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We propose an anisotropic diffusion method to denoise and aid the reconstruction of planar objects in three-dimensional images. The contribution of this paper is the development of a planarity function characterizing plate-like structures using an image Hessian's eigensystem. We then construct a diffusion tensor for anisotropically smoothing plates and satisfying necessary scale-space properties. Our method finds applications in improving the fidelity of highly noisy cell membrane images from confocal microscopy. In dense cellular regions, cell membranes assume linear shapes (planar) between neighbors. The imaging process makes cell membranes appear as diffuse structures owing to the non-uniform fluorescent marker distribution, point-spread function of the optics, and anisotropic voxel resolution which make automatic cell segmentation difficult. We apply diffusion filtering to identify and enhance membranes. We demonstrate the use of our methods on 3D cell membrane images of a zebrafish embryo acquired using fluorescent microscopy and quantify the improvement in image quality.