# Computer-Aided Diagnosis of Alzheimer's Disease via Deep Learning Models and Radiomics Method
## 4 Convolution Neural Network Assisting Diagnose
### Construction of Convolution Neural Network
#### 4.1.2 Improved Deformable U-Net (DeU-Net)  
U-Net is a special CNN [25] that excels in image segmentation. We use U-net to process data in this research, which is an approach of computer-aided diagnosis of Alzheimer's disease in PET images. Inspired by the network proposed by [PERSON] in 2019 [26], we replace the typical convolution kernel with deformable convolution in the network. Improved structure of network is shown in Figure 14.  
This CNN architecture has symmetrical structure and deformable convolution kernels. It consists path of the encoder and decoder, each formed by three layers. The encoder path has two \(3*3\) deformable convolution kernels and a down-sampling max-pooling operator; similarly, the base layer also formed by two deformable convolution kernels with this structure. Copy and crop between the encoder and the decoder helps the reservation of information for localization [27]. Activation function is Rectified Linear Unit (ReLU). By the Adam-Optimizer based on gradient decent, the parameters are relatively stable and promote their dynamic adjustment.  
Definition of deformable convolution kernel:  
\[Z_{l+1}(i,j)=\sum_{k=1}^{K1}\sum_{x=1}^{f}\sum_{y=1}^{f}[Z_{k}^{l}(s_{0}i+x,s_ {0}j+y)w_{k}^{l+1}(x+\Delta x,y+\Delta y)], \tag{15}\]  
Figure 14: Convolution neural network mode.  
\[Z_{l+1}(p)=\sum_{k=1}^{K1}\sum_{x=1}^{f}\sum_{y=1}^{f}\left[Z_{k}^{l}(p_{0})w_{k}^ {l+1}(\Delta p_{n})\right] \tag{16}\]  
where \(Z^{l}\) represents the input of convolution layer \(l\) and \(Z^{(l+1)}\) represents output. \(Z(p)\) is the value of pixels on the feature map. \(f,s_{0},K1\) represents size, stride, and padding layers size of the convolution kernel, respectively. In this equation \(\Delta x\), \(\Delta y\) can be fractional, the bilinear interpolated \(w_{k}^{l+1}\) as follows:  
\[w_{k}^{l+1}(p)=\sum_{x=1}^{q}\sum_{y=1}^{q}u(q_{w},p_{w})u(q_{z},p_{z})w(q) \tag{17}\]  
where \(p\) represents the arbitrary fractional pixels and \(q\) represents integral pixels in the feature map. The definition of Kernel \(u\) is:  
\[u(x,y)=\max(0,1-|m-n|) \tag{18}\]  
In every layer, we used deformable kernels to work in the same way rather than implementing different kernels for each layer, this will improve network performance. Using the back-propagation method, the network was trained end-to-end with labeled PET/MRI/fusion Images.  
Feature map pixels soft-max classifier with cross entropy loss function is the energy function, this classifier is:  
\[S_{k}(x,y)=\frac{e^{A_{k}(x,y)}}{\sum_{x=1}^{K}\sum_{y=1}^{K}e^{A^{K}(x,y)}} \tag{19}\]  
\(A_{k}(x,y)\) represents the activated feature pixels in layer \(k\), and the pixels number in the feature map is \(K\).  
The energy function is:  
\[E=\sum_{x=1}^{K}\sum_{y=1}^{K^{2}}\varpi(x,y)\log\left(s_{l(x,y)}(x,y)\right) \tag{20}\]  
where \(\varpi(x,y)\) is pixel weight map function that previously defined, inspired by [PERSON] (2015):  
\[\varpi(x,y)=w_{c}(x,y)+\varpi_{0}\exp\left(-\frac{(d_{1}(x,y)+d_{2}(x,y))^{2} }{2\sigma^{2}}\right) \tag{21}\]  
To avoid excessive activation of some pixels and reduce errors, the weight of U-net needs to initialize. We set \(\sigma=8\) and \(\varpi_{0}=10\) to obtain better performance. Epoch was set at 10,000, learning rate 0.01.