## 1 Introduction  
Image denoising is an essential issue in the field of image processing that aims to recover a clean image from a noisy image. A noisy image is generally formulated as follows:  
\[y=x+n \tag{1}\]  
where \(y\) is the observed noisy image, \(x\) is the corresponding clean image for recovery, and \(n\) is the additive noise. Here, \(n\) is typically assumed to satisfy the zero-mean normal Gaussian distribution, where the standard deviation reflects the level of image interference by noise. While traditional image denoising models (e.g., BM3D [1], NLM [2], and WNNM [3]) have achieved state-of-the-art peak signal-to-noise ratio (PSNR) performance, deep neural network (DNN)-based methods have achieved great success. In the past few years, DNN-based image denoising methods have shown a strong capability to learn natural image priors from a large number of example images, thereby significantly improving the denoising effect. Numerous DNN-based image denoisers have been proposed, such as DnCNN [4], FFDNet [5], GANID [6], and CBDNet [7]. In DnCNN [4], [PERSON] et al. introduced batch normalization and residual learning to enhance the network training performance and speed up the training process. DnCNN achieved superior performance compared with traditional non-learning-based methods for additive white Gaussian noise (AWGN) removal. Subsequently, numerous modified representative neural networks have been proposed for image denoising, such as U-Net [8], non-local networks [9], and wavelet transform networks [10]. FFDNet provides a flexible solution for dealing with spatially variant noise and AWGN with different noise levels. FFDNet uses both the noisy image and its tunable noise level map as the network input and achieves satisfactory results on both synthetic and real noisy images. Although the denoising performance of the aforementioned methods is significantly improved compared with non-learning-based methods [1; 2; 3], they generally suffer from the limitation of convolution operations. Convolution kernels cannot flexibly adapt to different image contents because of their static filter weights, and the convolution operator has a small receptive field, which limits its applicability to long-range dependency modeling. Therefore, transformer-based denoising models have been proposed to mitigate the shortcomings of DNN-based methods [11; 12]. SwinIR [11] employs a convolution layer to extract shallow features and a stack of residual Swin transformer blocks to extract deep features. Restormer [12] is also a transformer-based image restoration method that can process high-resolution images and model the global context using self-attention across channels instead of the spatial dimension to alleviate the computational bottleneck. The supervised learning studies mentioned above [4; 5; 11; 12] have improved the image denoising performance to some extent. However, these denoisers critically depend on the quantity and diversity of noisy-clean image pairs, which are extremely difficult and expensive to collect in real-world scenarios. For example, collecting high-quality noiseless images as training datasets is essential for medical image processing. However, high-quality labels are often derived from long scan durations and high doses, which are limited in clinical practice. Moreover, most supervised DNN methods learn the denoising model using abundant training data and only lead to promising denoising performance on specific datasets. The performance of these denoisers drops dramatically once the unseen noisy images are reconstructed. Compared with supervised methods, unsupervised learning is much more valuable for many real-world applications where no ground-truth image is available [13; 14]. Recently, several unsupervised [15; 16; 17; 18; 19; 20; 21; 22; 23] approaches have been proposed to eliminate the dependence on clean images. Rather than using noisy-clean pairs for training, the Noise2 Noise method [15] uses multiple noisy images of the same scene to train a denoising DNN.