Abstract
To improve the decomposition quality, it is very important to describe the local structure of the image in the proposed model. This fact motivates us to improve the Meyer’s decomposition model via coupling one weighted matrix with one rotation matrix into the total variation norm. In the proposed model, the weighted matrix can be used to enhance the diffusion along with the tangent direction of the edge and the rotation matrix is used to make the difference operator couple with the coordinate system of the normal direction and the tangent direction efficiently. With these operations, our proposed model owns the advantage of the local adaption and also describes the image structure robustly. Since the proposed model has the splitting structure, we can employ the alternating direction method of multipliers to solve it. Furthermore, the convergence of the numerical method can be efficiently kept under the framework of this algorithm. Numerical results are presented to show that the proposed model can decompose better cartoon and texture components than other testing methods.
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Notes
Here we set the W with the size \(16\times 16\). \(\tan ^{-1}\) is the arctangent function with output range of \((-\frac{\pi }{2},\frac{\pi }{2})\)
References
Rudin, L., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60(1–4), 259–268 (1992)
Kose, K., Cevher, V., Cetin, A.: Filtered variation method for denoising and sparse signal processing. In: IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 3329-3332 (2012)
Meyer, Y.: Oscillating patterns in image processing and nonlinear evolution equations. Univ. Lect. Ser. 22, 1–123 (2001)
Vese, L., Osher, S.: Modeling textures with total variation minimization and oscillating patterns in image processing. J. Sci. Comput. 19(1–3), 553–572 (2003)
Osher, S., Sole, A., Vese, L.: Image decomposition and restoration using total variation minimization and the \(H^{-1}\) norm. Multiscale Model. Simul. 1(3), 349–370 (2003)
Aujol, J., Aubert, G., Fraud, L., Chambolle, A.: Image decomposition into a bounded variation component and an oscillating component. J. Math. Imaging Vis. 22(1), 71–88 (2005)
Liu, X.: A new TGV-Gabor model for cartoon-texture image decomposition. IEEE Signal Process. Lett. 25(8), 1121–1125 (2018)
Gao, Y., Bredies, K.: Infimal convolution of oscillation total generalized variation for the recovery of images with structured texture. SIAM J. Imaging Sci. 11(3), 2021–2063 (2018)
Duan, J., Qiu, Z., Lu, W., Wang, G., Panc, Z., Bai, L.: An edge-weighted second order variational model for image decomposition. Digit. Signal Process. 49, 162–181 (2016)
Aujol, J., Chambolle, A.: Dual norms and image decomposition models. Int. J. Comput. Vis. 63(1), 85–104 (2005)
Wen, Y., Sun, H., Ng, M.: A primal-dual method for the Meyer model of cartoon and texture decomposition. Numer. Linear Algebra Appl. 26(2), e2224 (2019)
Tang, L., Zhang, H., He, C., Fang, Z.: Nonconvex and nonsmooth variational decomposition for image restoration. Appl. Math. Model. 69, 355–377 (2019)
Aujo, J., Gilboa, G., Chan, T., Osher, S.: Structure-texture image decompositionłmodeling, algorithms, and parameter selection. Int. J. Comput. Vis. 67(1), 111–136 (2006)
Jalalzai, K.: Some remarks on the staircasing phenomenon in total variation-based image denoising. J. Math. Imaging Vis. 54(2), 256–268 (2016)
Duan, J., Lu, W., Pan, Z., Bai, L.: An edge-weighted second order variational model for image decomposition. Digit. Signal Process. 49, 162–181 (2006)
Bayram, I., Kamasak, M.: Directional total variation. IEEE Signal Process. Lett. 19(12), 781–784 (2012)
Kongskov, R., Dong, Y.: Directional total generalized variation regularization for impulse noise removal. In: Scale Space and Variational Methods in Computer Vision, pp. 221–231 (2017)
Kongskov, R., Dong, Y., Knudsen, K.: Directional total generalized variation regularization. BIT Numer. Math. 59(4), 903–928 (2019)
Parisotto, S., Calatroni, L., Caliari, M., Schonlieb, C., Weickert, J.: Anisotropic osmosis filtering for shadow removal in images. Inverse Probl. 35(5), 54001 (2019)
Aujol, J., Kang, S.: Color image decomposition and restoration. J. Vis. Commun. Image Represent. 17(4), 916–928 (2006)
Rockafellar, R.: Convex Anaysis. Princeton University Press, Princeton (2015)
Wu, C., Tai, X.: Augmented Lagrangian method, dual methods, and split Bregman iteration for ROF, vectorial TV, and high order models. SIAM J. Imaging Sci. 3(3), 300–339 (2010)
Lu, W., Duan, J., Qiu, Z., n Pan, Z., Liu, W., Bai, L.: Implementation of high order variational models made easy for image processing. Math. Methods Appl. Sci. 39(14), 4208–4233 (2016)
Moura, C., Kubrusly, C.: The Courant–Friedrichs–Lewy (CFL) Condition: 80 Years After Its Discovery. Birkhauser, Basel (2013)
Chambolle, A., Pock, T.: A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imaging Vis. 40(1), 120–145 (2011)
Chambolle, A.: An algorithm for total variation minimization and applications. J. Math. Imaging Vis. 20(1–2), 89–97 (2004)
Aujol, J.: Some first-order algorithms for total variation based image restoration. J. Math. Imaging Vis. 34(3), 307–327 (2007)
Bresson, X., Chan, T.: Fast dual minimization of the vectorial total variation norm and applications to color image processing. Inverse Probl. Imaging 2(4), 455–484 (2008)
Steidl, G., Teuber, T.: Removing multiplicative noise by Douglas–Rachford splitting methods. J. Math. Imaging Vis. 36(2), 168–184 (2010)
Wu, C., Zhang, J., Tai, X.: Augmented Lagrangian method for total variation restoration with non-quadratic fidelity. Inverse Probl. Imaging 5(1), 237–261 (2011)
Zhang, J., Chen, R., Deng, C., Wang, S.: Fast linearized augmented method for Euler’s elastica model. Numer. Math.:Theory Methods Appl. 10(1), 98–115 (2017)
Gabay, D., Mercier, B.: A dual algorithm for the solution of nonlinear variational problems via finite element approximation. Comput. Math. Appl. 2(1), 17–40 (1976)
Glowinski, R., Marroco, A.: Sur 1’approximation,par elements finis dordre un, et la resolution, par enalisation-dualite, dune classe de problemes de Dirichlet non lineares. Rev. Francaise d’Autom. Inf. Recherche Operationelle 9(R–2), 41–76 (1975)
Gilles, J., Osher, S.: Bregman implementation of Meyer’s G-norm for cartoon+textures decomposition. UCLA cam, 11–73 (2011)
Yang, G., Burger, P., Firmin, D., Underwood, S.: Structure adaptive anisotropic image filtering. Image Vis. Comput. 14(2), 135–145 (1996)
Hong, L., Wan, Y., Jain, A.: Fingerprint image enhancement: algorithm and performance evaluation. IEEE Trans. Pattern Anal. Mach. Intell. 20(8), 777–789 (1998)
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This work was partially supported Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering (Changsha University of Science and Technology, China)
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Shi, B., Meng, G., Zhao, Z. et al. Image decomposition based on the adaptive direction total variation and \(\mathbb {G}\)-norm regularization. SIViP 15, 155–163 (2021). https://doi.org/10.1007/s11760-020-01734-z
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DOI: https://doi.org/10.1007/s11760-020-01734-z