An Accurate Disease Classification of COVID-19 and Pneumonia from Chest X-Ray Images Utilizing Mathematical Algorithm-Based Deep Learning Model

In early December 2019, there were reports of several patients in Wuhan City, China, who had pneumonia with unknown causes [1]. Some patients have exhibited severe acute respiratory syndrome (SARS) during the initial phases of this pneumonia, which has an unknown cause. However, it is important to note that only a small percentage of patients have exhibited symptoms of a rapid escalation in acute respiratory distress (ARD) disorder. Additionally, there have been reports of other worrisome complications as well. I'm sorry, but I need more context or information in order to provide a well-written response On January 7, 2020, the Chinese Centre for Disease Control and Prevention (CCDC) discovered a new strain of coronavirus (nCoV) from a throat swab sample collected from a patient in Wuhan. This virus was later designated as 2019-nCoV by the World Health Organisation (WHO) [2]. As of December 31, 2021, a global total of 284,992,606 cases of coronavirus infection have been reported, with 5,440,570 deaths and 252,735,264 recoveries [3]. The frontline experts used the real-time reverse transcription-polymerase chain reaction test as the first set for checking COVID-19 amid this tough time [4]. Reverse transcription method was applied to acquire the DNA from the individual infected. This DNA is subsequently subjected to PCR to amplify it before undergoing analysis. Hence, it is capable of detecting the coronavirus as this particular virus exclusively contains RNA patterns [5]. The results obtained from PCR kits for COVID-19 tests are currently being delayed due to increased demand. PCR kits are not considered reliable due to the presence of false-negative (FN) results [6]. Therefore, there is a need for alternative and robust diagnostic methods that can detect outbreaks of COVID-19 and pneumonia diseases at an early stage. Figure 1 illustrates a heat map showcasing the distribution of COVID-19 epidemic deaths, which have affected millions of people worldwide.

Chest X-rays are effective and cost-efficient diagnostic techniques that can detect outbreaks of COVID-19 and pneumonia diseases at an early stage. However, due to the significant similarities between COVID-19, viral pneumonia, and tuberculosis, it presents a challenge for physicians to distinguish between them solely based on chest X-rays [7-9]. Numerous research studies have been conducted to classify cases as either COVID-19 or healthy [10], COVID-19 or pneumonia [11], or COVID-19 or Tuberculosis (TB) [12]. However, these studies lacked the ability to accurately make a clear distinction between these diseases using chest x-rays.

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Figure 1. A heat map showcasing the distribution of COVID-19 over the worldwide [7]

Medical image classification is the act of image analysis and determination of its appropriate "category". This can be likened to a virtual machine designed to mimic human behavior. Classifying medical images has proven to be a challenging task for computers, despite its seemingly straightforward nature. Multiple classification systems were developed using various databases to determine the category to which the detected image belongs. These systems were then used to assess the accuracy percentage [13, 14].

In classification models for COVID-19 and pneumonia, the imaging techniques occasionally produce images that contain artefacts, exhibit low contrast, or fail to clearly depict the boundaries of the visceral organs. As a result, these challenges can impact the accuracy of a classification model's design [15]. Therefore, it is necessary to do some preprocessing such as resizing, normalisation, and augmentation on these types of medical images [16-18].

Mathematical algorithms, such as integral differentials and reference-based methods, have been used to classify medical images. However, these algorithms did not achieve the optimal resolution [19]. Fractional partial differential algorithms (FrPDAs) have been used as a new image processing technique in the medical sector. Fractional differentials, a theory of arbitrary order, bear similarities to integral differentials [20]. The use of fractional differentials in the processing of images has been found to improve edges, increase the visibility of texture details, and retain smooth areas in comparison to conventional integral differential methods [21].

The contrast and clarity of images that have undergone fractional differential processing are mostly enhanced. The conventional fractional differentials manage edges, textures, and smooth areas in an image by using the same fractional order even though this strategy has some drawbacks. It is possible to use higher fractional orders to efficiently improve edges, but they frequently overlook smooth sections and weak textures. Regarding the lower fractional orders, they can weaken the edges while preserving smooth sections and weaker textures. Therefore, achieving image enhancement in practice is challenging and to address these challenges, researchers have developed both conventional and enhanced fractional differential (FD) frameworks for usage in the processing of digital images [22, 23]. In earlier research [24], the application of adaptive fractional derivatives to solve image denoising issues was investigated. Sharma et al. [21] proposed the operator YiFeiPu-2, which has shown to have greater convergence and precision, as well as six fractional differential masks. The mentioned differential operators have fixed fractional orders determined by humans. Consequently, modifying the area features of the image would not optimise image enhancement.

