In this paper, we study the recovery of lost or erroneous transform coefficients in image communication systems employing block transform codecs. Previously, Wang et al. have developed a technique that exploits the smoothness property of image signals, and recovered the damaged blocks by maximizing a smoothness measure. There the first order derivative was used as the smoothness measure, which can lead to the blurring of sharp edges. In order to alleviate the edge blurring problem of this method, in this paper, we study the use of second order derivatives as the smoothness measure, including the quadratic variation and the Laplacian operators. Our simulation results show that a weighted combination of the quadratic variation and the Laplacian operator can significantly reduce the blurring across the edges while enforcing smoothness along the edges.