Freezing Sets Invariant-based Characteristics of Digital Images
EMAN ALMUHUR ¹, EMAN A. ABUHIJLEH ², GHADA ALAFIFI ³, ABDULAZEEZ ALKOURI4,
MONA BIN-ASFOUR5
1Department of Basic Science and Humanities, Applied Science Private University, Amman,
JORDAN
2Department of Basic Sciences, Al-Balqa Applied University, Zarqa, JORDAN
3Mathematics Department, Philadelphia University, Amman, JORDAN
4Mathematics Department, Ajloun National University, Ajloun, JORDAN
5Department of Mathematics and Statistics, Imam Mohammad Ibn Saud Islamic University, Riyadh,
SAUDI ARABIA
Abstract: - Due to the widespread use of digital images of real-world objects as mathematical models, this
research examines the freezing sets invariant-base properties of digital images. In contrast to earlier studies that
only covered a discrete or limited collection of points, fixed points of digitally continuous functions are
approved to deal with a variety of characteristics of digital images.
Key-Words: - Digital image, Freezing sets, Boundary, Irreducible
Received: September 22, 2022. Revised: April 11, 2023. Accepted: May 4, 2023. Published: May 18, 2023.
1 Introduction
Mathematical models commonly use illustrations of
the world’s objects. A digital representation of the
notion of a continuous function, which was drawn
from topology, is usually useful for the analysis of
digital images. However, the digital picture is
frequently a distinct, limited collection of points. As
a result, methods other than topology-based
methods for digital picture analysis are usually
needed. In this work, we examine a number of
digital picture features that are connected to the
fixed points of digitally continuous functions.
These characteristics include discrete
measurements that do not naturally correspond to
the characteristics of subsets. is a digital
image where for some integer n, and is an
adjacency on which is considered to be finite, [6].
If is a vertex set and is an edge set, then the pair
is a graph. Adjacency is a measure of how
"closedness" two points are to one another in .
When these conditions (finiteness of ) and
(closedness of adjacency points) are satisfied, the
digital image may be viewed as a model of a white-
and-black "real world" image, where white points in
the background are declared by elements of
and the black points in the foreground by
members of U, [1].
indicates that and are adjacent and
are adjacent or equal.
If is an integer such that and
, then
iff
(i) For at most indices i, | |.
(ii) For all indices j, | | implies .
The number of adjacent points is frequently used to
indicate the adjacencies.
Examples:
(i) The adjacency in is adjacency.
(ii) The adjacency is adjacency is and the
adjacency in is adjacency.
(iii) The adjacency is adjacency,
adjacency is adjacency, and
adjacency in is adjacency.
If and are two digital images, then
denotes the strong product adjacency or
normal adjacency, [2], on iff
and where
,
if one of the following conditions is
valid:
(i) α₀α₁ and
(ii) and .
(iii) ₁ and .
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.42
Eman Almuhur, Eman A. Abuhijleh,
Ghada Alafifi, Abdulazeez Alkouri, Mona Bin-Asfour
E-ISSN: 2224-2880
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Typically if and are two natural numbers
such that , (,) and
, then the adjacency
, [3], is defined as: For some
and in , if
, then:
q if for at least 1 and at most
indices , ′ and indices, .
In this paper, "digital images" is referred to as D.I.
2 Digitally Continuous
Function
Definition 1.1:
i. [4], If and are two D.I,
then is a digitally
continuous function, if ,
then is continuous.
ii. The path from to is the set
such that and
.
Now, if ≠ , then the length
of the path is .
iii. The path from to is a ,
where is a continuous
function and and
Theorem 1.2: [5], If and are two
D.Is', then:
i. is a digitally continuous
function iff , if , then
ii. If is a D.I and is
continuous, then is
continuous.
Definition 1.3:
i. [1], [6], Let and be two D.I
and, are two
continuous functions and
is defined as
and and
.
ii. A function h is a digital
homotopy, and are
digitally homotopic in (denoted by
.
iii. If , then holds
fixed.
iv. [1], If is a subset of and is a
continuous function, then is a
retraction. If , then is
a retract.
v. If is an inclusion function, and
then is a deformation
retract of .
iv. The function is an
isomorphisim (homeomorphisim) if is
a bijective continuous function and
is continuous.
v. If is a digital image, then
is continuous.
vi. If and ,
then is a fixed point.
vii. is the set of all fixed points of
.
