## 2 Materials and Methods
### Base Models' Training and Their Hyperparameter Optimization
DenseNet typically has fewer parameters than other deep CNN networks as it reuses the features across layers, resulting in a more efficient model. EfficientNet scales the network width, depth, and resolution and balances them with the help of compound scaling, as discussed in [30]. In the present study, the EfficientNet-B7 variant is chosen, as it turns out to be more accurate than the other versions of EfficientNet. Figure 4: Accuracy curves for training and validation of (**a**) ResNet-152, and (**b**) ResNet-200. Figure 5: Schematic diagram of core components of base models. (**a**) Skip connection; (**b**) Dense block; (**c**) MB Conv block. The mobile inverted bottleneck (MB) convolution block is the basic building block of EfficientNet-B7, which integrates the squeeze-and-excitation (SE) block with the inverted residual network, helping EfficientNet-B7 to emphasize the most relevant features. The schematic diagram of the MB convolution block is shown in Figure 5c. The chosen three base learners, provided with pre-trained ImageNet [33] weights are further trained with the chosen CT dataset. The probabilities predicted by them will be used in the next level of the proposed ensemble model. For this, the dataset is split into training, validation, and test sets by patients in the ratio of 70%, 15%, and 15%, respectively. For training, the input images are resized into different dimensions as per the requirements of the base models. Therefore, all the training images are resized into \(32\times 32\), \(224\times 224\), and \(256\times 256\) to feed into WRN-40, DenseNet-121, and EfficientNet-B7, respectively, followed by batch normalization and data augmentation. To obtain the best possible combination of hyperparameters for all three base models, a grid search approach along with 5-fold CV is employed. The hyperparameters, their reference ranges, and the chosen values through the grid search and 5-fold CV are given in Table 1. The initial learning rates are chosen as small values for better convergence. The batch size and number of epochs for all three models are selected as 32 and 200, respectively. Moreover, regularization techniques, such as weight decay, along with appropriate dropout rate are applied to avoid overfitting issues, which are indicated in Table 1. The growth rate, which indicates the number of feature maps added per layer within each dense block, and the compression factor, which determines the size of the feature maps that need to be reduced in the transition block, are two important parameters of DenseNet-121. The width coefficient and depth coefficient, the two fundamental parameters of EfficientNet-B7, which are chosen to be 1.5 and 2.5, respectively, represent the extent of scaling. As discussed in Section 2.1, the fracture class and the non-fracture class account for 47.72% and 52.27% of the total data, respectively, indicating the data are slightly imbalanced. To ensure that the model gives equal importance to both the classes during training, weighted binary cross-entropy (WBCE) is found suitable and chosen as the loss function, which is expressed as follows:  
\[L=-\frac{1}{N}\sum_{i=1}^{N}(\alpha_{1}y_{i}\log(\hat{y}_{i})+\alpha_{0}(1-y_{i })\log(1-\hat{y}_{i})) \tag{1}\] \begin{table}
\begin{tabular}{l c c c} \hline \hline
**Hyperparameter** & **Reference** & \multicolumn{3}{c}{**Chosen Value**} \\ \cline{3-4}  & & **WRN-40** & **DenseNet-121** & **EfficientNet-B7** \\ \hline Learning rate & [0.0001, 0.01] & 0.001 & 0.0001 & 0.0001 \\ Batch size & [16; 128] & 32 & 32 & 32 \\ Optimizer & - & Adam & SGD & Adam \\ Epoch & - & 200 & 200 & 200 \\ Model checkpoint & - & 15 epoch & 15 epoch & 15 epoch \\ Dropout rate & [0, 0.6] & 0.5 & 0.3 & 0.4 \\ Weight decay & [\(10^{-6},10^{-2}\)] & \(10^{-4}\) & \(10^{-4}\) & \(10^{-5}\) \\ Data augmentation & - & ✓ & ✓ & ✓ \\ Growth rate & [12; 48] & - & 32 & - \\ Compression factor & [0.1, 1.0] & - & 0.5 & - \\ Width coefficient & [1.0, 2.0] & - & - & 1.5 \\ Depth coefficient & [1.0, 3.0] & - & - & 2.5 \\ \hline \hline \end{tabular}
\end{table}