# An Improved Image Filtering Algorithm for Mixed Noise
## 1 Introduction
[38] proposed a denoising algorithm of sparse representation KSVD, which makes full use of the advantages of sparse representation and can well preserve the edge of the image while denoising. The algorithm assumes that the structural elements of the image can be expressed by a set of super-complete dictionary atoms. The dictionary atoms are used to reconstruct the image to remove redundancy and achieve the purpose of denoising. However, the algorithm is only suitable for removing slightly polluted noise images. In the case of serious noise pollution, the image denoising effect is not very satisfactory. In order to further improve the denoising effect, reference [39] proposed a Non-locally Centralized Sparse Representation (NCSR) algorithm. The algorithm introduces the concept of sparse coding noise to transform the solution target of the model from acquiring the original image to suppressing the sparse coding noise of the image, and it makes full use of the nonlocal self-similarity of the image to obtain a good estimate of the sparse coding coefficient of the original image. The algorithm can still achieve a good denoising effect when the noise pollution is serious. Reference [40] proposed a denoising algorithm, OCTOBOS, which can adaptively learn the structured incomplete sparse transform with block sparsity (or the equivalent union of square sparse transform), and cluster the data through sparse coding at the same time. Reference [41] proposed a Trilateral Weighted Sparse Coding (TWSC) scheme for real image denoising. TWSC introduces three weight matrices into the data and regularization terms of the sparse coding framework to describe the statistical characteristics of the real noise and image prior model, and it uses the alternating direction method of multiplier to solve the problem. TWSC also has a reasonable denoising performance. Although the denoising effect is good, the iteration and updating of dictionary learning lead to a large amount of calculation and high time cost. Considering the above problems, [PERSON] et al. [42] proposed a more efficient and higher sparsity transform domain sparse expression method, namely Block Matching and 3D Filtering (BM3D). BM3D utilizes the correlation between \"blocks\" to achieve a high degree of sparseness of images through joint filtering, with better denoising performance, higher operating efficiency, and more advantages than other methods. Due to the development of convex and non-convex optimization methods, matrix low-rank approximation has also attracted the attention of many scholars, and many important models and algorithms have been proposed. Rank minimization is one of the important research directions. Because rank minimization is an NP-hard problem [43], it is easy to solve using the nuclear norm of the matrix as the convex relaxation optimization target of the matrix rank, and this method is called Nuclear Norm Minimization (NNM). NNM has developed rapidly, both in theory and in application. [PERSON] et al. [44] proposed the Weighted Nuclear Norm Minimization (WNNM) algorithm to optimize NNM, which assigns different weights to singular values with different values, that is, the larger the singular value, the larger the weight, and the larger the proportion. The algorithm is more effective in removing the noise of natural images. In view of the mentioned problems above and the status quo of denoising algorithms, this paper proposes a new denoising algorithm for salt and pepper + Gaussian noise based on reference [10], reference [20], and reference [37]. The algorithm first uses the improved median filter algorithm to denoise the original image with mixed noise, which can filter out most of the salt and pepper noise, and then performs \(\mathbf{n}\)-scale wavelet decomposition on the filtered image. The high-frequency components of each subsequent layer are processed by the improved wavelet threshold algorithm to achieve preliminary removal of Gaussian noise. The processed high-frequency components and unprocessed low-frequency components are then reconstructed.