Image Restoration Using One-Dimensional Sobolev Norm Profiles of Noise and Texture

Abstract

This work is devoted to image restoration (denoising and deblurring) by variational models. As in our prior work [Inverse Probl. Imaging, 3 (2009), pp. 43--68], the image $\tilde f$ to be restored is assumed to be the sum of a cartoon component $u$ (a function of bounded variation) and a texture component $v$ (an oscillatory function in a Sobolev space with negative degree of differentiability). In order to separate noise from texture in a blurred noisy textured image, we need to collect some information that helps distinguish noise, especially Gaussian noise, from texture. We know that homogeneous Sobolev spaces of negative differentiability help capture oscillations in images very well; however, these spaces do not directly provide clear distinction between texture and noise, which is also highly oscillatory, especially when the blurring effect is noticeable. Here, we propose a new method for distinguishing noise from texture by considering a family of Sobolev norms corresponding to noise and texture. It turns out that the two Sobolev norm profiles for texture and noise are different, and this enables us to better separate noise from texture during the deblurring process.