## 4 Methodology
### Overview of the Deep Learning Models Used for Classification
#### 4.2.3 Xception  
The Inception architecture, renowned for its innovative Inception module, has significantly advanced neural network designs by enabling the capture of features at various scales. Developed with the goal of achieving both depth and width in networks without compromising computational efficiency, it introduced parallel pooling and convolutions of different sizes. Xception, or Extreme Inception, evolved from this model by employing depthwise separable convolutions, thereby enhancing the network's ability to learn spatial features and channel correlations more effectively [62]. This technique comprises two stages: depthwise convolution and pointwise convolution.  
In the first stage, depthwise convolution, a single filter is applied per input channel. Mathematically, for an input \(X\) of size \(W\times H\times D\) (width, height, depth), the depthwise convolution is defined as:  
\[Y_{i,j,d}=\sum_{m=0}^{k-1}\sum_{n=0}^{k-1}X_{i+m,j+n,d}\cdot F_{d,m,n}, \tag{1}\]  
where \(Y_{i,j,d}\) represents the output feature map, \(k\) is the filter size, \(F_{d,m,n}\) is the filter for depth \(d\), and \((i,j)\) are the spatial coordinates in the feature map. This step involves convolving each input channel separately with a corresponding filter, preserving the depth dimension, but reducing the spatial dimensions.  
In the second stage, pointwise convolution, the features obtained from the depthwise convolution are combined channelwise. This is mathematically expressed as:  
\[Z_{i,j,d^{\prime}}=\sum_{d=0}^{D-1}Y_{i,j,d}\cdot F^{\prime}_{d^{\prime},d^{ \prime}}, \tag{2}\]  
where \(Z_{i,j,d^{\prime}}\) is the final output after the pointwise convolution, \(D\) is the number of input channels, and \(F^{\prime}_{d^{\prime},d}\) is the pointwise filter. This operation effectively combines the information across different channels, allowing the model to learn more complex patterns.  
The Inception module's output, \(I(x)\), incorporates various convolutional and pooling operations as follows:  
\[I(x)=[f_{1\times 1}(x),f_{3\times 3}(x),f_{5\times 5}(x),f_{pool}(x)], \tag{3}\]  
where \(f_{1\times 1}(x)\), \(f_{3\times 3}(x)\), \(f_{5\times 5}(x)\), and \(f_{pool}(x)\) represent the outputs of the \(1\times 1\), \(3\times 3\), and \(5\times 5\) convolutions and the pooling operations, respectively. This structure allows the network to capture features at multiple scales, contributing to its robustness and effectiveness in complex image-recognition tasks.  
By refining the convolutional approaches, the Xception model demonstrates the power of deep learning techniques in achieving a more nuanced understanding of image data,  
Figure 4: VGG19 architecture.  
enhancing both parameter efficiency and overall performance in various computer vision applications. The Xception architecture is comprised of 3 sections, the entry flow, middle flow, and exit flow, highlighted in Figure 5. Here, the entry flow leverages separable convolutions and skip connections. This is followed by the middle flow, which repeats 8 times prior to the exit flow. Finally, there are another skip connection and separable convolution with a global max pooling.