## 2 Materials and Methods
### Deep Features and Fusion  
In the developed neural network architecture, which is meticulously crafted for the pivotal role of classifying lung nodules as either benign or malignant, we utilize 3D convolutional layers. The architecture incorporates 3D convolutional layers, ReLU activation functions, and max-pooling processes within a structure consisting of three distinct convolutional blocks, all implemented in the PyTorch framework. In the initial convolutional layer, we apply 64 filters with dimensions of \(3\times 3\times 3\), incorporating padding to maintain spatial integrity for single-channel input data. This is followed by layers utilizing 128 and 64 filters, respectively, in subsequent convolutional blocks ensuring the preservation of complex spatial relationships. Max-pooling operations with a kernel size of \(2\times 2\times 2\) are employed to reduce spatial dimensions while preserving critical features. To enhance the model's adaptability, the architecture incorporates fully connected layers with the initial dense layer containing 512 neurons. During the training phase, the model utilizes the widely recognized Adam optimization algorithm, setting the learning rate at 0.0001 to achieve a stable rate of convergence. For the purpose of binary classification tasks, Cross-Entropy Loss is selected as the loss function due to its appropriateness in handling such problems. The design of the constructed network plays a crucial role, having been tailored for processing volumetric data through linear transformations of flattened feature maps, resulting in a model well-suited for the classification of lung nodules. Characterized by its parameterization, which includes a single input channel and two output classes, this architecture aims to improve the accuracy of lung nodule classification significantly, thereby  
Figure 3: Illustrative example of computing the magnitude pattern. advancing diagnostics within the field of medical imaging. The outlined CNN architecture is summarized in Table 2.  
The differentiation between malignant and benign pulmonary nodules is significantly influenced by their geometric properties. It is commonly observed that nodules of smaller dimensions are predominantly benign, whereas larger nodules are more inclined to be malignant. In this investigation, we calculate a range of geometric features derived from the segmented nodules, encompassing:  
* Convex Volume (\(V_{\text{convex}}\)): Equation: \(V_{\text{convex}}=\sum_{i=1}^{N_{\text{convex}}}v_{i}\) Description: The sum of voxels (\(v_{i}\)) within the nodule's convex hull. * Volume (\(V\)): Equation: \(V=\sum_{i=1}^{N_{\text{module}}}v_{i}\) Description: The total count of voxels (\(v_{i}\)) composing the nodule. * Equivalent Diameter (\(D_{\text{eq}}\)): Equation: \(D_{\text{eq}}=2\times\left(\frac{3V}{4\pi}\right)^{\frac{1}{3}}\) Description: Diameter of an equivalent-volume sphere. * Surface Area (\(A_{\text{surface}}\)): Equation: [Complex calculations involving voxel neighborhood assessments] Description: Total area surrounding the nodule's boundary. * Solidity (\(S\)): Equation: \(S=\frac{V}{V_{\text{convex}}}\) Description: Ratio of voxel count within the nodule to that within its convex hull. * Principal Axis Length: Equation: Derived from the eigenvalues (\(\lambda_{i}\)) of the covariance matrix. Description: Major axes lengths of the corresponding ellipsoid. * Extent (\(E\)): Equation: \(E=\frac{V}{V_{\text{bbox}}}\) Description: Proportion of nodules' voxels to total voxels in the bounding box. For an in-depth exploration of these characteristics, the reader is referred to the work of [PERSON] et al. (2010). To facilitate a more efficient analysis, the volumes of the nodules are