Abstract
Image deblurring is a fundamental problem in imaging field which often needs to recover the important structure of images. This paper addresses the image deblurring problem by considering an image as a combination of its cartoon (the piecewise smooth part of the image) and texture (the oscillation part of the image) components. To recover both of these parts, we propose the use of coupled analysis-based sparse representations to regularize the cartoon structure and the texture part of the image. We apply anisotropic total variation with a quadratic term to enhance the edges existing in the cartoon part. Furthermore, we develop a multivariable Bregman optimization method to solve the proposed image restoration model by combining the alternating minimization method and the split Bregman iteration. The experiments show that the proposed algorithm not only performs well for image decomposition, but also outperforms the previously established methods in terms of the visual residual error, the structure similarity index and the peak signal-to-noise ratio for image deblurring.
Similar content being viewed by others
References
N. Ahmed, T. Natarajan, K. Rao, Discrete cosine transform. IEEE Trans. Image Comput. C-23 100(1), 90–93 (1974)
A. Buades, J. Lisani, Directional filters for color cartoon + texture image and video decomposition. J. Math. Imaging Vis. 55, 125–135 (2016)
A. Buades, B. Coll, J. Morel, Review of image denoising algorithms with a new one. Multiscale Modeling Simul. 4(2), 490–530 (2005)
A. Buades, T. Le, J.-M. Morel, L. Vese, Fast cartoon + texture image filters. IEEE Trans. Image Process. 19(8), 1978–1985 (2010)
J. Cai, Z. Shen, Framelet based deconvolution. J. Comput. Math. 282, 89–308 (2010)
J. Cai, S. Osher, Z. Shen, Split Bregman method and Frame based image restoration. SIAM Multiscale Modeling Simul. 8(2), 337–369 (2009)
A. Chai, Z. Shen, Deconvlution: a wavelet frame approach. Numerische athematik 106, 529–587 (2007)
A. Chambolle, An algorithm for total variation minimization and applications. J. Math. Imag. Vision 20(1–2), 89–97 (2004)
T. Chan, S. Esedoglu, Aspects of total variation regularized L1 function approximation. SIAM J. Appl. Math. 65(5), 1817–1837 (2005)
T. Chan, C. Wong, Convergence of the alternating minimization algorithm for blind deconvolution. Linear Algebra Appl. 316, 259–285 (2000)
T. Chan, S. Esedoglu, F. Park, A. Yip, et al., Recent developments in total variation image restoration, in Mathematical Models of Computer Vision, ed. by Nagy, J (Springer: New York, 2005)
H. Chen, K. Yan, J. Zhang, Z. Li, Simultaneous cartoon plus texture image decomposition by using variational image decomposition. Proc. SPIE 92732Q, 1–7 (2014)
H. Chen, C. Wang, Y. Song, Z. Li, Split Bregmanized anisotropic total variation model for image deblurring. J. Vis. Commun. Image Represent. R31, 282–293 (2015)
H. Chen, Y. Song, Z. Zhang, Z. Li, Motion blind deblurring method by using anisotropic total variation. J. Optoelectron·Laser 26(6), 1206–1214 (2015)
H. Chen, Q. Wang, C. Wang, Z. Li, Image decomposition based blind image deconvolution model by employing sparse representation. IET Image Proc. 10(11), 908–925 (2016)
H. Chen, X. Qu, Q. Wang, Z. Li, A cartoon-texture decomposition based image deblurring model by using framelet based sparse representation. Proc. SPIE 1002012, 1–13 (2016)
R. Choksi, Y. Gennipy, A. Obermanz, Anisotropic total variation regularized L1 approximation and denoising/deblurring of 2D bar codes. Inverse Problem Imaging 5(3), 591–617 (2011)