Furthermore, numerous studies have classified COVID-19 and related conditions using Convolutional Neural Networks (CNN) [25], Recurrent Neural Network - Long Short Term Memory (RNN-LSTM) [26], Visual Geometry Group (VGG-16) [27]. VGG-16 is a highly regarded CNN model in the field of computer vision, widely recognised as one of the most effective models available. It is specifically designed for tasks involving classification and localization. The VGG-16 model is renowned for its utilisation of 16 layers with weights, making it one of the most highly regarded vision model architectures currently available. Like many other popular networks such as GoogleNet and AlexNet, there are several well-known networks in the field of computer vision [28].

This study introduces a new robust model called FPD-VGG-16, which combines a deep learning technique known as VGG-16 with a mathematical method known as Fractional Partial Differential (FrPDA). The proposed model accurately detects and classifies pneumonia and COVID-19 from chest X-ray images. Fractional differentiation has started to play a very important role in various image and signal processing research fields. In image processing, fractional calculus can be rather interesting to improve edges, increase the visibility of texture details, and retain smooth areas in comparison to using only deep algorithms, which led the proposed model to high accuracy while maintaining the model's training complexity.

The paper has been organized thus: the investigation of the previous works is presented in Section 2 while Section 3 of this study presents a comprehensive overview of both the fractional difference algorithm and the VGG-16 network. Furthermore, this section discusses the datasets that were utilized and the performance metrics used to evaluate the performance of the proposed model. In Section 4 of the paper, an in-depth discussion is presented regarding the experiments conducted and the subsequent comparisons made. Finally, in Section 5, the study’s conclusion is made.

This section covers the techniques and phases of the suggested methodology that will be used to achieve the main objectives of this study. The used datasets, as well as the experimental environment, have been indicated.

3.1 An overview of the modified FrPDA

Fractional calculus is a branch of mathematics that generalizes the standard definitions of integral and derivative operators in the same upper line as do fractional powers allow to generalize the concept of exponent for real numbers. Fractional differential equations have become more popular in recent years as a powerful and organized mathematical tool for investigating various phenomena in the fields of science and engineering. Research in fractional differential equations spans across multiple disciplines and finds applications in a wide range of fields. These include continuum mechanics fluid mechanics, control systems, circuit systems, heat transfer, elasticity, electric drives, signal analysis, quantum mechanics, biomathematics, biomedicine, and social systems, bioengineering [37, 38].

To date, a universally accepted formula for defining fractional calculus has not yet been developed. Different definitions of fractional calculus have been developed as a result of the thorough analysis of the issue from numerous angles by mathematicians. The three traditional definitions of fractional calculus are the R-L, Capotu, and G-L definitions [39]. For the processing of medical images, we used the G-L definition, as it is less complex and requires only one coefficient. First, second and third order derivatives of f(t) were obtained by means of the L’Hospital rule:

$f^{\prime}(r)=\lim _{g \rightarrow 0} \frac{f(r+h)-f(r)}{g}$      (1)

$f^{\prime \prime}(r)=\left[f^{\prime}(t)\right]^{\prime} \lim _{g \rightarrow 0} \frac{f(r+2 h)-2 f(r+h)+f(r)}{g^2}$       (2)

$f^{\prime \prime \prime}(r)=\left[f^{\prime \prime}(r)\right]^{\prime} \lim _{g \rightarrow 0} \frac{f(r+3 h)-3 f(r+2 h)+f(r)}{g^3}$    (3)

The n-th order derivative (denoted as nN) of function f(r) is derived mathematically as follows:

$f^{(n)}(r)=\lim _{g \rightarrow 0} g^{-n} \sum_{j=0}^n(-1)^j\left(\begin{array}{l}n \\ j\end{array}\right) f(r-j g)$      (4)

The gamma function generates the fractional order, ranging from integer to fraction. The v-order FD of function f(t) is described as the derivative of order (n+1) on the interval [a, b], where function f(r) has (n+1)-order derivatives.

