Abstract
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 φ α , inherits many nice properties from φ α , 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.
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References
Astola, J., Kuosmanen, P.: Fundamentals of Nonlinear Digital Filtering. CRC, Boca Raton (1997)
Black, M., Rangarajan, A.: On the unification of line processes, outlier rejection, and robust statistics with applications to early vision. Int. J. Comput. Vis. 19, 57–91 (1996)
Bortoletti, A., Di Fiore, C., Fanelli, S., Zellini, P.: A new class of quasi-Newtonian methods for optimal learning in MLP-networks. IEEE Trans. Neural Netw. 14, 263–273 (2003)
Bouman, C., Sauer, K.: On discontinuity-adaptive smoothness priors in computer vision. IEEE Trans. Pattern Anal. Mach. Intell. 17, 576–586 (1995)
Bovik, A.: Handbook of Image and Video Processing. Academic Press, San Diego (2000)
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)
Chan, R.H., Ho, C.-W., Nikolova, M.: Convergence of Newton’s method for a minimization problem in impulse noise removal. J. Comput. Math. 22, 168–177 (2004)
Chan, R.H., Ho, C.W., Nikolova, M.: Salt-and-pepper noise removal by median-type noise detector and edge-preserving regularization. IEEE Trans. Image Process. 14, 1479–1485 (2005)
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)
Chan, R.H., Hu, C., Nikolova, M.: An iterative procedure for removing random-valued impulse noise. IEEE Signal Process. Lett. 11, 921–924 (2004)
Charbonnier, P., Blanc-Féraud, L., Aubert, G., Barlaud, M.: Deterministic edge-preserving regularization in computed imaging. IEEE Trans. Image Process. 6, 298–311 (1997)
Chen, T., Wu, H.R.: Space variant median filters for the restoration of impulse noise corrupted images. IEEE Trans. Circuits Syst. II 48, 784–789 (2001)
Dennis, J.E., Schnabel, R.B.: Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Prentice-Hall, Englewood Cliffs (1983)
Dong, Y., Chan, R., Xu, S.: A detection statistic for random-valued impulse noise. IEEE Trans. Image Process. 16, 1112–1120 (2007)
Di Fiore, C., Fanelli, S., Lepore, F., Zellini, P.: Matrix algebras in quasi-Newton methods for unconstrained minimization. Numer. Math. 94, 479–500 (2003)
Di Fiore, C., Lepore, F., Zellini, P.: Hartley-type algebras in displacement and optimization strategies. Linear Algebra Appl. 366, 215–232 (2003)
Green, P.J.: Bayesian reconstructions from emission tomography data using a modified EM algorithm. IEEE Trans. Med. Imag. MI-9, 84–93 (1990)
Huang, T.S., Yang, G.J., Tang, G.Y.: Fast two-dimensional median filtering algorithm. IEEE Trans. Acoust. Speech Signal Process. 1, 13–18 (1979)
Hwang, H., Haddad, R.A.: Adaptive median filters: new algorithms and results. IEEE Trans. Image Process. 4, 499–502 (1995)
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
Ng, M.K., Chan, R.H., Tang, W.C.: A fast algorithm for deblurring models with Neumann boundary conditions. SIAM J. Sci. Comput. 21, 851–866 (2000)
Nikolova, M.: A variational approach to remove outliers and impulse noise. J. Math. Imaging Vis. 20, 99–120 (2004)
Nocedal, J., Wright, S.J.: Numerical Optimization. Springer, Berlin (1999)
Nodes, T.A., Gallagher, N.C. Jr.: The output distribution of median type filters. IEEE Trans. Commun. 32, 532–541 (1984)
Pok, G., Liu, J.-C., Nair, A.S.: Selective removal of impulse noise based on homogeneity level information. IEEE Trans. Image Process. 12, 85–92 (2003)
Sun, J., Zhang, J.: Global convergence of conjugate gradient methods without line search. Ann. Oper. Res. 103, 161–173 (2001)
Vogel, C.R., Oman, M.E.: Fast, robust total variation-based reconstruction of noisy, blurred images. IEEE Trans. Image Process. 7, 813–824 (1998)
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This work was supported by HKRGC Grant CUHK 400405 and CUHK DAG 2060257.
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Cai, JF., Chan, R.H. & Di Fiore, C. Minimization of a Detail-Preserving Regularization Functional for Impulse Noise Removal. J Math Imaging Vis 29, 79–91 (2007). https://doi.org/10.1007/s10851-007-0027-4
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DOI: https://doi.org/10.1007/s10851-007-0027-4