## 2 Methodology
### Feature Extraction with Multi-Round 1D Convolutional Processes and 1D Pooling Process  
After image enhancement with the II-based convolutional process, we can extract the feature patterns from any mammographic image by using a 100 \(\times\) 100 bounding box, as seen in the red block in Figure 4. In the feature extraction layer, we first perform the normalization and flattening processes with a FLAT operator to convert a matrix form (100 \(\times\) 100) to a vector form (1 \(\times\) 10,000), which is presented as follows:  
\[I_{xy}(p,q)=ROI(IntegI(x,y)),\ FLATI_{\rm x}=FLAT(\frac{I_{xy}}{255}) \tag{8}\]  
where \(I_{xy}(p,\ q)\) are the II values within the ROI, \(p\) = 1, 2, 3,..., \(n\), and \(q\) = 1, 2, 3,..., \(n\) (\(n\) = 100 in this study); \(FLAT\) (\(\bullet\)) is the flattening operator; and \(FLATI_{\rm x}\) is the 1D data stream vector as a 1D feature signal. Multi-round 1D convolutional processes are used to deal with the incoming feature signal by using the convolutional operations, \(X_{\rm c}[i]\) = \(X_{\rm c-1}[i]\) * \(H_{\rm c}[j]\), and a discrete-time form [39, 40, 51] of convolutional operation is presented as follows:  
\[X_{\rm c}[i]=\sum_{j=0}^{M-1}H_{\rm c}[j]X_{\rm c-1}[i-j],\ H_{\rm c}[j]=\exp[ \frac{-1}{2}(\frac{j-1}{\sigma})^{2}] \tag{9}\]  
where vector \(X_{\rm c}[i]\) is the feature signal in the \(c\)th convolutional operation, index \(c\) = 1, 2,..., \(C_{th}\); \(X_{\rm 0}[i]\) = \(FLATI_{\rm x}[i]\), sampling point, \(i\) = 1, 2, 3,..., \(N\) (\(N\) = \(n^{2}\)); \(H_{\rm c}[j]\) is the discrete kernel mask with the sliding stride = 1 for feature signal process, which could be used to deal with \(c\)th round convolutional operation and sampling data, \(j\) = 0, 1, 2, 3,..., \(M_{\rm c}\) - 1, and index, \(M_{\rm c}\), is the data length of the 1D kernel mask (\(M_{\rm c}\) = 200 is default). In this study, we set the two-round 1D convolutional processes (\(C_{th}\) = 2) to deal with the feature signals, as seen in the flowchart in Figure 5.  
Figure 3: Two-dimensional spatial convolutional processes by using fractional-order-based convolutional and integral image operations. **(a)** Original mammographic image; **(b)** image enhancement with fractional-order-based spatial convolutional operation; **(c)** image enhancement with spatial integral imager-based spatial convolutional operation.  
After 1D convolutional processing, the 1D pooling process (downsampling) and normalization process can reduce the dimension of feature signals, which is presented as follows:  
\[x[i]=\frac{1}{\max(X_{c})}POOL(X_{c}[100i]),\ i=1,\ 2,\ 3,\ \ldots,\ n^{\prime} \tag{10}\]  
\[n^{\prime}\approx\frac{N+M_{c}-2}{200} \tag{11}\]  
where operator \(POOL\) (\(\bullet\)) is the 1D pooling process; vector \(x[i]\) is the downsampling 1D feature signal, which is obtained by using the stride = 100; and operator \(\max\) (\(X_{c}\)) can find the maximum value in vector \(X_{c}\). As shown in Figure 5, the convolutional-pooling layer performs the normalization and flattening processes (\(\textcircled{2}\)), two-round 1D convolutional processes (\(\textcircled{3}\) and \(\textcircled{4}\)), and 1D pooling and normalization processes (\(\textcircled{5}\)) to obtain the 1D feature signal and filter the noise, which can obtain the stable feature parameters for  
Figure 4: Structure of the proposed two-dimensional spatial and one-dimensional CNN-based classifier.  
Figure 5: Image enhancement and feature extraction with the integral image-based convolutional process (\(\textcircled{1}\)), normalization and flattening process (\(\textcircled{2}\)), two-round 1D convolutional processes (\(\textcircled{3}\) and \(\textcircled{4}\)), and pooling and normalization processes (\(\textcircled{5}\)).  
identifying different levels. Hence, the above-mentioned three processes can be combined into a feature extraction function for preliminarily separating the normal (Nor) from the abnormality (B and M classes), as seen in the downsampling 1D feature signals in Figure 6.