## 2 Basic Theories and Principles
### Untrained Neural Network Architecture  
In the field of single-pixel imaging (SPI), correlation algorithms or compressed sensing (CS) algorithms are typically used to reconstruct images of target objects. However, traditional methods often struggle to achieve high-quality reconstruction at low sampling rates, and the numerous iterative operations significantly increase both imaging time and computational cost. Data-driven deep learning (DL) algorithms have shown promise in addressing these challenges. Unfortunately, issues such as the difficulty of obtaining sufficient training data, limited model generalization, and lengthy training times remain significant hurdles. To overcome these challenges, computational imaging schemes based on the concept of deep image prior (DIP) have been proposed. These schemes integrate the physical processes of SPI into untrained neural networks to generate images of target objects, offering notable advantages in terms of generalizability and interpretability. In this approach, only the detected one-dimensional barrel signals are fed into the neural network, which then outputs optimal reconstruction results. This process adheres to the strict constraints imposed by the interactions between the neural network and the physical processes of SPI during image reconstruction. The image reconstruction process can be represented by Equation (10),  
\[R_{\theta*}=\arg\min_{\theta}\left\|P_{i}R_{\theta}(z)-y_{i}\right\|^{2}, \tag{10}\]Here, \(P_{i}\) is the modulation mask pattern, \(y_{i}\) is the measured one-dimensional bucket signal, \(\hat{T}=R_{\theta}(z)\) is the output of the neural network, and \(\left\|P_{i}R_{\theta}(z)-y_{i}\right\|^{2}\) denotes the error (loss function) between the actual measured and bucket signals estimated by the network. In addition, a physically enhanced DL framework for SPI is proposed to combine data-driven DL-based and physical model-driven untrained neural networks in order to further improve the generalization of the network to solve the computational imaging inverse problem. Compared with traditional iterative algorithms, data-driven DL-based reconstruction methods have been proven to be effective in avoiding huge computational burden and obtaining high-quality reconstruction results. Unfortunately, however, it is usually difficult to obtain sufficient training data in many tasks, and the limited generalization ability of the model and the long time for model training become a lingering haze. Recently, untrained convolutional neural network schemes based on DIP have attracted great attention in computational imaging, which is able to make an appropriate compromise between image quality and computational cost. DIP states that by stopping the network optimization in advance, the neural network can use its structure to solve the imaging inverse problem without the need of a large amount of data used to train the neural network. This feature compensates for the shortcomings of existing data-driven DL.The reconstruction process of DIP can be expressed as the function shown in Equation (11),  
\[R_{\theta*}=\arg\min_{\theta}\left\|R_{\theta}(z)-O^{\prime}\right\|^{2}, \tilde{O}=R_{\theta*}(z), \tag{11}\]  
In Equation (11), \(O^{\prime}\) is the degraded model of the target object, \(\tilde{O}\) is the image recovered by the untrained neural network, and \(z\) is a fixed random vector, \(\arg\min_{\theta}\) represents the process of solving the minimization problem. \(R_{\theta}\) is a convolutional neural network defined by a set of weights and a bias parameter \(\Theta\). Specifically, suppose that given a randomly initialized convolutional neural network, a function space is also defined. Assuming that the image of the object we are looking for is in this space, the process of reconstructing the image through the network is the process of updating the weights in the neural network and finding the appropriate parameter \(\theta*\in\Theta\).