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Single image blind deblurring based on the fractional-order differential. (English) Zbl 1442.94008

Summary: Image deblurring is a fundamental problem in image processing. When the blur type is unknown, the problem becomes more challenging, which is called blind deblurring. In this paper, we propose a novel variational method for single image blind deblurring based on the fractional-order differential, which can overcome the staircase effect produced by the total variation regularization and alleviate the ringing artifact in deblurring. The fractional-order gradient fidelity term is added in the cost functional to improve the restoration. Moreover, we make use of the edge detect function with the fractional-order gradient to preserve sharp edges, and the Bregman iteration to reconstruct more structures. The primal-dual algorithm is developed to solve the proposed model. Numerical experiments show that our method is able to get the sharp image without the staircase effect and correctly estimate the unknown blur kernel.

MSC:

94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
65K10 Numerical optimization and variational techniques
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[1] Fergus, R.; Singh, B.; Hertzmann, A.; Roweis, S. T.; Freeman, W. T., Removing camera shake from a single photograph, (SIGGRAPH 25 (2006)), 783-794 · Zbl 1371.94125
[2] Joshi, N.; Szeliski, R.; Kriegman, D., Psf estimation using sharp edge prediction, (IEEE Conference on Computer Vision and Pattern Recognition (2008)), 1-8
[3] Levin, A., Blind motion deblurring using image statistics, (International Conference on Neural Information Processing Systems (2006)), 841-848
[4] Kundur, D.; Hatzinakos, D., A novel blind deconvolution scheme for image restoration using recursive filtering, IEEE Trans. Signal Process., 46, 375-390 (1998)
[5] Wang, W.; Ng, M. K., Convex regularized inverse filtering methods for blind image deconvolution, Signal Image Video Process., 10, 7, 1353-1360 (2016)
[6] Lv, X.-G.; Li, F.; Zeng, T., Convex blind image deconvolution with inverse filtering, Inverse Problems, 34, 035003 (2018) · Zbl 1430.94027
[7] Chan, T. F.; Wong, C. K., Total variation blind deconvolution, IEEE Trans. Image Process., 7, 3, 370-375 (1998)
[8] Perrone, D.; Favaro, P., Total variation blind deconvolution: The devil is in the details, (IEEE Conference on Computer Vision and Pattern Recognition(CVPR) (2014)), 2909-2916
[9] Zhao, X.; Wang, W.; Zeng, T. Y., Total variation structured total least squares method for image restoration, SIAM J. Sci. Comput., 35, 1304-1320 (2013) · Zbl 1287.65014
[10] Wang, W.; Zhao, X.; Ng, M., A Cartoon-plus-texture image decomposition model for blind deconvolution, Multidimens. Syst. Signal Process., 27, 541-562 (2016) · Zbl 1380.94037
[11] You, Y. L.; Kaveh, M., Fourth order partial differential equations for noise removal, IEEE Trans. Image Process., 9, 10, 1723-1730 (2000) · Zbl 0962.94011
[12] Lysaker, M.; Lundervold, A.; Tai, X. C., Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time, IEEE Trans. Image Process., 3, 12, 1579-1590 (2003) · Zbl 1286.94020
[13] Guidotti, P.; Longo, K., Two enhanced fourth order diffusion models for image denoising, J. Math. Imaging Vision, 40, 2, 188-198 (2011) · Zbl 1255.68235
[14] Gilboa, G.; Osher, S., Nonlocal operators with applications to image processing, SIAM J. Multisc. Model. Simul., 7, 3, 1005-1028 (2008) · Zbl 1181.35006
[15] Lou, Y.; Zhang, X.; Osher, S.; Bertozzi, A., Image recovery via nonlocal operators, J. Sci. Comput., 42, 2, 185-197 (2010) · Zbl 1203.65088
[16] Gao, H.; Zuo, Z., An adaptive non-local total variation blind deconvolution employing split bregman iteration, Circuits Systems Signal Process., 32, 5, 2407-2421 (2013)
[17] Tang, S.; Gong, W.; Li, W.; Wang, W., Non-blind image deblurring method by local and nonlocal total variation models, Signal Process., 94, 1, 339-349 (2014)
[18] Bai, J.; Feng, X., Fractional-order anisotropic diffusion for image denoising, IEEE Trans. Image Process., 16, 10, 2492-2502 (2007) · Zbl 1119.76377
[19] Zhang, J.; Wei, Z., Fractional variational model and algorithm for image denoising, (IEEE Fourth International Conference on Natural Computation (2008)), 524-528
[20] Zhang, J.; Wei, Z., A class of fractional-order multi-scale variational models and alternating projection algorithm for image denoising, Appl. Math. Model., 35, 2516-2528 (2011) · Zbl 1217.94024
[21] Zhang, J.; Wei, Z.; Xiao, L., Adaptive fractional-order multi-scale method for image denoising, J. Math. Imaging Vision, 43, 1, 39-49 (2012) · Zbl 1255.68278
