JOURNAL OF L A T E X CLASS FILES, VOL. 14, NO. 8, AUGUST 2018 1 A Fast Iterative Method for Removing Impulsive Noise from Sparse Signals Sahar Sadrizadeh, Nematollah Zarmehi, Ehsan Asadi, Hamidreza Abin, and Farokh Marvasti Abstract—In this paper, we propose a new method to recon- struct a signal corrupted by noise where both signal and noise are sparse but in different domains. The problem investigated in this paper arises in different applications such as impulsive noise removal from images, audios and videos, decomposition of low- rank and sparse components of matrices, and separation of texts from images. First, we provide a cost function for our problem and then present an iterative method to find its local minimum. The analysis of the algorithm is also provided. As an application of this problem, we apply our algorithm for impulsive noise Salt- and-Pepper noise (SPN) and Random-Valued Impulsive Noise (RVIN)) removal from images and compare our results with other notable algorithms in the literature. Furthermore, we apply our algorithm for removing clicks from audio signals. Simulation results show that our algorithms is simple and fast, and it outperforms other state-of-the-art methods in terms of reconstruction quality and/or complexity. Index Terms—Adaptive thresholding, image denoising, itera- tive method, impulsive noise, sparse signal. I. I NTRODUCTION T HE problem considered in this paper can be modeled as: Y = D −1 (X 0 )+ N 0 , (1) where the original signal D −1 (X 0 ) ∈ R m×n is corrupted additively by sparse noise N 0 ∈ R m×n , and D is the domain in which the signal is sparse; in other words, the signal and the noise are both sparse but in different domains. We aim to reconstruct the original signal by removing the impulsive noise from the observed signal Y. One of the applications of this model is impulsive noise removal from images, videos and audios since these signals are sparse in some domains such as Discrete Cosine Transform (DCT), Wavelet and Contourlet; the impulsive noise is sparse in the image domain. Random missing samples, SPN noise and RVIN (common types of sparse noise) are common phe- nomenon in image processing, audio and video transition [1]– [3], and data transition over noisy communication channels [4], [5] such as underwater acoustic channels and power line channels. Additionally, in dictionary learning problems where impulsive noise exists [6], [7] (such as random missing sam- ples when no side information about the location of missing samples is available) the problem can be modeled by equation (1). The model stated in (1) also arises in low-rank and sparse matrix decomposition since the singular values of a low-rank matrices are sparse. [8], [9]. Another application of our model Author are with the Advanced Communication Research Institute (ACRI), Electrical Engineering Department, Sharif University of Technology, Tehran, Iran (email: ss.sadrizadeh@ee.sharif.edu; zarmehi n@ee.sharif.edu; ea460@cam.ac.uk; hamidreza.abin@ee.sharif.edu; marvasti@sharif.edu). E. Asadi is currently pursuing PhD degree at the University of Cambridge is separation of text from images since the text is sparse in the space domain while the image is sparse in the DCT domain. The algorithms related to the impulsive noise removal can be divided into two general categories: In the first category, methods first detect the position of the corrupted samples, i.e., the position of the impulsive noise, and then restore them from other clean samples. In the case of SPN, most of the research are from this category and usually result in a better reconstruction since they first find the mask matrix with which the signal is corrupted [2], [10]–[15]. These methods have two drawbacks: When the original signal is corrupted by RVIN, the detection of noisy pixels becomes very challenging. In addition, all these methods utilize the structure of the audio and image signals (mainly their low- pass characteristic) to detect the location of corrupted pixels and hence they are not applicable to signals other than audio and image signals. As examples of this category, inpainting of audio signals corrupted by impulsive noise is considered in [10]. It is assumed that the location of the distorted data is known and the audio signal is reconstructed through sparse recovery techniques. In [2], noisy pixels are detected through an impulse detector and the image is restored by applying a weighted-average filter. The authors of [11] present a two- step algorithm. In the first step the noisy pixels are detected by Support Vector Machine (SVM) classification, and then they are restored by applying an adaptive fuzzy filter. The second category consists of methods which detect and restore the noisy samples simultaneously [1], [16]–[20]. The method presented in this paper falls into this category and we compare our results with other algorithms of this class. Examples of this category are as follows: In [16], the Adaptive Median Filter (AMF) is introduced for impulsive noise removal from images. In this algorithm, the window size of the median filter is adjusted according to the impulsive noise density. An Adaptive Median Filter which utilizes the Center-Weighted median (ACWMF) is introduced in [21], and unlike other median-based filters, it performs well in the presence of RVIN. In [22], the Weighted Encoding with Sparse Nonlocal Regularization method (WESNR) is intro- duced which integrates a soft impulse detection and sparse non-local prior to remove mixed noise from images. The authors of [17] present a method based on Bayesian inference for impulsive noise removal from audio signals. For restoring images corrupted by impulsive noise, a method is suggested in [18] which utilizes particle swarm optimization and fuzzy filtering. The Structure-Adaptive Fuzzy Estimation (SAFE) algorithm is introduced in [20], in which RVIN is removed via Gaussian Maximum Likelihood Estimation. The structure information of the image is incorporated into this algorithm arXiv:1902.03988v2 [eess.SP] 30 Mar 2019