We propose here a class of restoration algorithms for color images, based upon the Mumford-Shah (MS) model and nonlocal image information. The Ambrosio-Tortorelli and Shah elliptic approximations are defined to work in a small local neighborhood, which are sufficient to denoise smooth regions with sharp boundaries. However, texture is nonlocal in nature and requires semilocal/non-local information for efficient image denoising and restoration. Inspired from recent works (nonlocal means of Buades, Coll, Morel, and nonlocal total variation of Gilboa, Osher), we extend the local Ambrosio-Tortorelli and Shah approximations to MS functional (MS) to novel nonlocal formulations, for better restoration of fine structures and texture. We present several applications of the proposed nonlocal MS regularizers in image processing such as color image denoising, color image deblurring in the presence of Gaussian or impulse noise, color image inpainting, color image super-resolution, and color filter array demosaicing. In all the applications, the proposed nonlocal regularizers produce superior results over the local ones, especially in image inpainting with large missing regions. We also prove several characterizations of minimizers based upon dual norm formulations.
Bibliographical noteFunding Information:
Manuscript received December 06, 2009; revised April 30, 2010; accepted September 20, 2010. Date of publication November 15, 2010; date of current version May 18, 2011. This work was supported by NSF-DMS Grants 031222 and 0714945, and by a UC Dissertation Year Fellowship. The associate editor coordinating the review of this manuscript and approving it for publication was Mr. Vishal Monga.
- Ambrosio-Tortorelli elliptic approximations
- Mumford-Shah (MS) regularizer
- impulse noise
- nonlocal operators
ASJC Scopus subject areas
- Computer Graphics and Computer-Aided Design