## Materials and Methods
### Generative Adversarial Network for Data Augmentation  
The target classifier we chose performs inferences on the 2D axial images, while the prediction for the A\(\beta\) distribution for a subject should reflect the 36 axial planes placed on the transverse axis. Therefore, to assist the target classifier, the GAN model must understand all of the A\(\beta\) deposition patterns for the 36 axial plane levels, with the anatomical information matched to each level. Thus, we followed the structure of the discriminator and generator of the Deep Convolutional Generative Adversarial Network (DCGAN) [52] to learn and infer the A\(\beta\) deposition patterns on a axial plane level, and trained each of the axial levels from 0 to 35 to which the input image belongs using additional condition labels [53]. Repeatedly stacked blocks were used to construct a network structure for both generator and discriminator, and the inputs for each module multiplied by the encoded vector for each axial level label stored in an embedding matrix and then entered the stacked network. A generator produced an image from a noise vector of size 300 sampled from a normal distribution, and a discriminator estimated a score for the similarity of the two distributions from real and generated images as a critic. The generator consisted of first hidden layer with 8192 nodes connected to the noise vector, and 5 layers of blocks which had up-sampling, convolution, batch normalization, and activation function. The activation function of the last block was tanh instead of ReLU which other blocks had. The discriminator had 4 layers of blocks which had convolution, batch normalization, leaky ReLU (\(\alpha\) = 0.2), and dropout layer (\(p\) = 0.25). And a global average pooling and a dense layer followed the blocks ahead. The structure of the GAN model used in this experiment is shown in Figure 3.  
The GAN learns a function that connects the target distribution directly from the input distribution without any estimation of the probability density function for the target domain. GAN has a mechanism in which the two models, Generator G and Discriminator D, learn from each other competitively [29] via Equation (1):  
\[\min_{G}\,\max_{D}V(D,\,G)\ =\ E_{x-\mathbb{P}_{r}}[\log D(x)]+E_{z-\mathbb{P}_{ z}(z)}[\log(1-D(G(z)))]. \tag{1}\]  
Discriminator D predicts the probability that the received data belongs to the real distribution \(\mathbb{P}_{r}\). To maximize V(D, G), D should ideally predict 0 for the generated data G(z) from Generator G, which learns the parameters such that D(G(z)) = 1 to minimize V(D, G). Figure 3: Structure of generative adversarial network to enhance data for a specific disease. The amyloid negative or positive dataset were used for training independent generator (G) and discriminator (D) with a condition indicating an axial level label (C\({}_{\text{G}}\) and C\({}_{\text{D}}\)). In our experiments implementing Wasserstein GAN (WGAN) and the loss, the input image X of the discriminator \(f_{w}\) parameterized by w, called critic in the original paper [54], and the input vector Z of Generator \(G_{\theta}\) parameterized by \(\theta\), are in the real spaces \(R^{DXD}\) and \(R^{d}\), respectively. Since the image synthesized by Generator \(G_{\theta}\) follows the distribution \(\mathbb{P}_{g}\), and \(\mathbb{P}_{g}\) is also the same as \(G_{\theta}(\mathbb{P}_{z})\), then  
\[X\in R^{DXD},\ Z\in R^{d}, \tag{2}\]  
\[f_{w}:X,\ c\rightarrow(0,\ 1), \tag{3}\]  
\[G_{\theta}:Z,\ c\to X, \tag{4}\]  
\[x\in X,\ \overline{x}\in G_{\theta}(z,\ c),\ z\in Z. \tag{5}\]  
Let the given PET image samples be \(S_{r}\ =\ \{x_{1}^{r},\ldots,x_{m}^{r},\ldots,x_{n}^{r}\}\), which is i.i.d, and the training set for the GAN and target model is the axial plane \(x_{p}^{r}\ =\ \{x_{axial}^{r},\ldots,x_{axial}^{r}\}\) extracted from \(S_{r}\) and are on \(\mathbb{P}_{r}\). \(x_{axial}^{r}\) has two labels, an A\(\beta\) class \(y_{A\beta}\in[0,\ 1]\) and an axial label class \(y_{Silic}\in[0,\ 35]\), respectively.