Reconstructing a signal corrupted by impulsive noise is of high importance in several applications, including impulsive noise removal from images, audios and videos, and separating texts from images. Investigating this problem, in this paper we propose a new method to reconstruct a noise-corrupted signal where both signal and noise are sparse but in different domains. 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. Simulation results indicate that our algorithm is not only simple and fast but also it outperforms the other state-of-the-art methods in terms of reconstruction quality and/or complexity.