Truncation artefacts are often present in many archived clinical magnetic resonance (MR) images due to the need of shortening the acquisition time by sampling a part of their k-space. This artificial information degrades the quality of the image and may hamper clinical diagnosis. In this paper, we propose a novel method to remove the artefacts by recovering the missing k-space or spectral data. The method consists of four steps: (a) estimating the truncated k-space from the images containing truncations artefacts, (b) computing the parameters of the sparse representation of the difference image of an image from the estimated truncated k-space, (c) recovering the missing spectral data using the parameters computed in (b), and (d) obtaining the artefact-removed image through inverse Fourier transform of the estimated and the recovered spectral data. Experiments on both simulated and real MR images have shown that the proposed method effectively removes truncation artefacts while preserving image quality and outperforms both the conventional Hamming window method and the popular TV method. (C) 2012 Elsevier Inc. All rights reserved.