$a D_b^v f(r)=\lim _{g \rightarrow 0} r^{-v} \sum_{j=0}^{[(b-a) / r]}(-1)^j\left(\begin{array}{c}v \\ j\end{array}\right) f(r-h g)$   (5)

where, the integer part of $\frac{b-a}{r}$ is $\left[\frac{b-a}{r}\right]$ and $\left(\begin{array}{l}v \\ j\end{array}\right)=\frac{v^i}{f^{!}(v-j)^{!}}$is binomial coefficient.

3.2 An overview of VGG-16 networks

One of the most heavily used deep learning CNN for vision tasks is the VGG-16 network [40]. VGG-16 is actually a specific implementation of the regular VGG network, but the difference is that if has a total of 16 layers (3 fully linked & 13 convolutional layers). Yet, the VGG-16 network has a specific outline as architecture. Initially, the input to be received by a VGG-16 network is a 224×224 image, which was kept standard as a result of cropping a 224×224 section from the center of each image in the ImageNet dataset. The receptive field on VGG is 3×3 which is used by the convolution filter is the smallest. A 1×1 convolution filter is also used by VGG to linearly transform the input. Next, a distinct linear function is applied, which is known as Rectified Linear Unit Activation Function to return an identical response on the input; while on the other hand, it would be a zero output for negative inputs. As a result of the fact that AlexNet, could be a classic CNN architecture and a Local Response Normalisation increases the time taken for training and the amount of memory used, and on the other hand also, it is not used by the hidden layers of VGG: the activation function for the hidden layer VGG is ReLu. Following the Convolution Phases, additionally, a Pooling layer are appended, which is used to reduce the dimensionality of the feature map, and it is due to reasons to reduce the parameters amount present in feature maps. Hence, a pooling layer is necessary, considering there were increasing filters for 64, then 128, and 256, and in the posterior levels 512 filters. You may also mention the fact that the VGG is made up of three interconnected layers with the first two combining 4096 channels:

In the $l$-th layer, we will consider transforming its inputs $x^l$, which form an order 3 tensor $x^l \in \mathbb{R}^{H^l \times W^l \times D^l}$. Where $x^l$ is the i-th row and j-th column and k-th depth of $x^l$ (in other words of $x^l$ is an element ${x}_{i,j,k}$ of $x^l$ ), it'll be typically be useful for us to know the triplet index set $((\mathrm{i}, \mathrm{j}, \mathrm{k}))$; i.e., the triplet $\left(i^l, j^l, d^l\right)$ with $0 \leq i^l<H^l, 0 \leq j^l<W^l$, and $0 \leq d^l<D^l$ that locates this particular element in $x^l$. Later in VGG training we'll use a kind of "minibatch" strategy where $N>1$, and so $x^l$ will typically become an order 4 tensor in $\mathbb{R}^{H^l \times W^l \times D^l \times N}$ where $N$ is the mini-batch size. But, for seriously issues sometimes it can be helpful to consider $N=1$. It turns out that in essentially all uses of our output tensor variables, we'll also use the zero-based indexing convention, so that a size $H^{l+1} \times W^{l+1} \times D^{l+1}$ output tensor is indexed by the triplets $\left(i^{l+1}, j^{l+1}, d^{l+1}\right), 0 \leq i^{l+1}<H^{l+1}, 0 \leq j^{l+1}<W^{l+1}$, $0 \leq d^{l+1}<D^{l+1}$.

In the $l$-th layer, our goal will be to transform the input $x^l$ to an output $y$, where $y$ and $x^{l+1}$ will refer to the same object. Thus, we expect $y$ to be of size $H^{l+1} \times W^{l+1} \times D^{l+1}$ (in other words, when we say that $y$ is the output of the $l$-th layer, we mean that it's the input to the (1+1)-st layer). An element in our output $y$ will be indexed by a triplet $\left(i^{l+1}, j^{l+1}, d^{l+1}\right), 0 \leq i^{l+1}<H^{l+1}, 0 \leq j^{l+1}<W^{l+1}$, $0 \leq d^{l+1}<D^{l+1}$.

