Anisotropic Image Sharpening via Well-Posed Sobolev Gradient Flows

We study well-posed perturbations of formally ill-posed diffusion equations which are used in image processing, such as the Perona–Malik equation. Our perturbation technique is to consider the diffusion equations as L gradient flows on integral functionals and then modify the inner product from L to a Sobolev inner product. We show that the functional differential equations obtained in this way are well-posed in both the forward and backward directions. We then show how to design a well-posed image sharpening algorithm via Sobolev gradient ascent on a Perona–Malik type functional. We provide full numerical implementation details and show experimental results on natural images which suggest that this method outperforms previous work by the authors as well as other sharpening algorithms such as the shock filter.