for digital images. □
We conclude now by showing that the fixed point property is a topological property; i.e., two topological spaces which are topologically equivalent to each other either both have or both lack the fixed point property. Because one of the main goals of topology is to discover what properties are maintained when moving between topologically equivalent spaces, this is an incredibly important proposition to make.
Theorem 3.6 Let $(X,\kappa )$ and $(Y,{\kappa}^{\prime})$ be digital images such that $X{\approx}_{(\kappa ,{\kappa}^{\prime})}Y$. If $(X,\kappa )$ has the fixed point property, then $(Y,{\kappa}^{\prime})$ has the fixed point property.
Proof Since $X{\approx}_{(\kappa ,{\kappa}^{\prime})}Y$, there exists a bijective function $f:(X,\kappa )\u27f6(Y,{\kappa}^{\prime})$ such that f is $(\kappa ,{\kappa}^{\prime})$continuous and that its inverse ${f}^{1}$ is $({\kappa}^{\prime},\kappa )$continuous. Also, X has the fixed point property; i.e., every $(\kappa ,\kappa )$continuous function $g:(X,\kappa )\u27f6(X,\kappa )$ has a fixed point. Now, let $h:(Y,{\kappa}^{\prime})\u27f6(Y,{\kappa}^{\prime})$ be $({\kappa}^{\prime},{\kappa}^{\prime})$continuous. Then $h\circ f:(X,\kappa )\u27f6(Y,{\kappa}^{\prime})$ and, consequently, ${f}^{1}\circ h\circ f:(X,\kappa )\u27f6(X,\kappa )$ are certainly $(\kappa ,{\kappa}^{\prime})$continuous and $({\kappa}^{\prime},\kappa )$continuous, respectively. However, since X has the fixed point property, ${f}^{1}(h(f(x)))=x$ for some $x\in X$. It follows that $f({f}^{1}(h(f(x))))=f(x)$, which implies that $h(f(x))=f(x)$, proving that h has a fixed point, namely $f(x)$. □
Corollary 3.7 The fixed point property is a topological invariant for digital images.
Theorem 3.8 Let $(X,\kappa )$ be any digital image, and let $f:(X,\kappa )\u27f6(X,\kappa )$ be any $(\kappa ,\kappa )$continuous map. If X is κcontractible, then f has a fixed point.
By Theorem 3.3, we have that f has a fixed point. □
which is simply the identity function and thus has a nonzero trace. From Theorem 3.3, every map on a digital image $(X,8)$ has at least one fixed point.
Corollary 3.10 Any digital image $(X,\kappa )$ with the same digital homology groups as a single point image always has a fixed point.
Theorem 3.11 Let $(X,\kappa )$ be a digital image and $(A,\kappa )$ be a digital κretract of $(X,\kappa )$. If $(X,\kappa )$ has the fixed point property, then $(A,\kappa )$ has the fixed point property.
Hence there is a point $x\in (A,\kappa )$ such that $h(x)=x$, where $h=r\circ f\circ i$. As a result, $(A,\kappa )$ has the fixed point property. □
Proposition 3.12 Let A, B be digital images with κadjacency. The $A\wedge B$ has the fixed point property if and only if both A and B have the fixed point property.
contrary to the assumption that $f(u)\in AB$. So, f has the fixed point. Necessary condition follows from Theorem 3.11. □
Converse of the Lefschetz fixed point theorem need not be true for digital images. Let us see this.
since ${\varphi}_{0},{\varphi}_{1}:\mathbb{Z}\u27f6\mathbb{Z}$ are the identity functions. Thus $\lambda (\varphi )=0$, but ϕ has the fixed point. As a consequence, we have the following corollary.
Corollary 3.14 Converse of the Lefschetz fixed point theorem need not be true for digital images.
Now we compute the Lefschetz number of ${S}_{2}$. We define and sketch it.
since ${f}_{0}:\mathbb{Z}\u27f6\mathbb{Z}$ is the identity function and ${f}_{1}:{\mathbb{Z}}^{23}\u27f6{\mathbb{Z}}^{23}$. As a result, f has a fixed point.
Proposition 3.16 Let $(X,\kappa )$ be a digital image. If a map $f:(X,\kappa )\u27f6(X,\kappa )$ is homotopic to the identity, then $\lambda (f)=\chi (X,\kappa )$.
□
By Theorem 3.3, f has a fixed point. The Euler characteristic of $MS{S}_{6}^{\prime}$ is $\chi (MS{S}_{6}^{\prime},6)=4$.
where $c\in {P}^{2}$. It is clear that this map is a 6deformation retract of ${P}^{2}$. So, ${P}^{2}$ is 6contractible image. As a result, ${P}^{2}$ has the same digital homology groups as a single point image. By Theorem 3.8, we have the following.
Corollary 3.18 Every $(6,6)$continuous map $f:{P}^{2}\u27f6{P}^{2}$ has a fixed point.
4 Conclusion
The main goal of this study is to determine fixed point properties for a digital image. Moreover, we study the relations between the Euler characteristic and the Lefschetz number. At the end of this work, we give a digital version of the real projective plane and calculate its homology groups and the Lefschetz number. We expect that these properties will be useful for image processing.
Declarations
Acknowledgements
The authors thank the editor and the referees for their valuable suggestions to improve the quality of this paper.
Authors’ Affiliations
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