Hostname: page-component-7bb8b95d7b-w7rtg Total loading time: 0 Render date: 2024-09-15T23:02:50.887Z Has data issue: false hasContentIssue false
  • English
  • Français

Hybrid Variational Model for Texture Image Restoration

Part of: Numerical analysis in abstract spaces

Published online by Cambridge University Press:  07 September 2017

Liyan Ma ,
Tieyong Zeng  and
Gongyan Li
Liyan Ma*
Affiliation:
Institute of Microelectronics of Chinese Academy of Sciences, Beijing, China
Tieyong Zeng*
Affiliation:
Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong, China HKBU Institute of Research and Continuing Education, Shenzhen Virtual University Park, Shenzhen 518057, China
Gongyan Li*
Affiliation:
Institute of Microelectronics of Chinese Academy of Sciences, Beijing, China
*
*Corresponding author. Email addresses:maliyan@ime.ac.cn (L. Ma), zeng@hkbu.edu.hk (T. Zeng), ligongyan@ime.ac.cn (G. Li)
*Corresponding author. Email addresses:maliyan@ime.ac.cn (L. Ma), zeng@hkbu.edu.hk (T. Zeng), ligongyan@ime.ac.cn (G. Li)
*Corresponding author. Email addresses:maliyan@ime.ac.cn (L. Ma), zeng@hkbu.edu.hk (T. Zeng), ligongyan@ime.ac.cn (G. Li)
Get access

Abstract

The hybrid variational model for restoration of texture images corrupted by blur and Gaussian noise we consider combines total variation regularisation and a fractional-order regularisation, and is solved by an alternating minimisation direction algorithm. Numerical experiments demonstrate the advantage of this model over the adaptive fractional-order variational model in image quality and computational time.

