## 3 Experimental Results and Discussion
### Multilayer CNN-Based Classifiers' Training and Validation  
Randomly selecting 80 mammography images (20 normal subjects and 20 abnormal subjects) from the MIAS image database, 40 tumor-free images and 40 tumor images were selected to extract the feature patterns for training and testing two classifiers, as shown in Table 2. All the breast lesions in the mammograms were labeled and agreed upon by expert radiologists for biomarkers. We extracted the feature patterns from the enrolled mammography images by using the 2D spatial convolutional process (integral image- and fractional-order-based convolutional processes), two-round 1D or 2D kernel convolutional processes, and a pooling process. For each mammography image (right or left hand side), with a specific bounding box, feature patterns were obtained from the enrolled images; then a total of 500 feature patterns were used to train and validate the multilayer CNN-based classifier. The 200 feature patterns (100 Nors, 50 Bs, and 50 Ms) were randomly selected to train the classifier, and another 150 feature patterns were also randomly selected to validate the classifier's performance. In the classification layer, our proposed classifier's structure was determined by \(K\) paired input-output training patterns (\(K\) = 200), with 200 comparative feature signals and 200 desired labeled patterns, including Class-Nor, Class-B, and Class-M, which were used to establish a GRA-based fully connected network, with 100 input nodes, 200 GRA nodes, 4 summation nodes, and 3 output nodes (for three classes). As seen in Table 1, in the 2 nd and 3 rd convolutional layers, two-round 1D convolutional processes used the discrete Gaussian function (with stride = 1) with 200 data length of convolutional mask to extract and enhance the feature signals (as seen in Table 2). In the downsampling layer, the dimension of the feature signal was reduced from 1 \(\times\) 10,000 to 1 \(\times\) 100 (with stride = 100). The dimension of 1 \(\times\) 100 pooling feature signal (as seen in Figure 6) was then fed into the inputs of the GRA-based fully connected network to perform the classification task. The GRA-based classifier carried out a multilayer network with straightforward mathematical operations by using Equations (12) to (16) and dealt with the incoming feature signals to perform the classification task. Its learning stage did not require complex iterative computations, such as the forward-pass and back-propagation algorithm, gradient descent-based algorithm, or swarm optimization algorithm [6; 54; 55; 56], to adjust the connecting weighted parameters between the input layer and output layer. We also established a 2D fractional-order CNN-based classifier, consisting of a 2D spatial fractional-order convolutional layer, two-round kernel convolutional-maximum pooling layers, a flattening layer, and a fully connecting classification network, as seen in the structure in Table 1. In the classification layer, the fully connecting network was a back-propagation multilayer network, consisting of an input layer (with 625 input nodes), two hidden layers (with 168 nodes and 64 nodes in the 1 st and 2 nd hidden layers, respectively), and an output layer (with 3 output nodes). In the training stage, the gradient descent-based ADAM algorithm was used to adjust the connecting weighted parameters between the input layer and the output layer with the forward-pass and back-propagation processes, which were used to minimize loss function by using iterative computations, such as the binary cross-entropy function for multi-label classification [49; 50]. In this study, we implemented the 2D fractional-order CNN-based classifier by using the open-source Tensorflow platform (Version 1.9.0) in the Python programming language [57; 58].