This paper studies the problem of image inpainting and image interpolation.By transforming the inpainting and interpolation model under the scheme of compressed sensing,we establish a new model of inpainting and interpolation,which is corresponding to the reconstruction model in compressed sensing.To solve this new reconstruction model,this paper gives an iterated hard threshold method based on the sparsity of images decomposed on the complex wavelet.Simulated and real image results demonstrate the effectiveness of the proposed method.