Minimization of a Detail-Preserving Regularization Functional for Impulse Noise Removal
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Recently, a powerful two-phase method for restoring images corrupted with high level impulse noise has been developed. The main drawback of the method is the computational efficiency of the second phase which requires the minimization of a non-smooth objective functional. However, it was pointed out in (Chan et al. in Proc. ICIP 2005, pp. 125–128) that the non-smooth data-fitting term in the functional can be deleted since the restoration in the second phase is applied to noisy pixels only. In this paper, we study the analytic properties of the resulting new functional ℱ. We show that ℱ, which is defined in terms of edge-preserving potential functions φα, inherits many nice properties from φα, including the first and second order Lipschitz continuity, strong convexity, and positive definiteness of its Hessian. Moreover, we use these results to establish the convergence of optimization methods applied to ℱ. In particular, we prove the global convergence of some conjugate gradient-type methods and of a recently proposed low complexity quasi-Newton algorithm. Numerical experiments are given to illustrate the convergence and efficiency of the two methods.
KeywordsImage Processing Variational Method Optimization
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- 1.Astola, J., Kuosmanen, P.: Fundamentals of Nonlinear Digital Filtering. CRC, Boca Raton (1997) Google Scholar
- 6.Cai, J.F., Chan, R.H., Morini, B.: Minimization of an edge-preserving regularization functional by conjugate gradient type methods. In: Image Processing Based on Partial Differential Equations: Proceedings of the International Conference on PDE-Based Image Processing and Related Inverse Problems, CMA, Oslo, August 8–12, 2005, pp. 109–122. Springer, Berlin (2007) CrossRefGoogle Scholar
- 9.Chan, R.H., Ho, C.W., Leung, C.Y., Nikolova, M.: Minimization of detail-preserving regularization functional by Newton’s method with continuation. In: Proceedings of IEEE International Conference on Image Processing, pp. 125–128. Genova, Italy (2005) Google Scholar
- 20.Leung, S., Osher, S.: Global minimization of the active contour model with TV-inpainting and two-phase denoising. In: Proceeding of the 3rd IEEE workshop on Variational, Geometric and Level Set Methods in Computer Vision, pp. 149–160, 2005 Google Scholar