3.2.1 The ReLU layer

The input size in the ReLU layer remains unchanged, and x^l and y have the same size. Additionally, there is no need for parameter learning. Sometimes, ReLU is considered a truncation that was performed separately for every component of the input:

$y_{i, j, d}=\max \left\{0, x_{i, j, d}^l\right\}$         (6)

where, $0 \leq i<H^l=H^{l+1}, 0 \leq j<W^l=W^{l+1}$, and $0 \leq d<D^l=D^{l+1}$.

From the above equation, it is obvious that:

$\frac{\mathrm{d} y_{i, j, d}}{\mathrm{~d} x_{i, j, d}^l}=\left[\left[x_{i, j, d}^l>0\right]\right]$    （7）

where, $\left[\left[x_{i, j, d}^l>0\right]\right]$ represents the indicator function (having a value of 1 or 0 if the argument is true or false respectively).

Hence,

$\left[\frac{\partial z}{\partial x^l}\right]_{i, j, d}=\left\{\begin{array}{cl}{\left[\frac{\partial z}{\partial y}\right]_{i, j, d}} & \text { if } \boldsymbol{x}_{i, j, d}^l>0 \\ 0 & \text { otherwise }\end{array}\right.$.   （8）

Noting that $y$ is an alias for $x^{l+1}$.

Note that the max (0, x) function cannot be differentiated at x=0, hence, Eq. (4) is somehow theoretically problematic. Therefore, it is not suitable in practice and as such, ReLU is safe to use.

3.2.2 The convolution layers

Using the following assumptions:

An input to layer l in the convolutional layers is a tensor of order 3 having size $H^l \times W^l \times D^l$.

A kernel of convolution is again a tensor of order 3 and its size is H×W×D^l.

Overlapping of the kernel at spatial position (0, 0, 0) on the input tensor makes the kernel slide over the input tensor generating product of related elements at all D^l channels and adding up the HWD^l products generates the result of convolution at the related spatial position; the kernel is moved to the bottom, left-to-right and alternated again to complete this process.

A simple case is considered in this section where no padding is used and the stride is 1 . Hence, we have $y$ (or $x^{1+1}$ ) in $\mathbb{R}^{H^{l+1} \times W^{l+1} \times D^{l+1}}$, with $H^{l+1}=H^l-\mathrm{H}+1, \quad W^{l+1}=W^l-\mathrm{W}+1$, and $D^{l+1}=\mathrm{D}$.

The mathematical expression of the convolution process is given thus:

$\begin{aligned} & y_{i^{l+1}}, j^{l+1}, d  =\sum_{i=0}^H \sum_{j=0}^W \sum_{d^{l=0}}^{D^l} f_{i, j, d^l, d} \times x_{{i^{l+1}}+{i, j}^{l+1}}+j, d^l \\ & \end{aligned}$    (9)

A repeat of this equation is performed for all 0≤d≤D=D^l+1, as well as for any other spatial location $\left(i^{l+1}, j^{l+1}\right)$ that satisfies the condition $0 \leq i^{l+1}<\bar{H}^l-H+1=H^{l+1}, 0 \leq j^{l+1}<$ $W^l-W+1=W^{l+1}$. The notation $x_{i^{l+1}+i, j^{l+1}+j, d^l}$ in the equation describes the component of $x^l$ indexed by the triplet $\left(i^{l+1}+\mathrm{i}, j^{l+1}+\mathrm{j}, d^1\right)$.

Even though a bias term $b_d$ is often added to $y_{i^{l+1}, j^{l+1}, d}$, this term was omitted in this case for better presentation.

Let’s now consider a pooling layer where the input to the l-th layer is $x^l \in \mathbb{R}^{H^l \times W^l \times D^l}$. The pooling operation has no parameters, so it doesn’t need to worry about learn any w’s. So, the w’s are going to be 0, and in this layer what we need to specify is the shape of the pooling layer. So, in the architecture of a ConvNet the dimensionality of a volume goes through a series of changes and in the very end it is Ho×Wo×Do, where we C the depth dimension). Let’s say that the spatial extent of the pooling is H×W. More so, the pooling operates with a stride of S, where H divides H^l and W divides W^l, then the output of the pooling O(l) will be an order 3 tensor and the size will be H^l+1×W^l+1×D^l+1, with:

$H^{l+1}=\frac{H^l}{H}, W^{l+1}=\frac{W^l}{W}, D^{l+1}=D^l$      (10)

The operation of a pooling layer upon x^l is independently channel-wise. The matrix with H^l×W^l elements in each channel is partitioned into H^l+1×W^l+1 sub-regions that cannot overlap, and the size of each of the sub-region is H×W. Then, a sub-region is mapped by the pooling operator into a single number.