[22] Chan, R. H.; Lanza, A.; Morigi, S.; Sgallari, F., An adaptive strategy for the restoration of textured images using fractional order regularization, Numer. Math.: Theory Methods Appl., 6, 1, 276-296 (2013) · Zbl 1289.68196
[23] Zhang, J.; Wei, Z., A fast adaptive reweighted residual feedback iterative algorithm for fractional-oeder total variation multiplicative noise removal of partly-textured images, Signal Process., 98, 5, 381-395 (2014)
[24] Zhang, Z.; Zhang, J.; Wei, Z.; Xiao, L., Cartoon-texture composite regluarization based non-blind deblurring method for partly-textured blurred images with Possion noise, Signal Process., 116, 127-140 (2015)
[25] Zhou, L.; Tang, J., Fraction-order total variation blind image restoration based on L1-norm, Appl. Math. Model., 51, 469-476 (2017) · Zbl 1480.94014
[26] Cai, J.-F.; Ji, H.; Liu, C.; Shen, Z., Blind motion deblurring from a single image using sparse approximation, (IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (2009)), 104-111
[27] Krishnan, D.; Tay, T.; Fergus, R., Blind deconvolution using a normalized sparsity measure, (IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (2011)), 233-240
[28] Carlavan, M.; Blanc-Feraud, L., Sparse Poission noisy image deblurring, IEEE Trans. Image Process., 21, 4, 1834-1846 (2012) · Zbl 1373.94060
[29] Xu, L.; Zheng, S.; Jia, J., Unnatural L0 sparse representation for natural image deblurring, (IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (2013)), 1107-1114
[30] Yin, M.; Gao, J.; Tien, D.; Cai, S., Blind image deblurring via coupled sparse representation, J. Vis. Commun. Image Represent., 25, 5, 814-821 (2014)
[31] Pan, J.; Hu, Z.; Su, Z.; Yang, M.-H., L0-regularized intensity and gradient prior for deblurring text images and beyond, IEEE TPAMI, 39, 2, 342C355 (2017)
[32] Pan, J.; Sun, D.; Pfister, H., Blind image deblurring using dark channel prior, (2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (2016))
[33] Pan, J.; Liu, R.; Su, Z.; Liu, G., Motion blur kernel estimation via salient edges and low rank prior, (ICME (2014)), 1-6
[34] Ren, W.; Cao, X.; Pan, J.; Guo, X.; Zuo, W.; Yang, M., Image deblurring via enhanced low-rank prior, IEEE Trans. Image Process., 25, 7, 3426-3437 (2016) · Zbl 1408.94557
[35] Dong, J.; Pan, J.; Su, Z., Blur kernel estimation via salient edges and low rank prior for blind image deblurring, Signal Process., Image Commun., 58, 134-145 (2017)
[36] Osher, S.; Burger, M.; Goldfarb, D.; Xu, J.; Yin, W., An iterative regularization method for total variation-based image restoration, SIAM J. Multisc. Model. Simul., 4, 2, 460-489 (2005) · Zbl 1090.94003
[37] M. Zhu, T.F. Chan, An efficient primal – dual hybrid gradient algorithm for total variation image restoration, UCLA CAM Report 08-34, 2008.; M. Zhu, T.F. Chan, An efficient primal – dual hybrid gradient algorithm for total variation image restoration, UCLA CAM Report 08-34, 2008.
[38] E. Esser, X. Zhang, T. Chan, A general framework for a class of first order primal – dual algorithms for TV minimization, UCLA CAM report 2009.; E. Esser, X. Zhang, T. Chan, A general framework for a class of first order primal – dual algorithms for TV minimization, UCLA CAM report 2009.
[39] Dong, F.; Chen, Y., A fractional-order derivative based variational framework for image denoising, Inverse Probl. Imaging, 10, 1, 27-50 (2016) · Zbl 1335.49048
[40] Loverro, A., Fractional Calculus: History, Definitions and Applications for the Engineer (2004), Univeristy of Notre Dame: Department of Aerospace and Mechanical Engineering
[41] Chen, Y.; Lan, G.; Ouyang, Y., Optimal primal – dual methods for a class of saddle pont problems, SIAM J. Optim., 24, 4, 1779-1814 (2014) · Zbl 1329.90090
[42] Chambolle, A.; Pock, T., On the ergodic convergence rates of a first-order primal – dual algorithm, Math. Program., 159, 1-2, 253-287 (2015) · Zbl 1350.49035
[43] Bahouri, H.; Chemin, J. Y.; Danchin, R., Fourier Analysis Nonlinear Partial Differential Equations, Vol. 343 (2011), Springer: Springer Berlin · Zbl 1227.35004
[44] Dong, J.; Pan, J.; Su, Z.; Yang, M. H., Blind image deblurring with outlier handling, (IEEE International Conference on Computer Vision (2017)), 2497-2505
[45] Wang, Z.; Bovik, A. C.; Sheikh, H. R.; Simoncelli, E. P., Image quality assessment: From error visibility to structural similarity, IEEE Trans. Image Process., 13, 4, 600-612 (2004)
[46] Levin, A.; Weiss, Y.; Durand, F.; Freeman, W. T., Understanding and evaluating blind deconvolution algorithms, (IEEE Conference on Computer Vision and Pattern Recognition (2009)), 1964-1971
[47] Cho, S.; Lee, S., Fast motion deblurring, ACM Trans. Graph., 28, 5, 145:1-145:8 (2009)
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