## 2 Methodology
### Integral Image (II)-Based Convolutional Process  
The II-based convolutional process can rapidly evaluate and calculate the summations in a specific rectangular region (as seen in the summed area table (SAT) [43; 44; 45] in Figure 2), which performs the convolutional process irrespective of the convolutional mask sizes. Its process facilitates summation of pixels over axis-aligned rectangular regions in constant time, regardless of the neighborhood size. For a discrete image \(I\) at the pixel \((x,y)\), its II, \(\textit{Integl}\) (\(\bullet\)), is defined as the sum of the pixel values of \(I(x,y)\) of the upright rectangle ranging from the location \((0,0)\) (top left corner) to the location \((x,y)\) (bottom right corner), as the expression form [43; 44; 45]:  
\[\textit{Integ}I(x,y)=\sum_{x^{\prime}\leq x,y^{\prime}\leq y}I(x^{\prime},y^{ \prime}),\ x^{\prime}=0,\ 1,\ 2,\ \ldots,\ x,\ y^{\prime}=0,\ 1,\ 2,\ \ldots,\ y \tag{1}\]  
where the summed area is a rectangle region with four array references, and as seen in the summed area table (SAT) in Figure  
Hence, with four values from the SAT, we can simply compute the II's value by using Equation (3),  
\[Integl(x,y)=I(x,y)-Integl(x-1,y-1)+Integl(x,y-1)+Integl(x-1,y) \tag{7}\]  
where \(Integl(x,\,-1)=Integl(-1,\,y)=0\). An II-based convolutional process can rapidly compute the summations over the image's subregions, as a linear combination of four pixels. Hence, for a matrix \(I\) of size \(n\times n\), each column and each row need \((n-1)\) operations of addition in both the column-wise and row-wise prefix sums, which can reduce the computational processes (takes \(\approx 2n\)\((n-1)\) additions) and computational complexity level at the convolutional layer using the SAT.  
For example, Figure 3b,c indicate the preliminarily results of 2D spatial convolutional processes by using the fractional-order-based convolutional and II-based convolutional operations, respectively. It can be seen that the 2D spatial convolution process can enhance the edge information while the gray-level values are significantly changed. Their spatial convolutional processes act as a low-pass frequency filter [50] and then remove the high-spatial-frequency components, which can enhance the contour of a possible abnormal object and retain non-characteristic information. Hence, the desired object can be easily localized within the specific region. Then, the ROI can be identified in a 2D mammographic image (Figure 3a,c) for further feature extraction in right or left breast, as seen in Figure 3. Hence, we can locate the possible lesion and then automatically locate the ROI and extract the feature patterns from an enhanced mammographic image with a \(n\times n\) bounding box (\(n=100\) in this study). The fractional-order-based convolution has promising denoising and sharpening capabilities; however, its convolutional mask needs to select the appropriate mask sizes and fractional-order parameters, \(v\in(0,1)\). The SAT enables the rapid calculation of the sum of pixel values (Figure 2) for II-based convolutional processes in arbitrarily sized and axis-aligned rectangle sections, allowing the implementation of real-time computations and a reduction in the computational complexity level.  
Figure 2: Summed area table (SAT) to indicate the generating sum of values in a rectangular section for integral image (II) process at the location \((x,y)\).