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.

Pawlak’s rough set theory [

Differential equations and diffusion equations may both be used to simulate many real-world issues in the fields of science and engineering [

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 [

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

Note that

Dilation and erosion are basic concepts in mathematical morphology and image processing, where any image set

It is clear that

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

Let

Let

Let

If

If

If

If

If

If

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.

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

One of the symptoms of severe SARS-CoV-2 coronavirus diseases [

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

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

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.

Rough Sets

Structure Element

Rough Dilation

Rough Erosion

Rough Closing

Rough Opening

Positive integers

Discrete topological space

D-dimensional product of E

Images as sets of pixels

Structure element

Topological space

Closure of X

Interior of X