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# Mathematical Morphology View of Topological Rough Sets and Its Applications

1 Department of Mathematics, Faculty of Science and Arts in Al-Mandaq, Al-Baha University, Kingdom of Saudi Arabia

2 Department of Mathematics, Faculty of Science, Tanta University, Tanta, Egypt

3 Department of Electrical Engineering, Faculty of Engineering, Kafrelsheikh University, Kafrelsheikh, 33516, Egypt

4 Department of Physics and Engineering Mathematics, Faculty of Engineering, Kafrelsheikh University, Kafrelsheikh, 33516, Egypt

* Corresponding Author: Tamer Medhat. Email:

*Computers, Materials & Continua* **2023**, *74*(3), 6893-6908. https://doi.org/10.32604/cmc.2023.033539

**Received** 20 June 2022; **Accepted** 20 October 2022; **Issue published** 28 December 2022

## Abstract

This article focuses on the relationship between mathematical morphology operations and rough sets, mainly based on the context of image retrieval and the basic image correspondence problem. Mathematical morphological procedures and set approximations in rough set theory have some clear parallels. Numerous initiatives have been made to connect rough sets with mathematical morphology. Numerous significant publications have been written in this field. Others attempt to show a direct connection between mathematical morphology and rough sets through relations, a pair of dual operations, and neighborhood systems. Rough sets are used to suggest a strategy to approximate mathematical morphology within the general paradigm of soft computing. A single framework is defined using a different technique that incorporates the key ideas of both rough sets and mathematical morphology. This paper examines rough set theory from the viewpoint of mathematical morphology to derive rough forms of the morphological structures of dilation, erosion, opening, and closing. These newly defined structures are applied to develop algorithm for the differential analysis of chest X-ray images from a COVID-19 patient with acute pneumonia and a health subject. The algorithm and rough morphological operations show promise for the delineation of lung occlusion in COVID-19 patients from chest X-rays. The foundations of mathematical morphology are covered in this article. After that, rough set theory ideas are taken into account, and their connections are examined. Finally, a suggested image retrieval application of the concepts from these two fields is provided.## Keywords

Nomenclature

Abbreviations

RS | Rough Sets |

SE | Structure Element |

RD | Rough Dilation |

RE | Rough Erosion |

RC | Rough Closing |

RO | Rough Opening |

Symbols | |

Positive integers | |

Discrete topological space | |

D-dimensional product of E | |

X,Y | Images as sets of pixels |

B | Structure element |

Topological space | |

CL(X) | Closure of X |

Int(X) | Interior of X |

1 Introduction and Preliminaries

Pawlak’s rough set theory [1] is a mathematical tool that is useful for studying uncertain or incomplete data in information systems and is based on the classification of data into Objects [2]. However, Pawlak’s definition of equivalence classes of an equivalence relation for the universe set is not valid for all incomplete data, particularly various real-life problems. By studying Pawlak’s rough set theory from a topological point of view, Lashin et al. [3] and Salama et al. [4] derived a general relation and observed that most of Pawlak’s properties are not held in a general sense. This led to the study of new kinds of rough sets and their applications by many researchers [5–7]. Neighborhood systems and rough sets have recently been used to represent structures, such as self-similar fractals [8] and the human heart [9], with potential applications in physics and medicine, respectively. One of the most important achievements in rough set theory is knowledge reduction in information systems, by which a membership function is used to reduce the data. Pawlak et al. [10] expanded the membership function into an initial rough membership function, while El Atik et al. [11] used this similarity as a membership function. Allam et al. [12] and Salama [13] presented new approaches for basic rough set concepts. In [14], Yao et al. address the problem of overlapping classes in rough sets and introduce object class membership, and in [15] Yu et al. introduce approximations to measure accuracy. Polkowski et al. introduced some of important mechanisms for rough sets in [16,17].

Differential equations and diffusion equations may both be used to simulate many real-world issues in the fields of science and engineering [18]. Recently, four novel mutant SARS-COV-2 strains that are thought to be 70% more lethal than the currently circulating COVID-19 virus were found in several locations [19]. By using a mathematical model, it is crucial to understand the SARS-CoV-2 dynamics in the context of immune surveillance [20]. In the COVID-19 outbreak investigation, asymptomatic transmission of the coronavirus illness and the prediction of infected individuals have become crucial [21]. For biomathematicians and medical professionals, controlling these acute illnesses has been a major priority in recent years [22].

The main aim of the present paper is to study rough set theory from another angle, that is, from the viewpoint of mathematical morphology. Mathematical morphology provides a range of techniques for image processing and analysis based on basic algebraic and geometric principles. Matheron [23] and Serra [24] first introduced the concepts of mathematical morphology, in application to petrography and mineralogy where they studied the physical properties of certain types of rocks. Mathematical morphology has since been explored in some detail (see, e.g., [25–30]) and has been applied in engineering and medicine [31–34]. A new method for scene classification from the remote sensing images is investigated in [35], but diagnosis and testing of COVID-19 chest are investigated in [36,37].

Here we propose new forms of rough morphological structures: rough dilation, rough erosion, rough closing, and rough opening. These forms are defined for application in topological and digital image processing and applied specifically for the delineation of lung occlusion from a chest x-ray of a patient with acute COVID-19 pneumonia.

