A fourth order PDE is proposed as the regularization operator for image restoration in order to alleviate the "blocky" effects that frequently mar restored images regularized with anisotropic diffusion (a second order PDE). This is motivated by its desirable property of evolving toward an image consisting of piecewise planar areas which is a less blocky and better approximation to natural images. In order to mitigate speckle artifacts that it frequently brings about, image gradient magnitude is added as an additional variable of the nonlinearity function that controls its behavior. A numerical implementation method is presented and simulation results indicate that the proposed method tend to produce restored images which are smoother in smooth areas and sharper in feature-rich areas. However, speckle artifacts need to be carefully addressed.