×

A new TV-Stokes model for image deblurring and denoising with fast algorithms. (English) Zbl 06805216

Summary: The famous TV-Stokes models, which improve the restored images comfortable, have been very successful in image denoising. In this paper, we propose a new TV-Stokes model for image deblurring with a good geometry explanation. In the tangential field smoothing, the data fidelity term is chosen to measure the distance between the solution and the orthogonal projection of the tangential field of the observation image onto the range of the conjugate of the blurry operator, while the total variation of the solution is chosen as the regularization term. In the image reconstruction, we compute the smoothing part of the image from the smoothed tangential field for the first step, and use an anisotropic TV model to obtain the “texture” part of the deblurred image. The solvability properties for the minimization problems in two steps are established, and fast algorithms are presented. Numerical experiments demonstrate that the new deblurring model can capture the details of images hidden in the blurry and noisy image, and the fast algorithms are efficient and robust.

MSC:

65-XX Numerical analysis

Software:

RecPF; FTVd
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ballaster, C., Bertalmio, M., Caselles, V., Sapiro, G., Verdera, J.: Filling in by joint interpolation of vector fields and gray levels. IEEE Trans. Image Process. 10, 1200-1211 (2000) · Zbl 1037.68771 · doi:10.1109/83.935036
[2] Bar, L., Brook, A., Sochen, N., Kiryati, N.: Deblurring of color images corrupted by salt-and-pepper noise. IEEE Trans. Image Process. 16, 1101-1111 (2007) · doi:10.1109/TIP.2007.891805
[3] Buades, A., Coll, B., Morel, J.M.: A review of image denoising algorithms, with a new one. Multiscale Model. Simul. 4, 490-530 (2005) · Zbl 1108.94004 · doi:10.1137/040616024
[4] Bertalmio, M., Bertozzi, A.L., Sapiro, G.: Navier-Stokes, fluid dynamics, and image and video inpaiting. In: Proceedings of the 2001 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2001, pp. 355-362 (2001) · Zbl 1035.65065
[5] Burchard, P., Tasdizen, T., Whitaker, R., Osher, S.: Geometric surface processing via normal maps, Tech. Rep. 02-3, Applied Mathematics, UCLA (2002) · Zbl 1217.65071
[6] Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, New York (2004) · Zbl 1058.90049 · doi:10.1017/CBO9780511804441
[7] Cai, J., Chan, R., Shen, L., Shen, Z.: Restoration of chopped and nodded images by framelets. SIAM J. Sci. Comput. 30(3), 1205-1227 (2008) · Zbl 1161.94303 · doi:10.1137/040615298
[8] Cai, J., Chan, R., Shen, Z.: A framelet-based image inpainting algorithm. Appl. Comput. Harmon. Anal. 24(2), 131-149 (2008) · Zbl 1135.68056 · doi:10.1016/j.acha.2007.10.002
[9] Cai, J., Osher, S., Shen, Z.: Linearized Bregman iterations for compressed sensing. Math. Comput. 78(267), 1515-1539 (2009) · Zbl 1198.65102 · doi:10.1090/S0025-5718-08-02189-3
[10] Cai, J., Osher, S., Shen, Z.: Linearized Bregman iterations for frame-based image deblurring. SIAM J. Imaging Sci. 2(1), 226-252 (2009) · Zbl 1175.94010 · doi:10.1137/080733371
[11] Cai, J., Chan, R.H., Nikolova, M.: Fast two-phase image deblurring under impulse noise. J. Math. Imaging Vis. 36, 46-53 (2010) · doi:10.1007/s10851-009-0169-7
[12] Chambolle, A.: An algorithm for total variation minimization and applications. J. Math. Imaging Vis. 20, 89-97 (2004) · Zbl 1366.94048 · doi:10.1023/B:JMIV.0000011320.81911.38
[13] Chan, T.F., Shen, J.: Image processing and analysis: Variational, PDE, Wavelet, and Stochastic Methods. SIAM, Philadelphia (2005) · Zbl 1095.68127
