Computers, Materials & Continua DOI:10.32604/cmc.2022.020384 | |

Article |

A Study of Cellular Neural Networks with Vertex-Edge Topological Descriptors

1Country College of Computer Science & Information Technology, Jazan University, Jazan, Saudi Arabia

2Department of Mathematical Sciences, United Arab Emirates University, Al Ain, United Arab Emirates

3Deanship of E-learning and Information Technology, Jazan University, Jazan, Saudi Arabia

*Corresponding Author: Muhammad Imran. Email: imrandhab@gmail.com

Received: 22 May 2021; Accepted: 23 June 2021

Abstract: The Cellular Neural Network (CNN) has various parallel processing applications, image processing, non-linear processing, geometric maps, high-speed computations. It is an analog paradigm, consists of an array of cells that are interconnected locally. Cells can be arranged in different configurations. Each cell has an input, a state, and an output. The cellular neural network allows cells to communicate with the neighbor cells only. It can be represented graphically; cells will represent by vertices and their interconnections will represent by edges. In chemical graph theory, topological descriptors are used to study graph structure and their biological activities. It is a single value that characterizes the whole graph. In this article, the vertex-edge topological descriptors have been calculated for cellular neural network. Results can be used for cellular neural network of any size. This will enhance the applications of cellular neural network in image processing, solving partial differential equations, analyzing 3D surfaces, sensory-motor organs, and modeling biological vision.

Keywords: Cellular neural networks; degree; topological indices

Graph theory is a vast field and is used to solve real problems and natural phenomena. That’s why it has applications in molecular chemistry, robotics, physics, networks, computer science, statistics, biological activities, and data science. It represents real scenarios in the graph based on vertices and edges.

A topological descriptor is a single value that characterizes the whole graph [1–3]. In chemical graph theory, they are used to estimate biological activities and atomic movements [4–6]. The first topological descriptor Wiener index was introduced by Wiener in 1947 [7]. Hosoya polynomial, Schultz index, atom bond connectivity, geometric-arithmetic index are other famous topological descriptors [4,8,9]. Topological descriptors can be categorized on the basis of the mechanism of calculation involved [10–17]. Nowadays vertex-edge topological descriptors are gaining importance in applied sciences [18].

Let

The

The Cellular Neural Network (CNN) is an array of cells that are interconnected locally. It is an analog paradigm with various applications including image processing, parallel processing, and high-speed computations. Each cell has an input, an output, and its state. A cell can interact with neighbor cells only. A neighbor cell of a cell is in its radius. A neighborhood includes the cell itself and its eight neighboring cells [28,29]. Fig. 1, consists of two diagrams (a) and (b). Diagram (a), is a graph of two-dimensional CNN, in which a neighborhood is representing red and blue colors. Diagram (b), is the internal structure of the red cell of diagram (a), this cell can interact with all blue cells in this neighborhood. In this article, vertex-edge topological descriptors have been calculated for CNN. The results are generalized and can be used for CNN of any structure and size. This will enhance the applications of CNN in image processing, parallel processing, image processing, non-linear processing, geometric maps, high-speed computations, solving partial differential equations, analyzing 3D surfaces, sensory-motor organs, and modeling biological vision [30].

In 2018, new degree-based topological indices are considered and the analytical sharp bounds has been derived for neural networks in [31]. In 2019, Imran et al. [32] has been calculated the degree based topological indices of cellular neural network. Topology optimization and Zagreb connection indices for cellular neural network are studied in [30,33]. Recently in 2021, the comparison and analysis between the dominating topological indices are determined for the cellular neural network [34]. Some topological indices have been calculated for cellular neural network, and are prominence their importance, but still many topological indices have not been calculated. Their calculation will provide an analytical study of cellular neural network in different applications. The cellular neural network is also known as the strong product of two paths, which has applications in different area of research, see [35,36].

3 The Graph of Cellular Neural Networks

The Figs. 2 and 3 show cellular neural network for different values of

The cellular neural network can be arranged either linearly or in a sheet form. A cellular neural network of

We partitioned the edges of CNN, based on

In Tab. 5, we partitioned the vertices of CNN, based on

We partitioned the edge of CNN with respect to

Now we calculated

3.1 The

The values of

3.2 The First

The

3.3 The First

The edge partition of cellular neural network (CNN) with respect to

3.4 The Second

Using Tab. 6 we compute the second

3.5 The

Using Tab. 6 we compute the

3.6 The

Using Tab. 4 we compute the

3.7 The

Using Tab. 6 we compute the

3.8 The

Using Tab. 6 we compute the

3.9 The

Using Tab. 6 we compute the

3.10 The

Using Tab. 6 we compute the

4 Numerical and Graphical Representation

In this section, we determine the numerical values of the

As a machine learning algorithm, CNN can take image as an input, highlight image key features, and differentiate one image from the other [37]. CNN has powerful function-fitting capabilities and has great potential to study partial differential equations [38,39]. CNN has been widely used in analyzing 3D surfaces and facial images applications [40]. The design components underlying the implementation of the physiologically faithful retina and other topographic sensory organ models on CNN universal chips. The results obtained through the proposed technique can better understand the features and characteristics of CNN. It can enhance image processing applications, solve partial differential equations, analyze 3D surfaces, sensory-motor organs, and model biological vision. Combinations of CNN and artificial intelligence provide enhanced human-level performance to computer architectures [41]. This article is vital in the implementation of Human-level visual recognition.

The calculation of vertex-edge topological descriptors carried out in this research are specifically derived for cellular neural networks. However, with some modifications they can be applied in other fields like image processing, biological modelling, 3D surface analyzing, and complex imaging. The applicability of this research still needs to be validated for these fields to identify the full potential of this research. These open problems can be further studied to get full benefit of this research.

A cell in Cellular Neural Network (CNN), can communicate with neighbor cells only. Due to its architecture, it is used to manage hierarchical levels, and it has variety of applications. In applied sciences, graph theory provides different tools and methods to remedy real-world problems. To solve these problems, it represents them in the form of graphs. Topological descriptors in graph theory are used to study and characterize biological activities in the form of graphs. In this article, the vertex-edge topological descriptors have been calculated for the graphic representation of CNN. Results can be used for the CNN of any size. The proposed technique can enhance the CNN’s applications in image processing, parallel processing, non-linear processing, geometric maps representations, high-speed computations, solving partial differential equations, analyzing 3D surfaces, sensory-motor organs, and modeling biological vision.

Funding Statement: This research is supported by the University program of Advanced Research (UPAR) and UAEU-AUA grants of United Arab Emirates University (UAEU) via Grant No. G00003271 and Grant No. G00003461.

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

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