# Automatic Breast Tumor Screening of Mammographic Images with Optimal Convolutional Neural Network
## 2 Materials and Methods
### Design of the Multilayer Deep-Learning-Based CNN
This study applies the 2D spatial fractional-order convolutional processes in the fractional convolutional layer, selects the appropriate fractional order parameters, and performs convolution in the x and y directions, thus yielding a combination of 2D weight values in space, the general formula being [35; 36; 37; 38]:  
\[C^{v}I_{xy}=conv(I_{xy},M(i,j),v)^{T} \tag{1}\]  
\[C^{v}_{x}I_{xy}=\sum_{i=-\frac{h-1}{2}}^{\frac{h-1}{2}}\sum_{j=-\frac{h-1}{2}}^ {\frac{h-1}{2}}M_{x}(i,j)I(x+i,y+j) \tag{2}\]  
\[C^{v}_{y}I_{xy}=\sum_{j=-\frac{h-1}{2}}^{\frac{h-1}{2}}\sum_{i=-\frac{h-1}{2}}^ {\frac{h-1}{2}}M_{y}(j,i)I(x+j,y+i) \tag{3}\]  
where \(h=3\) is the dimension of the convolution window, \(v\) is a fractional parameter and \(v\in(0,\,1)\), and \(I(x,\,y)\in[0,\,255]\) is the pixel value at point \((x,\,y)\) in a 2D image. Each fractional-order convolutional mask multiplies each element, \(M(i,j)\) or \(M(j,\,i)\), by the corresponding input pixel values, \(I(x,\,y)\), and then obtains an enhanced feature pattern containing spatial features in the \(x\)-axis and \(y\)-axis directions. These 2D spatial convolutional processes act as two low-pass frequency filters [39] and then remove the high-spatial-frequency components from a breast mammogram. In this study, the image dimension is \(n\times n\), \(x=1\), \(2\), \(3\),..., \(n\), and \(y=1\), \(2\), \(3\),..., \(n\). \(M_{x}\) and \(M_{y}\) are \(3\times 3\) convolutional windows that can be written as follows [35; 36; 37; 38]:  
\[M_{x}=\left[ \begin{array}{ccc}0&\frac{v^{2}-v}{2}&0\\ 0&-v&0\\ 0&1&0\end{array} \right],\,M_{y}=M_{x}^{T}=\left[ \begin{array}{ccc}0&0&0\\ \frac{v^{2}-v}{2}&-v&1\\ 0&0&0\end{array} \right] \tag{4}\]  
where \(v\in(0,1)\) is the fractional-order parameter. A sliding stride = 1 is selected for spatial domain-based convolution operations in the horizontal and vertical directions. The results of the convolution operation of (1) and (2) are combined and normalized, and the approximate formula is written below:  
\[\
abla^{v}I_{xy}=\left[ \begin{array}{c}C^{v}_{x}I_{xy}\\ C^{v}_{y}I_{xy}\end{array} \right]^{T},\,\left|\
abla^{v}I_{xy}\right|\cong \frac{\left|C^{v}_{x}I_{xy}\right|+\left|C^{v}_{y}I_{xy}\right|}{255} \tag{5}\]  
These multilayer convolutional layers are also called the perception layers of the CNN network for feature enhancement and extraction. After feature extraction, the \(2\times 2\) sliding window is used to perform maximum pooling (MP), as shown in the general formula (6):  
\[MP\Big{|}_{\frac{h}{2}\times\frac{h}{2}}=\underset{M_{2\times 2}}{\max}( \Big{|}\
abla^{v}I_{xy}\Big{|}_{|n\times n}) \tag{6}\]  
After MP, the number of feature patterns is reduced to 25% of the total number of original feature images. This reduction in the dimensions of the feature patterns can overcome the overfitting problem for training a multilayer classifier. * Rapid screening of breast tumors: Breast tumors can be identified at the image classification layer, which includes the flattening process (FP) and a multilayer classifier, as seen in Figure 1. The FP can convert a 2D feature matrix into a 1D feature vector, which is then fed as the input vector of the classifier for further pattern recognition. After two MP treatments, the FP treatment may be written as shown in the general formula (7):  
\[X\Big{|}_{1\times(\frac{1}{4})^{2}}=FP(MP\Big{|}_{\frac{\theta}{4}\times\frac{ \theta}{4}}) \tag{7}\]  
where \(X\) is the 1D feature vector of the multilayer classifier used as input. In this study, the multilayer classifier includes an input layer, two hidden layers (i.e., the first and second hidden layers), and an output layer. In two hidden layers, the Gaussian error linear unit (GeLU) function [40; 41; 42] is used as the activation function in each hidden node.