A class of fourth-order partial differential equations (PDEs) are proposed to optimize the trade-off between noise removal and edge preservation. The time evolution of these PDEs seeks to minimize a cost functional which is an increasing function of the absolute value of the Laplacian of the image intensity function. Since the Laplacian of an image at a pixel is zero if the image is planar in its neighborhood, these PDEs attempt to remove noise and preserve edges by approximating an observed image with a piecewise planar image. Piecewise planar images look more natural than step images which anisotropic diffusion (second order PDEs) uses to approximate an observed image. So the proposed PDEs are able to avoid the blocky effects widely seen in images processed by anisotropic diffusion, while achieving the degree of noise removal and edge preservation comparable to anisotropic diffusion. Although both approaches seem to be comparable in removing speckles in the observed images, speckles are more visible in images processed by the proposed PDEs, because piecewise planar images are less likely to mask speckles than step images and anisotropic diffusion tends to generate multiple false edges. Speckles can be easily removed by simple algorithms such as the one presented in this paper.