K. Dabov, A. Foi, V. Katkovnik, K. Egiazarian, Image restoration by sparse 3D transform-domain collaborative filtering. Proc. SPIE 6812, 681207 (2008)
I. Daubechies, B. Han, A. Ron, Z. Shen, Framelets: MRA-based constructions of wavelet frames. Appl. Comput. Harmon. Anal. 14, 1–46 (2003)
S. Esedoḡlu, S. Osher, Decomposition of images by the anisotropic Rudin-Osher-Fatemi model. Commun. Pure Appl. Math. 57(12), 1609–1626 (2004). https://doi.org/10.1002/cpa.20045
G. Gilboa, S. Osher, Nonlocal operators with applications to image processing. Multiscale Modeling Simul. 7(3), 1005–1028 (2008)
T. Goldstein, S. Osher, The split Bregman method for L1 regularized problems. SIAM J. IMAGING SCIENCES 2(2), 323–343 (2009)
Y. Meyer, Oscillating patterns in image processing and nonlinear evolution equations, University Lecture series, 22 (American Mathematical Society, Providence, 2002)
M. Ng, X. Yuan, W. Zhang, Coupled variational image decomposition model for blurred cartoon and texture images with missing pixels. IEEE Trans. Image Process. 22(6), 2233–2246 (2013)
J.P. Oliveria, J.M. Bioucas-Dias, M. Figueiredo, Adaptive total variation image deblurring: a majorization-minimization approach. Sig. Process. 89, 1683–1693 (2009)
R. Puetter, T. Gosnell, A. Yahil, Digital image reconstruction: deblurring and Denoising. Astronomy and astrophysics review 43, 139–194 (2005)
A. Ron, Z. Shen, Affine systems in L2(Rd): the analysis of the analysis operator. J. Funct. Anal. 148, 408–447 (1997)
L.I. Rudin, S. Osher, E. Fatemi, Nonlinear total variation based noise removal algorithms. Physics D: Nonlinear Phenomena 60, 259–268 (1992)
J.-L. Starck, M. Elad, D. Donoho, Image decomposition via the combination of sparse representation and a variational approach. IEEE Trans. Image Process. 14(10), 1570–1582 (2005)
S. Tang, W. Gong, W. Li, W. Wang, Non-blind image deblurring method by local and nonlocal total variation models. Sig. Process. 94, 339–349 (2014)
Z. Wang, A. Bovik, H. Sheikh, E. Simoncelli, Image quality assessment: from error visibility to structural similarity. IEEE Trans. Image Process. 13(4), 600–612 (2004)
Y. Wang, J. Yang, W. Yin, Y. Zhang et al., A new alternating minimization algorithm for total variation image reconstruction. SIAM Journal of Imaging Sciences 1(3), 248–272 (2008)
S. Wang, Z. Liu, W. Dong, L. Jiao, Q. Tang, Total variation based image deblurring with nonlocal self-similarity constraint. Electron. Lett. 47(16), 916–918 (2011)
C. Wu, X.C. Tai, Augmented Lagrangian method, dual methods, and split Bregman iteration for ROF, Vectorial TV, and high order models. Siam Journal on Imaging Sciences 3(3), 300–339 (2012)
X. Zhang, M. Burger, X. Bresson, S. Osher, Bregmanized nonlocal regularization for deconvolution and sparse reconstruction. SIAM Journal of Imaging Sciences 3(3), 253–276 (2010)
Acknowledgements
This work is supported by the National Natural Science Fund of China (No. 61371167). We thank Prof. Cai, Prof. Zhang, Prof. Vese et al. for providing the corresponding software online. We also thank Irina Entin, M.Eng., from Liwen Bianji, Edanz Editing China (www.liwenbianji.cn/ac), for editing the English text of a draft of this manuscript.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Chen, H., Fan, Y., Wang, Q. et al. Morphological Component Image Restoration by Employing Bregmanized Sparse Regularization and Anisotropic Total Variation. Circuits Syst Signal Process 39, 2507–2532 (2020). https://doi.org/10.1007/s00034-019-01268-x
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00034-019-01268-x