Many sensing techniques and image processing applications are characterized by noisy, or corrupted, image data. Anisotropic diffusion is a popular, and theoretically well understood, technique for denoising such images. Diffusion approaches however require the selection of an "edge stopping" function, the definition of which is typically ad hoc. We exploit and extend recent work on the statistics of natural images to define principled edge stopping functions for different types of imagery. We consider a variety of anisotropic diffusion schemes and note that they compute spatial derivatives at fixed scales from which we estimate the appropriate algorithm-specific image statistics. Going beyond traditional work on image statistics, we also model the statistics of the eigenvalues of the local structure tensor. Novel edge-stopping functions are derived from these image statistics giving a principled way of formulating anisotropic diffusion problems in which all edge-stopping parameters are learned from training data.