Average pooling and max pooling are the two commonly used types of pooling operators. For the average pooling, a sub-region is mapped by the pooling operator to its average value while in the max pooling, a sub-region is mapped by the pooling operator to its maximum value.

Mathematically,

max:

$y_{{i^{l+1}}, j^{l+1}, d}=\max _{0 \leq i<H, 0 \leq j<W} x_{i^{l+1} \times H+i, j}^{l+1} \times W+j, d^{\prime}$    (11)

Average:

$y_{i^{l+1}, j^{l+1}, d} \frac{1}{H W} \sum_{0 \leq i<H, 0 \leq j<W} x_{i^{l+1} \times H+i, l^{l+1} \times W+i, d^{\prime}}^l$      (12)

where, $0 \leq i^{l+1}<H^{l+1}, 0 \leq j^{l+1}<W^{l+1}$, and $0 \leq d<D^{l+1}=D^l$.

As a local operator, forward computation of pooling is a straightforward task; hence, attention is given to the back propagation and discussion is only focused on max pooling; let’s revisit the indicator matrix. For this indicator matrix, all that needs to be encoded is: for every element in y, where does it come from in $x^l$. A triplet $\left(i^l, j^l, d^l\right)$ is needed to identify one component in the input $x^l$, while another triplet $\left(l^{l+1}, j^{l+1}, d^{l+1}\right)$ is needed to find one component in $y$. The output $y_{i^{l+1}, j^{l+1}, d^{l+1}}$ of the pooling process is derivable from $x_{i^l, j^l, d^l}$, if and only if:

i. They are in the same channel;

ii. The $\left(i^l, j^l\right)$-th spatial entry belongs to the $\left(i^{l+1}, j^{l+1}\right)$-th subregion;

iii. The $\left(i^l, j^l\right)$-th spatial entry is the largest one in that sub-region.

A translation of these conditions gives the following equations:

$\left|\frac{i^l}{H}\right|=i^{l+1},\left|\frac{j^l}{W}\right|=j^{l+1}$     (13)

$\begin{aligned} & x_{i^l, j^l, d^l} \geq y_{i+i^{l+1} \times H, j+j^{l+1} \times W, d^l} \\ & \forall 0 \leq i<H, 0 \leq j<W\end{aligned}$       (14)

where, the floor function is given as $[\cdot]$. In the vertical or horizontal planes, if the stride is not $H(W)$, Eq. (11) must accordingly be altered.

Consider a given triplet $\left(i^{l+1}, j^{l+1}, d^{+1+1}\right)$ where only one $\left(i^l, j^l\right.$, $d^l$) triplet can satisfy the whole of these conditions, the indicator matrix can then be defined as:

$S\left(\boldsymbol{x}^l\right) \in \mathbb{R}^{\left(H^{l+1} W^{l+1} D^{l+1}\right) \times\left(H^l W^l D^l\right)}$   (15)

where, $S$ a row is specified by one triplet of indexes $\left(i^{l+1}, j^{l+1}\right.$, $d^{l+1}$ ) while a column is specified by $\left(i^l, j^l, d^l\right)$. Collectively, both triplets identify one component in $S\left(x^l\right)$ and that element is set to 0 if Eqs. (10) to (12) are not satisfied simultaneously, and 1 if satisfied. One row and one column of $S\left(x^l\right)$ corresponds to one element in $y$ and one element in $x^l$, respectively.

Considering this indicator matrix, it applies that:

$\operatorname{vec}(\boldsymbol{y})=S\left(\boldsymbol{x}^l\right) \operatorname{vec}\left(\boldsymbol{x}^l\right)$    (16)

Then, it is obvious that:

$\frac{\partial \operatorname{vec}(\boldsymbol{y})}{\partial\left(\operatorname{vec}\left(\boldsymbol{x}^l\right)^T\right)}=S\left(\boldsymbol{x}^l\right), \frac{\partial z}{\partial\left(\operatorname{vec}\left(\boldsymbol{x}^l\right)^T\right)}=\frac{\partial z}{\partial\left(\operatorname{vec}(\boldsymbol{y})^T\right)} S\left(\boldsymbol{x}^l\right)$     (17)

and consequently,

$\frac{\partial z}{\partial \operatorname{vec}\left(x^l\right)}=S\left(x^l\right)^T \frac{\partial z}{\partial \operatorname{vec}(\boldsymbol{y})}$ （18）

$S\left(x^l\right)$ is highly sparse and its entry in every row is exactly one non-zero entry. Hence, the entire matrix is not going to be used in the computation, rather, the locations of those nonzero entries are to be recorded - there are only $H^{l+1} W^{l+1} D^{l+1}$ such entries in $S\left(x^l\right)$.

