# Self-Adaptive Alternating Direction Method of Multipliers for Image Denoising
## 1 Introduction  
Image denoising occupies a pivotal role within the realm of digital image processing, functioning as a foundational task crucial for refining and accentuating the intrinsic qualities inherent in an image [1; 2; 3; 4]. Noise, stemming from a multitude of sources, such as image acquisition devices, transmission channels, or storage modalities, presents a substantial impediment to maintaining image fidelity. The presence of noise can obfuscate intricate details, introduce artifacts, and degrade the overall visual fidelity of an image. Through the efficacious elimination of noise via denoising methodologies, the quality, integrity, and interpretability of images can be significantly augmented [5; 6; 7; 8]. In domains like medical imaging, surveillance, remote sensing, and scientific inquiry, where accurate and lucid visual information holds paramount importance, the indispasability of image denoising cannot be overstated. It not only enhances the aesthetic allure of images but also safeguards the preservation of critical details, fostering more dependable analysis, interpretation, and decision-making grounded in visual data [9; 10; 11]. Common image denoising techniques include statistical modeling, transformation methods, and deep learning strategies [12; 13; 14]. Compared to other methods, transformation methods have efficient algorithm implementations, capable of processing large-scale data in a short time. They also exhibit good adaptability to various signal types, making them suitable for a variety of image processing tasks. Low-rank matrix recovery techniques model low-rank matrices within the image domain to restore the original low-rank structure from corrupted image data [15; 16; 17; 18]. This approach leverages the latent low-rank properties within images, representing the image as a sum of a low-rank matrix capturing structural information and a sparse matrix representing noise and fine details. Through low-rank matrix recovery techniques, noise in images can be removed while preserving essential structural features, resulting in clearer and more accurate image restoration [19]. Low-rank matrix recovery techniques include singular value soft thresholding, low-rank nuclear norm regularization, alternating direction method of multipliers, and low-rank representation methods [20; 21; 22]. These methods operate by manipulating singular values,introducing penalties based on low-rank nuclear norms, employing alternating optimization schemes, or utilizing low-rank representations to restore the low-rank structure of images while eliminating noise. In recent years, several works have garnered significant attention. For example, in [23], [PERSON] et al. utilized sparse signal representation methods and designed overcomplete dictionaries to explore the advantages and limitations of sparse signal representation algorithms in the processing of ultrasonic non-destructive evaluation signals. In [24], [PERSON] et al. reconstructed images from their under-sampled Fourier coefficients using infimal convolution regularizations, and an advanced Generalized Structured Low-Rank Algorithm is introduced. Due to its applicability to various constraints and regularization terms, the ADMM has garnered more attention among these low-rank matrix recovery algorithms. The traditional ADMM methodology employs the minimization of the (1,1)-norm of a matrix to enforce sparsity and control the count of non-zero elements, thereby approximating the optimization of the matrix's 0-norm to a certain degree. However, neighboring pixels in images typically exhibit correlations. By enforcing sparse constraints on individual pixels, there is a risk of losing model intricacies and structural nuances. Through the utilization of collaborative sparse regression, a shared sparse representation can be identified, preserving structured sparsity within the image and enabling a deeper analysis of spatial interrelations among pixels [25].