New Possibilities in Image Diffusion and Sharpening via High-Order Sobolev Gradient Flows
We study the use of high-order Sobolev gradients for PDE-based image smoothing and sharpening, extending our previous work on this problem. In particular, we study the gradient descent equation on the heat equation energy functional obtained by modifying the usual metric on the space of images, which is the L(2) metric, to a weighted H(k) Sobolev metric. We present existence and uniqueness results which show that the Sobolev diffusion PDE are well-posed both in the forward and backward direction. Furthermore, we perform a Fourier analysis on the scale space generated by the Sobolev PDE and show that as the order of the Sobolev metric tends to infinity, the Sobolev gradients converge to a Gaussian smoothed L(2) gradient. We then present experimental results which exploit the theoretical stability results by applying the various Sobolev gradient flows in the backward direction for image sharpening effects. Furthermore, we show that as the Sobolev order is increased, the sharpening effects become more global in nature and more immune to noise.
image diffusion gradient descent image sharpening partial differential equations sobolev spaces