# A Triple Deep Image Prior Model for Image Denoising Based on Mixed Priors and Noise Learning
## 1 Introduction  
The number of digital images has increased significantly with the popularization of imaging devices. Because image quality is often degraded by noise in the process of acquisition or transmission, image-denoising technology, which aims to extract noise while preserving image detail [1], has been used in various high-level visual tasks such as object detection [2], image generation [3], and semantic segmentation [4]. In most studies [5; 6; 7; 8; 9], degradation by noise is modeled by additive Gaussian white noise (AGWN) [10]. Its expression is:  
\[y=x+n \tag{1}\]  
where \(y\) represents the corrupted image, \(x\) denotes the undistorted image, and \(n\) denotes an additive noise that obeys a Gaussian distribution. Previously, various denoising algorithms [11; 12; 13; 14; 15; 16] with different performance have been proposed, and they can be divided into three categories: filtering-based, model-based, and learning-based. The non-local means (NLM) [17], block-matching3D (BM3D) [18] and weighted nuclear norm minimization (WNNM) [19] represent the standard filter-based denoising algorithms, which utilize the intrinsic statistics of the image patch for denoising. Typically, model-based methods define the task of denoising as an optimization problem using the maximum a posteriori (MAP) approach including nonlocally centralized sparse representation (NCSR) [20] and multivariate sparse regularization (MUSR) [21]. Although these methods can significantly decrease the image's noise, they have to encounter the problem of computational efficiency, over-smoothness, and parameter selection. Compared with conventional methods, the learn-based approaches can achieve more satisfactory noise fitting ability and denoising performance by extracting the shallow pixel-level features and deep semantic features in training. Based on whether clean images are needed to train models, the learn-based method can be categorized into two groups: supervised and unsupervised. In 2017, [PERSON] et al. [22] proposed a denoising convolutional neural network called DnCNN, which can be regarded as milestone work in the supervised denoising model. Distinct from considerable discriminative denoising models, DnCNN utilized residual learning [23] and batch normalization techniques [24] to expedite the training process, and achieve superior performance with simpler architectures. Nevertheless, the model is unable to achieve the desired results when the level of noise is unknown. To improve the adaptability of the model to diverse noise level, a fast and flexible denoising network (FFDNet) was proposed by [PERSON] et al. [25]. It can manage a significant range of noise levels effectively and efficiently by incorporating an adjustable noise level map as an input. However, the generalization ability of FFDNet can improve under complex noise. [PERSON] et al. incorporated synthetic images with real noise and actual images with that are practically noiseless to train their convolutional blind denoising network (CBDNet) [26] with a noise level estimation subnetwork. This improved the generalization ability of the model significantly. However, interactive denoising and separate subnets degrade the model's efficiency and flexibility, and are not conducive to perceiving the model's optimal solution. Recently, numerous innovative network architectures have been introduced and have achieved the desired effect. For example, the RIDNet [8] proposed by [PERSON] et al. applied feature attention in denoising and utilized the modular architecture to alleviate vanishing gradients while improving performance. Inevitably, the CNN-based denoisers mentioned above frequently suffer from a limited receptive field and lack long-range pixel perception ability. The Transformer [27], which was originally applied in natural language processing, has been broadened to the realm of visual tasks. In 2021, [PERSON] et al.