Sobolev Image Sharpening

Image lena-grayscale Image lena-sharpen

As part of my Master's thesis, I proposed new partial differential equation (PDE)-based algorithms for image sharpening based on Sobolev gradient flows. Many existing algorithms for image processing, such as total variation (TV) or Perona-Malik anisotropic diffusion, are gradient flows on an energy under the L2 metric. We considered gradient flows on the same energies with respect to various Sobolev metrics. The resulting gradient flows are well-posed in the forward and backward directions, allowing one to use the PDE for image diffusion or sharpening. The figures above show the original Lena image and the result of our Sobolev anisotropic image sharpening. In the figures below we applied a Gaussian blur and noise to the baboon image (left) and then anisotropic Sobolev sharpening (right). This illustrates the advantage of Sobolev sharpening; it can sharpen large scale image features while supressing noise.

Image baboon-corrupted Image baboon-sharpen


  1. J. Calder and A.-R Mansouri. Anisotropic image sharpening via well-posed Sobolev gradient flows. SIAM Journal on Mathematical Analysis, 43(4):1536-1556, 2011. pdf

  2. J. Calder, A.-R Mansouri, and A. Yezzi. New possibilities in image diffusion and sharpening via high-order Sobolev gradient flows. Journal of Mathematical Imaging and Vision, 40(3):248-258, 2011. pdf

  3. J. Calder, A.-R Mansouri, and A. Yezzi. Image sharpening via Sobolev gradient flows. SIAM Journal on Imaging Sciences, 3(4):981-1014, 2010. pdf