The meaning of these equations can be explained using a simple example. Assume a $2 \times 2$ max pooling with a stride value of 2; the spatial subregion for each given channel $d^l$ has 4 elements in the input, where $(i, j)=(0,0),(1,0),(0,1)$ and $(1$, $1)$;also assume that the largest element among all the elements is the one at spatial location $(0,1)$. Then, in the forward pass of the input, the indexed value by $\left(0,1, d^l\right)$, i.e., $x_{0,1, d^l}^l$, is going to be allocated to the element occupying the $\left(0,0, d^l\right)$-th position in the output (i.e., $y_{0,0, d^l}$ ).

It is expected that in $S\left(x^l\right)$ one column can only contain a maximum of one non-zero entry if the strides are H & W. In the considered case, the $S\left(x^l\right)$ columns indexed by (0, 0, d^l), (1, 0, d^l) & (1, 1, d^l) are all 0 vectors. The (0, 1, d^l) column only contains one non-zero entry and the row index of this entry is only determined by (0, 0, d^l). Therefore, the back-propagation process will give:

$\left[\frac{\partial z}{\partial \operatorname{vec}\left(x^l\right)}\right]_{\left(0,1, d^l\right)}=\left[\frac{\partial z}{\partial \operatorname{vec}(y)}\right]_{\left(0,0, d^l\right)}$       （19）

$\left[\frac{\partial z}{\partial \operatorname{vec}\left(\boldsymbol{x}^l\right)}\right]_{\left(0,0, d^l\right)}=\left[\frac{\partial z}{\partial \operatorname{vec}\left(\boldsymbol{x}^l\right)}\right]_{\left(1,0, d^l\right)}=\left[\frac{\partial z}{\partial \operatorname{vec}\left(\boldsymbol{x}^l\right)}\right]_{\left(1,1, d^l\right)}=0$     （20）

But if the pooling strides are comparatively less than H in the y-axis and less than W in the x-axis, then, the largest element in most of the pooling sub-regions could be one element in the input tensor. Therefore, one column of S(x^l) may have more than one non-zero entries. If the stride is 1 in both directions and a 2×2 max pooling is applied, the element 9 will become the largest entry in 2 different pooling sub-regions as follows: $\left[\begin{array}{ll}5 & 6 \\ 8 & 9\end{array}\right] \&\left[\begin{array}{ll}6 & 1 \\ 9 & 1\end{array}\right]$. Hence, there are 2 non-zero entries in the S(x^l) column that corresponds to the element 9 (indexed by (2, 2, d^l) in the input tensor), with row indexes corresponding to (i^l+1, j^l+1, d^l+1)=(1, 1, d^l) & (1, 2, d^l). Therefore, this example provides that:

$\left[\frac{\partial z}{\partial \operatorname{vec}\left(x^l\right)}\right]_{\left(2,2, d^l\right)}=\left[\frac{\partial z}{\partial \operatorname{vec}(y)}\right]_{\left(1,1, d^l\right)}+\left[\frac{\partial z}{\partial \operatorname{vec}(y)}\right]_{\left(1,2, d^l\right)}$   (21)

3.3 Proposed methodology

Image enhancement is a process that typically uses several algorithms to enhance the contrast and precise details of an image. The outcome of this research is going to be a new and effective model that will be FPD-VGG-16. The proposed model exhibits a high degree of accuracy in pneumonia and COVID-19 detection and classification using chest X-ray images. This goal will require the selection of a dataset appropriate for this activity before it can be achieved. To ensure the accurate identification of both pneumonia and COVID-19, two separate datasets are utilized in the new model. Each dataset corresponds to a specific disease infection and consists of Chest X-ray data. Figure 2 illustrates the acquisition of two separate datasets: a pneumonia dataset and a COVID-19 dataset. Both datasets undergo a data pre-processing stage to prepare them for input into the proposed FPD-VGG-16 model. Once the PD-VGG-16 model is created, it is trained using pre-processed data to learn how to accurately identify COVID-19 and Pneumonia. Following that, the FPD-VGG-16 network undergoes testing using new data to assess its effectiveness in accurately classifying the infection.

