## 2 Methodology
### Proposed Signature Image Generation Model  
The overall framework of the proposed signature image generation model is shown in Figure 1. In the diffusion forward process of the DDPM, Gaussian noise \(\mathcal{N}\) is used to create noise \(\beta_{t}\) for time step \(t\), and noise is added to the training data \(x_{t-1}\) to create the noisy image \(x_{t}\). During the restoration process, the neural network model, trained to predict the noise \(\beta_{t}\) added during the diffusion forward process, generates the predicted noise \(\tilde{\beta}_{t}\). This predicted noise is then used to restore \(x_{t}\) to \(\tilde{x}_{t-1}\). The key aspect of the DDPM training process is to train the neural network model in order to generate \(\tilde{\beta}_{t}\), which is most similar to \(\beta_{t}\) for all time steps \(t\). As mentioned earlier, neural network models based on U-Net architecture are primarily used for noise prediction. U-Net [25] is a neural network architecture that is commonly used in computer vision, particularly excelling in tasks such as image restoration and segmentation. U-Net architecture involves downsampling the input image through multiple stages and then upsampling the information, allowing it to leverage local and global information within the image. Additionally, DDPM employs time embedding to provide the noise prediction model with information about time step \(t\). This type of time embedding informs the model with respect to which time step it is predicting noise for, guiding it to perform appropriate noise removal at each step. Typically, a positional encoding method based on sine-cosine periodic functions is used to transform integer time step \(t\) into an embedding vector. The transformation process is as follows: \(d\) represents the size of the embedding vector, and \(i\) is the index corresponding to each dimension of the embedding vector. Figure 1: Overall framework of the proposed signature generation model. \[\text{TimeEmbedding}(t)=\left[\sin\!\left(\frac{t}{10000^{\frac{2}{2}}}\right),\cos \!\left(\frac{t}{10000^{\frac{2}{2}}}\right)\right]_{i=1}^{d/2} \tag{11}\]  
DDPM can generate images from a trained domain, starting with Gaussian noise, by incorporating time embedding into a U-Net-based noise prediction model. In the proposed method, three strategies were employed to develop a DDPM-based signature image generation model. The first strategy is to guide the generation model to produce signatures for specific individuals. A typical DDPM learns and generates data without any special conditions; hence, when used as is, it can generate signatures arbitrarily for all individuals in the trained dataset without distinguishing between signers. However, since signature images are generally used to identify the signer, there are often cases where only the signatures of specific individuals are needed. When considering each signature as a distinct class, to enable the DDPM to generate signatures for a particular class, conditional predictions must be carried out to predict noise differently based on the class. By concatenating the class code with the time embedding calculated from time step \(t\), we created a new time-class embedding and fed it into the prediction model to facilitate conditional generation. The time-class embedding is generated by a relatively small neural network layer, as depicted in Figure 3a. The second strategy is the improvement of the U-Net-based noise prediction module. In a typical U-Net structure, only the outputs of the last layers of the downsampling and upsampling paths at the same resolution are concatenated. In the proposed method, the noise prediction module contains an equal number of residual blocks at each resolution. In addition to the output of the last layer, concatenation is added between all corresponding blocks of the other residual blocks, increasing the number of skip connections.