Deblurring and denoising of images by nonlocal functionals. (English) Zbl 1161.68827

Kindermann, Stefan; Osher, Stanley; Jones, Peter W.

Multiscale Model. Simul. 4, No. 4, 1091-1115 (2005).

Summary: This paper investigates the use of regularization functionals with nonlocal correlation terms for the problem of image denoising and image deblurring. These functionals are expressed as integrals over the Cartesian product of the pixel space. We show that the class of neighborhood filters can be described in this framework. Using these functionals we can consider the functional analytic properties of some of these neighborhood filters and show how they can be seen as regularization terms using a smoothed version of the Prokhorov metric. Moreover, we define a nonlocal variant of the well-known bounded variation regularization, which does not suffer from the staircase effect. We show existence of a minimizer of the corresponding regularization functional for the denoising and deblurring problem, and we present some numerical examples comparing the nonlocal version to the bounded variation regularization and the nonlocal mean filter.

MSC:

68U10	Computing methodologies for image processing
94A08	Image processing (compression, reconstruction, etc.) in information and communication theory
47G20	Integro-differential operators

Keywords:

nonlocal functionals; image processing; bounded variation; nonlocal mean filter; neighborhood filter; Prokhorov metric; denoising; deblurring; energy method