A new iterative algorithm for mean curvature-based variational image denoising

• Published in: BIT Numerical Mathematics Vol. 54; no. 2; pp. 523 - 553
• Main Authors: Sun, Li, Chen, Ke
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.06.2014
• Subjects: Computational Mathematics and Numerical Analysis; Mathematics; Mathematics and Statistics; Numeric Computing; 68U10; 65F10; Mean curvature-based model; Staircasing effect; Fixed point iteration method; 65K10; Nonlinear multigrid; Denoising
• ISSN: 0006-3835; 1572-9125; 1572-9125
• DOI: 10.1007/s10543-013-0448-y

The total variation semi-norm based model by Rudin-Osher-Fatemi (in Physica D 60, 259–268, 1992 ) has been widely used for image denoising due to its ability to preserve sharp edges. One drawback of this model is the so-called staircasing effect that is seen in restoration of smooth images. Recently several models have been proposed to overcome the problem. The mean curvature-based model by Zhu and Chan (in SIAM J. Imaging Sci. 5(1), 1–32, 2012 ) is one such model which is known to be effective for restoring both smooth and nonsmooth images. It is, however, extremely challenging to solve efficiently, and the existing methods are slow or become efficient only with strong assumptions on the formulation; the latter includes Brito-Chen (SIAM J. Imaging Sci. 3(3), 363–389, 2010 ) and Tai et al. (SIAM J. Imaging Sci. 4(1), 313–344, 2011 ). Here we propose a new and general numerical algorithm for solving the mean curvature model which is based on an augmented Lagrangian formulation with a special linearised fixed point iteration and a nonlinear multigrid method. The algorithm improves on Brito-Chen (SIAM J. Imaging Sci. 3(3), 363–389, 2010 ) and Tai et al. (SIAM J. Imaging Sci. 4(1), 313–344, 2011 ). Although the idea of an augmented Lagrange method has been used in other contexts, both the treatment of the boundary conditions and the subsequent algorithms require careful analysis as standard approaches do not work well. After constructing two fixed point methods, we analyze their smoothing properties and use them for developing a converging multigrid method. Finally numerical experiments are conducted to illustrate the advantages by comparing with other related algorithms and to test the effectiveness of the proposed algorithms.