This article is organized as follows; Section 2 for morphological definitions of roughness and Section 3 introduces the basic properties of rough dilation and rough erosion. Section 4 presents the application of rough morphological structures for differential analysis of chest x-ray images. Finally, Section 5 presents the conclusion and future work.

Throughout this paper

Definition 1. [30] Mathematical morphology allows the extraction and analysis of discrete quantal image structures. There are two essential components: the image as a set of objects and a structure element (SE). Each object is represented by binary digits, e.g.,

Definition 2. [30] For any two images,

Note that

Definition 3. [30] The dilation of image called X by structure element called A is given by

Definition 4. [1] Let

where

2 Morphological Definitions of Roughness

Dilation and erosion are basic concepts in mathematical morphology and image processing, where any image set X can be dilated (eroded) by a structure element B. Here, we consider rough set theory by scanning

Definition 5. A rough dilation (RD) and rough erosion (RE) are defined by

It is clear that

Example 1. Move and scan

Example 2. Move and scan

Definition 6. Let X and B be subsets of a discrete space

Example 3. Move and scan

Example 4. Move and scan

3 Basic Properties of Rough Dilation and Rough Erosion

In this section, we consider some topological properties based on rough dilations and rough erosions. In a topological space

Lemma 1. Let X and Y be image sets in

Proposition 1. Let X be an image set and

Proof. Let

Proposition 2. Let

Proof. From Proposition 1, it is easy to prove (i) and (ii).

Let

Let

Proposition 3. Let X be an image set and

Proof. Let

Proposition 4. Let

Proof. From Proposition 3, it is easy to prove (i) and (ii).

Let

Remark 1. One can easily note that

Example 5. Move and scan

Proposition 5. The following properties hold

Proof. It is sufficient to prove only (i), as (ii) holds by similarity. As

Corollary 1. The following properties are held,

If

If

Proposition 6. The following properties hold

If

If

Proof. As

Corollary 2. The following properties hold

If

If

Proposition 7. Let X and Y be image sets. The following hold:

Proof. By Propositions 1 and 3, proofs of (i) and (ii) are obvious.

Since

Since

Since

Let

Let

Remark 2 shows that the equalities do not hold in general. This can be seen from Examples: example 6, example 7 and example 8.

Remark 2. Let X and Y be image sets in

(i)

(ii)

(iii)

(iv)

(v)

(vi)

Example 6. Let B in Fig. 7b move and scan in

Example 7. Let B in Fig. 8b move and scan in

The property (iv) in Remark 2 can also be seen to be satisfied in Example 8.

Example 8. Let B in Fig. 9b be a move and scan in

Proposition 8. Let X and Y be images in

Proof. Since

Proposition 9. Let X and Y be images in

Proof. Since

4 Application of Rough Morphological Structures for Differential Analysis of Chest X-ray Images

One of the symptoms of severe SARS-CoV-2 coronavirus diseases [38] is the development of pneumonia and acute respiratory distress syndrome (ARDS) [39]. Admitted patients suspected of such severe COVID-19 disease typically undergo a radiological examination of the lungs for ARDS. While computed tomography offers the most sensitive and accurate imaging of lung condition [40,41], chest X-rays are often the front-line approach employed by many hospitals and can be performed using portable equipment [42], which can reduce patient movements and thereby lower the risk of infection [43–45]. The importance of chest X-ray imaging for the diagnosis of ARDS in COVID-19 pneumonia prompted us to examine whether rough morphological structures could be used to aid differential diagnostics.

Figs. 10 and 11 show two typical chest X-ray images, one from a health subject and another from a COVID-19 patient with ARDS.

Now we consider the rough boundary (rough opening and rough closing) using the original-rough opening and rough closing–original transforms, as shown in Figs. 12 and 13.

An algorithm for differential analysis of these two images is provided below: algorithm 1. Here, RD and RE are operators. The input image is the binary image of the chest X-ray (image 1; IM1) with size

The main steps for finding the rough opening (RO) and rough closing (RC) of the chest X-ray images are shown in the following flowchart at Fig. 14.

Although mathematical morphology and rough set theory are two different fields in terms of their initial domains and implementations, there are relations between the two systems as shown in this article. Specifically, we have shown that the lower and upper approximations of rough set theory are similar to opening/erosion and closing/dilation in mathematical morphology. This principle can be used to find similarity among images with a lower approximation. The topology of the partition can be defined in images as part of the universe set using four features defined using color and image indices. Subspace topologies can also be used to model each image type. We proposed an algorithm using these rough morphological operations that could be used to delineate lung occlusion (ARDS) in COVID-19 patients from chest X-ray images. In future work, we will add the detection accuracy measured.

Authors’ Contributions: All authors read and approved the final manuscript.

Availability of Data and Materials: The datasets used and analyzed during the current study are public and available from the corresponding author on request.

Funding Statement: The authors received no specific funding for this study.

Conflicts of Interest: The authors declare that they have no conflicts of interests to report regarding the present study.

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*74*

*(3)*, 6893-6908. https://doi.org/10.32604/cmc.2023.033539

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