Keywords

Image deblurringtotal variationfractional-order regularisation

MSC classification

Secondary: 65J22: Inverse problems 65K10: Optimization and variational techniques 68U10: Image processing
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 7 , Issue 3 , August 2017 , pp. 629 - 642
Copyright
Copyright © Global-Science Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Afonso, M.V., Bioucas-Dias, J.M. and Figueiredo, M.A.T., Fast image recovery using variable splitting and constrained optimization, IEEE Trans. Image Process. 19, 1057–7149 (2010).Google Scholar
[2] Bai, J. and Feng, X.C., Fractional-order anisotropic diffusion for image denoising, IEEE Trans. Image Process. 16, 2492–2502 (2007).CrossRefGoogle ScholarPubMed
[3] Beck, A. and Teboulle, M., A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM J. Imaging Sci. 2, 183–202 (2009).CrossRefGoogle Scholar
[4] Berkels, B., Burger, M., Droske, M., Nemitz, O. and Rumpf, M., Cartoon extraction based on anisotropic image classification vision, modeling, and visualization, Vision, Modeling, and Visualization, pp. 293–300, Akademische Verlagsgesellschaft (2006).Google Scholar
[5] Boyd, S. and Vandenberghe, L., Convex Optimization, Cambridge University Press (2004).Google Scholar
[6] Bredies, K., Kunisch, K. and Pock, T., Total generalized variation, SIAM J. Imaging Sci. 3, 492–526 (2010).Google Scholar
[7] Buades, A., Coll, B. and Morel, J.M., A review of image denoising algorithms, with a new one, Multiscale Model. Simul. 4, 490–530 (2005).Google Scholar
[8] Buades, A., Le, T.M., More, J.M. and Vese, L., Fast cartoon+texture image filters, IEEE Trans. Image Process. 19, 1978–1986 (2010).Google Scholar
[9] Chambolle, A., An algorithm for total variation minimization and applications, J. Math. Imag. Vis. 20, 89–97 (2004).Google Scholar
[10] Chambolle, A. and Lions, P.L., Image recovery via total variation minimization and related problems, Numer. Math. 76, 167–188 (1997).CrossRefGoogle Scholar
[11] Chambolle, A. and Pock, T., A first-order primal-dual algorithm for convex problems with applications to imaging, J. Math. Imaging Vis. 40, 120–145 (2011).Google Scholar
[12] Chan, R.H., Lanza, A., Morigi, S. and Sgallari, F., An adaptive strategy for the restoration of textured images using fractional order regularization, Numer. Math. Theor. Meth. Appl. 6, 276–296 (2013).CrossRefGoogle Scholar
[13] Chan, T. and Zhou, H.M., Total variation wavelet thresholding, J. Sci. Comput. 32, 315–341 (2007).CrossRefGoogle Scholar
[14] Chen, D., Chen, Y. and Xue, D., Three fractional-order TV-l2 models for image denoising, J. Comput. Inform. Syst. 9, 4773–4780 (2013).Google Scholar
[15] Chen, D., Chen, Y. and Xue, D., Fractional-order total variation image denoising based on proximity algorithm, Appl. Math. Comput. 257, 537–545 (2015).Google Scholar
[16] Daubechies, I. and Teschke, G., Variational image restoration bymeans of wavelets: Simultaneous decomposition, deblurring, and denoising, Appl. Comput. Harmon. A. 19, 1–16 (2005).CrossRefGoogle Scholar
[17] Dong, B., Ji, H., Li, J, Shen, Z. and Xu, Y., Wavelet frame based blind image inpainting, Appl. Comput. Harmon. A. 32, 268–279 (2012).Google Scholar
[18] Dong, Y., Hintermüler, M. and Neri, M., An efficient primal-dual method for L1TV image restoration, SIAM J. Imag. Sci. 2, 1168–1189 (2009).CrossRefGoogle Scholar
[19] Gilboa, G., Sochen, N. and Zeevi, Y.Y., Variational denoising of partly textured images by spatially varying constraints, IEEE Trans. Image Process. 15, 2281–2289 (2006).Google Scholar
[20] Goldstein, T. and Osher, S., The split Bregman algorithm for L1 regularized problems, SIAM J. Imag. Sci. 2, 323–343 (2009).Google Scholar
[21] Grasmair, M., Locally adaptive total variation regularisation, in Scale Space and Variational Methods in Computer Vision, Tai, X., Morken, K., Lysaker, O.M. and Lie, K. (Eds.), pp. 331–342, Springer (2009).Google Scholar
[22] Guo, X., Li, F. and Ng, M.K., A fast l1-TV algorithm for image restoration, SIAM J. Sci. Comput. 31, 2322–2341 (2009).CrossRefGoogle Scholar
[23] Lou, Y., Zeng, T., Osher, S. and Xin, J., A weighted difference of anisotropic and isotropic total variation model for image processing, SIAM J. Imag. Sci. 8, 1798–1823 (2015).Google Scholar
[24] Lou, Y., Zhang, X., Osher, S. and Bertozzi, A., Image recovery via nonlocal operators, J. Sci. Comput. 42, 185–197 (2010).Google Scholar
[25] Ma, L., Ng, M., Yu, J. and Zeng, T., Efficient box-constrained TV-type-l1 algorithms for restoring images with impulse noise, J, Comput. Math. 31, 249–270 (2013).Google Scholar
[26] Meyer, Y., Oscillating Patterns in Image Processing and in Some Nonlinear Evolution Equations. The Fifteenth Dean Jacquelines B. Lewis Memorial Lectures, American Mathematical Society (2001).Google Scholar
[27] Miller, K.S. and Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley Interscience (1993).Google Scholar
[28] Ng, M., Ngan, H., Yuan, X. and Zhang, W., Patterned fabric inspection and visualization by the method of image decomposition, IEEE Trans. Aotum. Sci. Eng. 11, 943–947 (2013).Google Scholar
[29] Ng, M., Yuan, X. and Zhang, W., A coupled variational image decomposition and restoration model for blurred cartoon-plus-texture images with missing pixels, IEEE Trans. Image Process. 22, 2233–2246 (2013).Google Scholar
[30] Oldham, K.B.A. and Spanier, J.A., The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order, Dover (2006).Google Scholar
[31] Osher, S., Sole, A. and Vese, L., Image decomposition and restoration using total variation minimization and the H1, Multiscale Modeling & Simulation 1, 349–370 (2003).Google Scholar
[32] Rudin, L., Osher, S. and Fatemi, E., Nonlinear total variation based noise removal algorithms, Physica D 60, 259–268 (1992).CrossRefGoogle Scholar
[33] Setzer, S., Steidl, G. and Teuber, T., nfimal convolution regularizations with discrete L1-type functionals, Comm. Math. Sci. 9, 797–872 (2011).Google Scholar
[34] Tai, X.-C. and Wu, C., Augmented Lagrangian method, dual methods and split Bregman iteration for ROF model, SSVM 2009, LNCS 5567, 42, pp. 502–513, Springer (2009) .Google Scholar
[35] Tikhonov, A. and Arsenin, V., Solution of Ill-Posed Problems, Winston and Sons (1977).Google Scholar
[36] Wang, W. and Ng, M.K., On algorithms for automatic deblurring from a single image, J. Comput. Math. 30, 80–100 (2012).Google Scholar
[37] Wen, Y-W., Ng, M.K. and Huang, Y., Efficient total variation minimization methods for color image restoration, IEEE Trans. Image Process. 17, 2081–2088 (2008).Google Scholar
[38] Vese, L. and Osher, S., Modeling textures with total variation minimization and oscillating patterns in image processing, J. Sci. Computing 19, 553–572 (2003).Google Scholar
[39] Wu, C., Zhang, J. and Tai, X.-C., Augmented lagrangian method for total variation restoration with non-quadratic fidelity, Inverse Problems Imag. 5, 237–261 (2011).Google Scholar
[40] Yang, F., Chen, K. and Yu, B., Homotopy curve tracking for total variation image restoration, J. Comput. Math. 30, 177–196 (2012).CrossRefGoogle Scholar
[41] Yang, J., Zhang, Y. and Yin, W., An efficient TVL1 algorithm for deblurring multichannel images corrupted by impulsive noise, SIAM J. Sci. Comput. 31, 2842–2865 (2009).Google Scholar
[42] You, Y.L. and Kaveh, M., Fourth-order partial differential equations for noise removal, IEEE Trans. Image Process. 9, 1723–1730 (2000).Google Scholar
[43] Zeng, T., Li, X. and Ng, M., Alternating minimization method for total variation based wavelet shrinkage model, Commun. Comput. Phys. 8, 976–994 (2010).Google Scholar
[44] Zhang, X., Burger, M., Bresson, X. and Osher, S., Bregmanized nonlocal regularization for deconvolution and sparse reconstruction, SIAM J. Imag. Sci. 3, 253–276 (2010).Google Scholar
[45] Zhang, J. and Chen, K., A total fractional-order variation model for image restoration with non-homogeneous boundary conditions and its numerical solution, SIAM J. Imag. Sci. 8, 2487–2518 (2015).Google Scholar
[46] Zhang, J., Wei, Z. and Xiao, L., Adaptive fractional-order multi-scale method for image denoising, SIAM J. Imag. Sci. 43, 39–49 (2012).Google Scholar