## 4 Results and Discussion
### Time Complexity and Order of the Proposed Method  
We can determine the efficiency of the proposed algorithm using time complexity. The complexity is computed for each operation involved in the algorithm using Big-Oh notation. This technique measures the maximum amount of time taken by the algorithm. In HOG feature extraction, the calculations are performed on each cell and block for each image. It involves various stages of operations having the complexity of O (\(p^{2}\)), where p is the number of pixels of an image. The algorithm consists of computing gradients and orientations for each pixel, which requires iterating over all pixels in the image. For CBAM-based CNN, the time complexity is O \(\left(\sum_{i=1}^{L}S_{L}^{2}*K_{L}^{2}*C_{L-1}*C_{L}\right)\), where S, K, and C are the dimension of the feature maps, convolutional kernel size, and channel count for  
Figure 9: Training and validation accuracy and loss curves for both datasets: (**a**) Massey ASL, (**b**) ISL dataset. each convolutional layer L. The batch normalization layer, max pooling layer, and CBAM module layer have a time complexity of O (1) hand and have minor operations that do not depend on the size of the input feature. The flattening layer has an O (n \(\ast\) m) complexity. The concatenation layer is a linear operation with an O (n) complexity. The fully connected layer has the complexity of \(O\left(\sum_{i=1}^{f}F_{f}\right)\), where \(F_{f}\) is the number of neurons and f is the number of fully connected layers. The last classification layer has complexity O (\(k\ast h\)), where k is the total sign classes and h is the neuron count in the hidden layer [35]. So, we can express the overall complexity as  
\[O\left(p^{2}\right)+O\left(\sum_{i=1}^{L}S_{L}^{2}\ast K_{L}^{2}\ast C_{L-1} \ast C_{L}+\sum_{i=1}^{f}F_{f}\right)\ +O(k\ast h) \tag{13}\]  
Figure 10: Confusion matrix for classification of Massey ASL database. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Data** & **Year** & **Method (Year)** & **Accuracy** (\%)** & **Training Parameters** & **Training Epochs** & **Inference Time (s)** \\ \hline  & [30] & 2-level ResNet-50 (2020) & 99.03 & 25,636,712 & 200 & 0.025 \\ \hline Massey & [31] & VGG16+Attention (2022) & 98.02 & 138.35 M & 30 & - \\ \hline  & [32] & Canny + HOG+ KNN (2022) & 97.6 & - & - & 0.39 \\ \hline  & [33] & Spatial feature learning network (2023) & 80.44 & 3.8 M & - & 2.23 \\ \hline  & **Proposed** & **HOG+ CBAM-based CNN** & 99.22\% & 2.3 M & 30 & 0.023 s \\ \hline Static & [34] & Vision Transformer based approach (2023) & 99.29\% & 7 M & 5 & - \\ \hline  & [21] & Ensemble (ResNet50 +Attention) (2022) & 98.20\% & 23.5 M & 50 & - \\ \hline  & **Proposed** & **HOG+ CBAM-based CNN** & 99.79\% & 2.3 M & 30 & 0.023 s \\ \hline \hline \end{tabular}
\end{table} 
Table 3: State-of-the-art comparison of Massey ASL and ISL dataset. We have discussed the comparison of time complexity with other algorithms. The time complexity of two-level ResNet50 [30] is \(2*\text{O}\) (\(\sum_{i=1}^{L}S_{L}^{2}*K_{L}^{2}*C_{L-1}*C_{L}+\sum_{i=1}^{f}F_{f}\))+ \(\text{O(k}*\text{h)}\) since there are two branches of the CNN model for feature extraction for each layer of the model. In [31], the authors have used fused features of pre-trained VGG16 and attention-based VGG16. The model has a computational complexity of \(2*\text{O}\) (\(\sum_{i=1}^{L}S_{L}^{2}*K_{L}^{2}*C_{L-1}*C_{L}+\sum_{i=1}^{f}F_{f}\)) +\(\text{O(k}*\text{h)}\). The study [32] involves canny edge detection, HOG feature extraction, and KNN for classification, which has an overall time complexity of O (m \(*\) n log m \(*\) n) + \(\text{O}\) (\(n^{2}\)) + \(\text{O}\)(d \(*\) F), where m \(*\) n is the image size, d, and F is the number of data points and features, respectively. In [34], authors have used a vision transformer encoder with a time complexity of O (\(8*s^{2}*\text{d}\)) + \(\text{O}\) (\(8*s*d^{2}\)), where s is the sequence length and d is the depth.