2.png

Figure 2. General diagram of the proposed FPD-VGG-16 classification model

3.3.1 Datasets description

Two different datasets were chosen for this study, particularly one for Pneumonia and one for COVID-19. The Dataset containing Pneumonia images is actually called the "Chest X-ray" dataset, and it was obtained from the Mendeley Data repository, comprising 5856 images in total [41]. Chest X-ray datasets normally contain two types/classes of data, namely the normal class (NC) and the pneumonia class (PC), where each class contains several images that are partitioned into training, testing, and validation sets. The NC category contains 1341 training images, 8 validation images, and 234 testing images, whereas the PC contains 3875 training images, 8 validation images, and 390 testing images.

On the one hand, the pictures in the “CovidX” dataset were utilized to train the FPD-VGG-16 model in detecting and classifying cases of COVID-19 infection from chest X-ray images. The CovidX dataset has 1300 images or 13975 chest x-ray images. These images assist in identifying whether the diagnosis of COVID-19 is accurate. This dataset is one of the most extensive chest X-ray datasets for diagnosing COVID-19, and it is publicly available. The CovidX data is compiled from five data sets, all of which are publicly available [42]. Table 1 shows the summary of the distribution of the selected datasets.

Furthermore, Figure 3 displays a compilation of chest X-ray images retrieved from selected databases. It demonstrates both normal chest X-ray images and a sample of a diseased chest X-ray, specifically from the COVID-19 class.

3.3.2 Data pre-processing

There are three main stages applied in these current studies before the implementation of any machine learning or deep learning model, and the first one is data pre-processing, and this is the most vital stage that must be conducted. The programmer should ensure that the data collected has a high quality to implement the model during the pre-processing.

The first pre-processing step is the merging of the two datasets (Chest X-ray and COVIDX). The merging took place by appending the datasets and assigning labels to each image. Thus, the merged dataset would contain three labels: Normal, COVID-19, and Pneumonia. The following equation represents the merging procedure. Eq. (22) illustrates this action.

merged dataset $=$ pneumonia dataset + COVIDx dataset        (22)

The second step involves resizing the images in order to achieve an identical dimension for all of them. The chosen size is 224×224 pixels. Image resizing is necessary because if the images have different sizes, it can have a negative impact on the performance of the deep learning model. Image resizing ensures that the inputs are consistent. Eq. (23) represents the procedure for image resizing.

resized image $=\operatorname{resize}($ image,$(224,224))$      (23)

The final step involves data Normalization, which is necessary to ensure that the pixel values are within an equivalent range. During the normalization step, it is standard practise to ensure that the pixel values of the data have a mean value of zero and a standard deviation of one. Eq. (24) represents the normalization step.

normalized image $=($ image - mean $) /$ standard deviation      (24)

After the pre-processing procedure is finished, the data is ready to be fed to the model for training.

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3.3.3 Training environment

The training of the model was done using the Adam Optimizer [43] and the categorical cross-entropy loss function [44]. During the training process of the model, recall, precision, accuracy, and F1-score are used as performance metrics to evaluate performance. In addition, callbacks like Model Check point [45], Early Stopping [46], and ReduceLROnPlateau [47] functions are employed to mitigate the issue of overfitting. The purpose of Model Check point is to save the model with the highest performance achieved during training. The Early Stopping function is responsible for continuously monitoring the training accuracy. If the accuracy does not show improvement over a specified number of epochs, the training process is halted. The ReduceLROnPlateau function is responsible for decreasing the learning rate if there is no improvement in accuracy after certain numbers of epoch.

In this scenario, the patience parameter is set to 1. This means that if the accuracy does not improve after 1 epoch, the learning rate will decrease. Furthermore, the stop patience parameter is set to 3, indicating that the training will halt if the accuracy does not improve after 3 epochs. The factor parameter is now set to 0.5, which indicates a decline in the learning rate by a factor of 0.5 if there is no improvement observed after one epoch.