# A Triple Deep Image Prior Model for Image Denoising Based on Mixed Priors and Noise Learning
## 3 Methodology  
### Basic Idea  
To address the deficiencies of the original DIP mentioned above, we improve DIP's overall denoising performance in the following three aspects. (1) Utilizing internal and external mixed priors;  
To introduce complementary external priors, we select the supervised denoising model based on Transformer (i.e., Restormer) for the initial denoising of noisy image. Subsequently, we used noisy and pre-denoised images as the first and second targets to guide the denoising process. The network parameter values for the supervised denoising model are usually static after training on significant image pairs, which cannot flexibly adapt to changes in image content but have a high perception of the image's general information. In contrast, the DIP performs the denoising task in a plug-and-play manner, and the update of the network parameters is the final image generation process. Therefore, the DIP is an unsupervised model, the network parameters are closely related to the image, and the local detail restoration ability is excellent. The combination of unsupervised and supervised denoising models can exploit their respective advantages for denoising. Considering the unique capture capability for long-distance features and the efficient global computing efficiency, we select a Transformer-based supervised denoising model called Restormer as the generator of the preprocessed image. Compared with the DIP, which only used the internal image prior, our TripleDIP introduced the preprocessed image as the external prior to constrain the noise reduction process. The internal and external mixed priors are conducive for the model to repair the image complementarily and achieve satisfactory denoising performance. (2) Two-branch independent noise learning;  
To constrain the model's denoising process, we add a noise learning module composed of two branches and the corresponding compound loss function. As shown in Equation (1), a noisy image can be represented as the sum of its content and noise [41; 42]. One denoising strategy based on deep learning directly learns the mapping from noisy images to content image:  
\[I_{Content}=f_{content}(I_{Noisy}) \tag{10}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Methods**} & **Number of** & **Regularization** & **Noise Learning** & **Denoising** \\  & **Iteration** & **Method** & **Performance** & **Performance** \\ \hline \multirow{2}{*}{DIP [34]} & \multirow{2}{*}{Fixed} & Internal prior & \multirow{2}{*}{without} & \multirow{2}{*}{Baseline} \\  & & (particular & network) & & \\ \hline \multirow{2}{*}{DIP-TV [36]} & \multirow{2}{*}{Fixed} & Internal prior & \multirow{2}{*}{without} & \multirow{2}{*}{Enhanced} \\  & & (total variation) & & \\ \hline \multirow{2}{*}{DIPRecon [37]} & \multirow{2}{*}{Fixed} & Internal prior & \multirow{2}{*}{without} & \multirow{2}{*}{Enhanced} \\  & & (similar images) & & \\ \hline \multirow{2}{*}{DIP-ES [38]} & \multirow{2}{*}{Automatic} & Internal and & \multirow{2}{*}{without} & \multirow{2}{*}{Enhanced} \\  & & external priors & & \\  & & (auto-encoder) & & \\ \hline \multirow{2}{*}{DeepRED [39]} & \multirow{2}{*}{Fixed} & Internal prior & \multirow{2}{*}{without} & \multirow{2}{*}{Enhanced} \\  & & (pre-set & denoiser) & \\ \hline \multirow{2}{*}{PnP-DIP [40]} & \multirow{2}{*}{Fixed} & Internal prior & \multirow{2}{*}{without} & \multirow{2}{*}{Enhanced} \\  & & (pre-set & denoiser) & \\ \hline \multirow{2}{*}{TripleDIP} & \multirow{2}{*}{Automatic} & Internal and & \multirow{2}{*}{with} & \multirow{2}{*}{Significant} \\  & & external priors & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Performance comparison of different DIP variants. where \(I_{Content}\) is the output content image and \(f_{content}\) denotes the content generator (i.e., DIP1), \(I_{Noisy}\) represents the inputted noisy image.