In this paper, we introduce a new method to remove random-valued impulse noise in an image. Random-valued impulse noise replaces the pixel value at a random position by a random value. Due to the randomness of the noisy pixel values, it is difficult to detect them by comparison with neighboring pixels, which is used in many conventional methods. Then we improve the recent noise detector which uses a non-local search of similar structure. Next we propose a new noise removal algorithm by sparse representation using DCT basis. Furthermore, the sparse representation can remove impulse noise by using the neighboring similar image patch. This method has much more superior noise removal performance than conventional methods at images. We confirm the effectiveness of the proposed method quantitatively and qualitatively.