[14] Chan, T.F., Marquina, A., Mullet, P.: High-order total variation-based image restoration. SIAM J. Sci. Comput. 22, 503-516 (2000) · Zbl 0968.68175 · doi:10.1137/S1064827598344169
[15] Chan, R.H., Ho, C.W., Nikolova, M.: Salt-and-pepper noise removal by median-type noise detectors and edge-preserving regularization. IEEE Trans. Image Process. 14, 1479-1485 (2005) · doi:10.1109/TIP.2005.852196
[16] Chan, R.H., Tao, M., Yuan, X.: Constrained total variation deblurring models and fast algorithms based on alternating direction method of multipliers. SIAM J. Imaging Sci. 6(1), 680-697 (2013) · Zbl 1279.68322 · doi:10.1137/110860185
[17] Chang, Q., Chern, I.: Acceleration methods for total variation-based image denoising. SIAM J. Sci. Comput. 25(3), 982-994 (2003) · Zbl 1046.65048 · doi:10.1137/S106482750241534X
[18] Chang, Q., Huang, Z.: Efficient algebraic multigrid algorithm and their convergence. SIAM J. Sci. Comput. 24, 597-618 (2002) · Zbl 1027.65166 · doi:10.1137/S1064827501389850
[19] Chang, Q., Jia, Z.: New fast algorithms for a modified TV-Stokes model (in Chinese). Sci. Sin. Math. 44(12), 1323-1336 (2014) · Zbl 1488.94021 · doi:10.1360/012014-55
[20] Chen, D., Cheng, L., Su, F.: A new TV-Stokes model with augmented Lagrangian method for image denoising and deconvolution. J. Sci. Comput. 51, 505-526 (2012) · Zbl 1258.94017 · doi:10.1007/s10915-011-9519-x
[21] Combettes, P., Wajs, V.: Signal recovery by proximal forward-backward splitting. Multiscale Model. Simul. 4(4), 1168-1200 (2005) · Zbl 1179.94031 · doi:10.1137/050626090
[22] Dong, F., Liu, Z., Kong, D., Liu, K.: An improved LOT model for image restoration. J. Math. Imaging Vis. 34, 89-97 (2009) · doi:10.1007/s10851-008-0132-z
[23] Durand, S., Fadili, J., Nikolova, M.M.: Multiplicative noise cleaning via a variational method involving curvelet coefficients. In: X.-C. Tai, K. Morken, M. Lysaker, K.-A. Lie (eds.) Scale Space and Variational Methods in Computer Vision, Lecture Notes in Comput. Sci. 5567, pp. 282-294. Springer, Berlin (2009) · Zbl 1027.65166
[24] Elo, C.A., Malyshev, A., Rahman, T.: A dual formulation of the TV-Stokes algorithm for image denoising. In: SSVM 2009. LNCS 5567, pp. 307-318 (2009) · Zbl 1206.90245
[25] Giusti, E.; Borel, A. (ed.); Moser, J. (ed.); Yau, S-T (ed.), Minimal surfaces and functions of bounded variation, No. 80 (1984), Boston · Zbl 0545.49018
[26] Girault, V., Raviart, P.A.: Finite Element Approximation of the Navier-Stokes Equations. Springer, New York (1981) · Zbl 0441.65081
[27] Goldstein, T., Osher, S.: The split Bregman method for L1-regularized problems. SIAM J. Imaging Sci. 2(2), 323-343 (2009) · Zbl 1177.65088 · doi:10.1137/080725891
[28] Hahn, J., Wu, C., Tai, X.-C.: Augmented Lagrangian method for generalized TV-Stokes model. J. Sci. Comput. 50, 235-264 (2012) · Zbl 1253.94012 · doi:10.1007/s10915-011-9482-6
[29] Jia, R., Zhao, H.: A fast algorithm for the total variation model of image denoising. Adv. Comput. Math. 33, 231-241 (2010) · Zbl 1191.94017 · doi:10.1007/s10444-009-9128-5
[30] Litvinov, W.G., Rahman, T., Tai, X.-C.: A modified TV-Stokes model for image processing. SIAM. J. Sci. Comput. 33(4), 1574-1597 (2011) · Zbl 1234.94012 · doi:10.1137/080727506
[31] Lysaker, M., Lundervold, A., Tai, X.-C.: Noise removal using fourth-order partial differential equations to medical magnetic resonance images in space and time. IEEE Trans. Image Process. 12(12), 1579-1590 (2003) · Zbl 1286.94020 · doi:10.1109/TIP.2003.819229
[32] Lysaker, M., Osher, S., Tai, X.-C.: Noise removal using smoothed normals and surface fitting. IEEE Trans. Image Process. 13(10), 1345-1357 (2004) · Zbl 1286.94022 · doi:10.1109/TIP.2004.834662
