Abstract
In this paper, we present a novel model for recovering color images that have been affected by mixed Gaussian Cauchy noise and blur. Our approach utilizes the \(l_1\)-norm of the wavelet base as the regularization term and combines the Gaussian and Cauchy noise in a summation term as the data fidelity term. To solve this minimization model, we employ the alternating direction method of multipliers (ADMM). We evaluate the performance of our model by conducting experiments with different types of blur and mixed Gaussian–Cauchy noise. The experimental results demonstrate that our method surpasses other existing approaches in terms of PSNR values, SSIM values, and visual quality.
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References
Achim A, Kuruoglu E (2005) Image denoising using bivariate K-stable distributions in the complex wavelet domain. IEEE Signal Process Lett 12:17–20
Aubert G, Aujol J-F (2008) A variational approach to removing multiplicative noise. SIAM J Appl Math 68:925–946
Aubert G, Kornprobst P (2006) Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations, Appl. Math. Sci. 147, Springer, New York
Brownrigg D (1984) The weighted median filter. Commun. Ass. Comput. Mach. 27:807–818
Bonettini S, Ruggiero V (2012) On the convergence of primal-dual hybrid gradient algorithms for total variation image restoration. J. Math. Imaging Vis. 44:2361–253
Bovik A (2000) Handbook of Image and Video Processing. Academic Press, New York
Boyd S, Parikh N, Chu E, Peleato B, Eckstein J (2010) Distributed optimization and statistical learning via the alternating direction method of multipliers, Found. Trends. Mach Learn 3:1–122
Cai MC, Jin XQ (2005) BCCB preconditioners for solving linear systems from delay differential equations, Pergamon Press, Inc
Chambolle A (2004) An algorithm for total variation minimization and applications. J. Math. Imaging Vision 20:89–97
Chambolle A, Pock T (2011) A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imaging Vision 40:120–145
Chan R, Dong Y, Hintermuller M (2010) An effcient two-phase \(L_1\)-TV method for restoring blurred images with impulse noise. IEEE Trans Image Process 19:1731–1739
Chan TF, Golub GH, Mulet P (1999) A nonlinear primal-dual method for total variation-based image restoration. SIAM J Sci Comput 20:1964–1977
Chan R, Jin XQ (2007) An introduction to iterative Toeplitz solvers. Fundamentals of Algorithms, 5. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA
Chang Y, Kadaba S, Doerschuk P, Gelfand S (2001) Image restoration using recursive Markov random field models driven by Cauchy distributed noise. IEEE Signal Process Lett 8:65–66
Condat L (2013) A primal-dual splitting method for convex optimization involving Lipschitzian, proximable and linear composite terms. J Optim Theory Appl 158:460–479
Dong B, Ji H, Shen ZW, Xu YH (2012) Wavelet frame based blind image inpainting. Appl Comput Harmon Anal 32:268–279
Dong Y, Zeng T (2013) A convex variational model for restoring blurred images with multiplicative noise. SIAM J. Imaging Sci. 6:1598–1625
Elad M, Aharon M (2006) Image denoising via sparse and redundant representations over learned dictionaries. IEEE Trans Image Process 15:3736–3745
Federica S, Dong YQ, Zeng TY (2015) Variational Approach for Restoring Blurred Images with Cauchy Noise. SIAM J. Imaging Sci. 2:1894–1922
Figueiredo B, Bioucas-Dias J (2010) Restoration of Poissonian images using alternating direction optimization. IEEE Trans Image Process 19:3133–3145
Figueiredo M, Nowak R (2003) An EM algorithm for wavelet-based image restoration. IEEE Trans Image Process 12(8):906–916
Gilboa G, Osher S (2008) Nonlocal operators with applications to image processing. Multiscale Model Simul 7:1005–1028
Goldstein T, Osher S (2009) The split Bregman method for L1-regularized problems. SIAM J. Imaging Sci. 2:323–343
Grimmett G, Welsh D (1986) Probability: An Introduction. Oxford Science Publications, London
Hintermuller M, Langer A (2013) Subspace correction methods for a class of nonsmooth and nonadditive convex variational problems with mixed l1 l2 data-fidelity in image processing. SIAM J. Imaging Sci. 6:2134–2173
Huang Y-M, Ng MK, Wen Y-W (2009) A new total variation method for multiplicative noise removal. SIAM J. Imaging Sci. 2:20–40
Hwang H, Haddad R (1995) Adaptive median filters: New algorithms and results. IEEE Trans Image Process 4:499–502
Idan M, Speyer J (2010) Cauchy estimation for linear scalar systems. IEEE Trans. Automat. Control 55:1329–1342
Kuruoglu E, Fitzgerald W, Rayner P (1998) Near optimal detection of signals in impulsive noise modeled with asymmetric alpha-stable distribution. IEEE Commun Lett 2:282–284
Langer A (2017) Automated parameter selection in the l1-l2-tv model for removing gaussian plus impulse noise. Inverse Prob 33:1–40
Le T, Chartrand T, Asaki T (2007) A variational approach to reconstructing images corrupted by Poisson noise. J. Math. Imaging Vision 27:257–263
Mei J, Dong Y, Huang T, Yin W (2018) Cauchy Noise Removal by Nonconvex ADMM with Convergence Guarantees. J Sci Comput 74:743–766
Nikolova M (2004) A variational approach to remove outliers and impulse noise. J. Math. Imaging Vision 20:90–120
Nolan J (1997) Numerical calculation of stable densities and distribution functions. Comm. Statist. Stochastics Models 13:759–774
Setzer S, Steidl G, Teuber T (2010) Deblurring Poissonian images by split Bregman techniques. J Visual Commun Image Represent 21:193–199
Wan T, Canagarajah N, Achim A (2011) Segmentation of noisy colour images using Cauchy distribution in the complex wavelet domain. IET Image Process. 5:159–170
Wu C, Tai X-C (2010) Augmented Lagrangian method, dual methods, and split Bregman iteration for ROF, vectorial TV, and high order models. SIAM J. Imaging Sci. 3:300–339
Yang J, Yin W, Zhang Y, Wang Y (2009) A fast algorithm for edge-preserving variational multi-channel image restoration. SIAM J. Imaging Sci. 2:569–592
Yang J, Zhang Y, Yin W (2009) An effcient tvl1 algorithm for deblurring multichannel images corrupted by impulsive noise. SIAM J Sci Comput 31:2842–2865
Yang J, Zhang Y, Yin W (2010) A fast alternating direction method for tvl1-l2 signal reconstruction from partial fourier data. IEEE J. Selected Topics Signal Process 4:288–297
Zhou W, Bovik A, Sheikh H, Simoncelli E (2004) Image quality assessment: From error visibility to structural similarity. IEEE Trans Image Process 13:600–612
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Communicated by Antonio José Silva Neto.
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Ai, X., Ni, G. & Zeng, T. Color image restoration with mixed Gaussian–Cauchy noise and blur. Comp. Appl. Math. 42, 347 (2023). https://doi.org/10.1007/s40314-023-02461-0
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DOI: https://doi.org/10.1007/s40314-023-02461-0