## 2 Proposed Ensemble Model for 8-Class BC Image Classification
### Pre-Trained DCNNs for Image Feature Extraction  
To create the multi-scale pooled image feature representation (MPIFR) representation, features are extracted from four resolutions (40\(\times\), 100\(\times\), 200\(\times\), and 400\(\times\)) of histopathological breast cancer images using four (4) state-of-the-art DCNN-based models (1) (Efficientnet-b0), (2) DenseNet201, (3) ResNet50, and (4) ResNet18. ResNet18 [17] : ResNet has three variants according to its number of layers: ResNet18, ResNet50, and ResNet101 with 18, 50, and 101 layers, respectively. A transfer learning algorithm based on ResNet has been successfully used to classify biomedical images. ResNet18, shown in Figure 5, was used for feature extraction in this paper. The deep neural network layers learn low or high-level features during training, whereas the ResNet layer learns residuals instead. EfficientNetB0 [19]: architecture and scaling method utilizes a compound coefficient to uniformly scale all depth, width, and resolution dimensions of the CNN using a set of fixed scale coefficients.In a principled manner, EfficientNet scales network width, depth, and resolution based on a single \(\delta\) compound coefficient as expressed in Equation (10). \[ \begin{split}\text{depth: }& d=\alpha^{\phi}\\ \text{width: }& w=\beta^{\phi}\\ \text{resolution: }& r=\gamma^{\phi}\\ \text{s.t. }&\alpha\cdot\beta^{2}\cdot\gamma^{2}\approx 2 \\ \alpha&\geq 1,\beta\geq 1,\gamma\geq 1\end{split}  \tag{10}\]  
For instance, in order to utilize 2N times more computational resources, the network depth can simply be increased by \(\alpha N\), the width by \(\beta N\), and the image size by \(\gamma N\), where \(\alpha\), \(\beta\), and \(\gamma\) are constant coefficients determined by a small grid search on the original small model. In order to capture more fine-grained patterns from a larger input image, the compound scaling method uses more layers to increase the receptive field and more channels to capture a larger number of channels. MobileNet-V2's inverted bottleneck residual blocks along with squeeze-and-excite blocks are the basis of EfficientNet-B0's base network. Figure 6 is the architecture for the EfficientNet B0 model. DenseNet201 [18]: Compared with conventional CNNs, DenseNet requires fewer parameters because it does not learn redundant feature maps. The DenseNet layers are  
Figure 5: ResNet18 architecture. Figure 6: EfficientNetB0 architecture. relatively narrow, (12 filters), which adds a fewer number of new featuremaps. Four variants of DenseNet exist; DenseNet121, DenseNet169, DenseNet201, and DenseNet264. Our paper uses DenseNet201, which has 201 layers, to extract features. In DenseNet (see Figure 7), the input image and gradients from the loss function are accessible directly to each layer. DenseNet is therefore a good choice for image classification due to its reduced computational cost. ResNet50 [17]: introduced a deep residual learning structure, which reformulates the CNN's layers as learning residual functions of the layer inputs. Correctly denoting the desired underlying mapping as \(K(i)\), the stacked non-linear layers were made to fit another mapping of \(E(i):=K(i)-i\). ResNet solved the vanishing gradient, whereby the value of the neural network's gradient decreases significantly during backpropagation until its weights barely change. ResNet solved the vanishing gradient problem using a skip connection, by adding the original input to the output of the convolutional block. A skip connection is a direct connection that skips over some of the model layers and can be expressed as \(\mathbf{y}=\mathcal{F}(\mathbf{x},\{W_{i}\})+W_{i}\mathbf{x}\), where \(\mathcal{F}(\mathbf{x},\{W_{i}\})\) represents the residual mapping to be learned.