[33] Mumford, D., Shah, J.: Optimal approximations by piecewise smooth functions and associated variational problems. Commun. Pure Appl. Math. 42, 577-585 (1989) · Zbl 0691.49036 · doi:10.1002/cpa.3160420503
[34] 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 (1999) · Zbl 0951.65038 · doi:10.1137/S1064827598341384
[35] Ng, M.K., Weiss, P., Yuan, X.: Solving constrained total-variation image restoration and reconstruction problems via alternating direction methods. SIAM J. Sci. Comput. 32, 2710-2736 (2010) · Zbl 1217.65071 · doi:10.1137/090774823
[36] O’Connor, D., Vandenberghe, L.: Primal-dual decomposition by operator splitting and applications to image deblurring. SIAM J. Imaging Sci. 7(3), 1724-1754 (2014) · Zbl 1309.65069 · doi:10.1137/13094671X
[37] Osher, S., Sole, A., Vese, L.: Image decomposition and restoration using total variation minimization and the \[H^{-1}H-1\] norm. Multiscale Model. Simul. SIAM Interdiscip. J. 1(3), 1579-1590 (2003) · Zbl 1051.49026
[38] Rahman, T., Tai, X.-C., Osher, S.: A TV-Stokes denoising algorithm. In: F. Sgallari et al. (eds.) Lecture Notes in Comput. Sci. 4485, pp. 473-483. Springer, New York (2007) · Zbl 0962.94011
[39] Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Phys. D 60, 259-268 (1992) · Zbl 0780.49028 · doi:10.1016/0167-2789(92)90242-F
[40] Shi, J., Osher, S.: A nonlinear inverse scale space method for a convex multiplicative noise model. SIAM J. Imaging Sci. 1, 294-321 (2008) · Zbl 1185.94018 · doi:10.1137/070689954
[41] Shi, Y., Wang, L.-L., Tai, X.-C.: Geometry of total variation regularized \[L^p\] Lp-model. J. Comput. Appl. Math. 236, 2223-2234 (2012) · Zbl 1259.94020 · doi:10.1016/j.cam.2011.09.043
[42] Tai, X.-C., Osher, S., Holm, R.: Image inpainting using TV-Stokes equation. In: Tai, X.-C., Lie, K.A., Chan, T.F., Osher, S. (eds.) Image Processing Based on Partial Differential Equations. Springer, Heidelberg (2007)
[43] Tikhonov, A., Arsenin, V.: Solution of Ill-Posed Problems. V. H. Winston & Sons, Washington, DC (1977) · Zbl 0354.65028
[44] Vese, L., Osher, S.: Numerical methods for \[PP\]-Harmonic flows and applications to image processing. SIAM J. Numer. Anal. 40(6), 2085-2104 (2002) · Zbl 1035.65065 · doi:10.1137/S0036142901396715
[45] Wang, Y., Yang, J., Yin, W., Zhang, Y.: A new alternating minimization algorithm for total variation image reconstruction. SIAM J. Imaging Sci. 1, 248-272 (2008) · Zbl 1187.68665 · doi:10.1137/080724265
[46] Wen, Y., Chan, R., Zeng, T.: Primal-dual algorithms for total variation based image restoration under Poisson noise. Sci. China Math. 59(1), 141-160 (2016) · Zbl 1403.65029 · doi:10.1007/s11425-015-5079-0
[47] Wu, C., Tai, X.-C.: Augmented Lagrangian method, dual methods, and split Bregman iteration for ROF, vectorial TV, and high order models. SIAM J. Imaging Sci. 3, 300-339 (2010) · Zbl 1206.90245 · doi:10.1137/090767558
[48] Wu, C., Zhang, J., Tai, X.-C.: Augmented Lagrangian method for total variation restoration with non-quadratic fidelity. Inverse Probl. Imaging 5, 237-261 (2011) · Zbl 1225.80013 · doi:10.3934/ipi.2011.5.237
[49] Yang, J., Yin, W., Zhang, Y., Wang, Y.: A fast algorithm for edge-preserving variational multi-channel image restoration. SIAM J. Imaging Sci. 2, 569-592 (2009) · Zbl 1181.68304 · doi:10.1137/080730421
[50] Yang, J., Zhang, Y., Yin, W.: An efficient TVL1 algorithm for deblurring multichannel images corrupted by impulsive noise. SIAM J. Sci. Comput. 31, 2842-2865 (2009) · Zbl 1195.68110 · doi:10.1137/080732894
[51] You, Y.-L., Kaveh, M.: Fourth-order partial differential equation for noise removal. IEEE Trans. Image Process. 9, 1723-1730 (2000) · Zbl 0962.94011 · doi:10.1109/83.869184
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.