Theorem 1.4: [3], If (,) and ( ) are D.I
, : (,) →( ) and
:
→
given by
which is (
continuous iff is (, ψontinuous
Definition 1.5, [1]:
i. A continuous function
is rigid if there is no continuous
map homotopic to f except itself.
ii. is rigid if is rigid.
iii. [7], If a finite image is homotopy
equivalent to an image with fewer
points, it is said to be reducible.
Otherwise, U is irreducible.
iv. [1], If is irreducible, then for
some point such
that id and , is a reduction
point.
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DOI: 10.37394/23206.2023.22.42
Eman Almuhur, Eman A. Abuhijleh,
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Remark 1.6: [7], A finite image is
reducible if is homotopic to a
non-surjective function.
Definition 1.7: [8], For the D.I and
:
i. The set is
the homotopy fixed point spectrum of the
function .
ii. The set
holding
fixed} is the pointed homotopy fixed
point spectrum of the function f for some
.
iii. The set
is the fixed point spectrum of
.
iv. The set
is the pointed fixed point
spectrum of (U,κ,).
Theorem 1.8: i. [8], If is a retract of the D.I
, then .
ii. If is a retract of , then
3 Freezing Sets
Definition 2.1: [1], If is a D.I, then is a
freezing subset for if for
some
Theorem 2.2: If is a D.I and is a freezing
subset for , then:
i. has a unique extension of to a
member of .
ii. If is an isomorphisim,
is continuous and
h=then .
iii. A continuous function
has one extension to an isomorphism
.
Lemma 2.3: Freezing sets are topological
invariants.
Theorem 2.4: If is a D.I and is a freezing
subset for and is an
isomorphism, then is a freezing set for .
Proof: Suppose that and
=|. Now,
and by theorem
3.2,
, then
Thus is a freezing set for .
Theorem 2.5: If is a D.I for
, and
, then
Proof: If and
then .
Hence , but , so
.
Therefore, (
Theorem 2.6:
i. If is a D.I and is a retract
of , then has no freezing sets for
.
ii. If is a reducible digital image and
is a freezing subset for , then if
is a reduction point of , .
Proof: i. If is an inclusion
function, then is a retraction and
is continuous.
Now, =, but f is not the identity function.
ii. If is a reduction point of ,
then where has no
freezing sets for .
4 Boundaries of Freezing Sets
Definition 3.1:
i. [1], If is a D.I and is a
freezing set for , then is minimal if
no proper subset of is a freezing set
for .
ii. [9], If , then the boundary of
is for some
.
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DOI: 10.37394/23206.2023.22.42
Eman Almuhur, Eman A. Abuhijleh,
Ghada Alafifi, Abdulazeez Alkouri, Mona Bin-Asfour
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iii. [1], The interior of is
.
Theorem 3.2: Let be finite, is a subset
of , . If
and is a freezing set for , then
Proof: Let but is
an interior point of .
Now, such that , and
because of the finiteness, there exists a path
For and , the path belongs to and
for , the path belongs to .
By theorem 3.5, the path does not belong to
for and m which contradicts the
assumption.
Thus,
Theorem 3.3: [10], If
ℤⁿ such that
>, then is a minimal freezing set for
.
Proof: Let for
some proper set of .
For some index , we have . If =0,
then for the function given by
and
we have
where = and .
Now, if , then and
where
where = and
Theorem 3.4: If is a set of D.I
, U=
and a subset of is a freezing
set for
, then for the
projection function
:
→
given by
we have is a freezing set for
.
Proof: Suppose and is
defined as
then
.
Now, ()= , but is a freezing set
for , hence ,
Therefore, =.
5 Conclusion
Freezing sets are topological invariants. So, If
is a D.I, is a freezing subset for and
is an isomorphism, then is a freezing set
for . is an inclusion
function, then is a retraction and
is continuous. Moreover, if
is finite, is a subset of ,
. If and is a freezing
set for , then
Acknowledgments:
The authors are grateful to the anonymous referees
for their insightful criticism and recommendations,
which helped to strengthen the paper's presentation.
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Eman Almuhur, Eman A. Abuhijleh,
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
-Eman Almuhur is in charge of both conceptualizing
the research challenge and overseeing the effort.
-Eman A. AbuHijleh and Ghada Alafifi are in
charge of doing the formal analysis and composing
the paper's initial draft.
The initial draft of the paper was amended and
modified by Alkouri and Bin-Asfour.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
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