Hostname: page-component-8448b6f56d-jr42d Total loading time: 0 Render date: 2024-04-19T23:55:57.042Z Has data issue: false hasContentIssue false
  • English
  • Français

Wavelet Based Restoration of Images with Missing or Damaged Pixels

Published online by Cambridge University Press:  28 May 2015

Hui Ji ,
Zuowei Shen  and
Yuhong Xu
Hui Ji*
Affiliation:
Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543
Zuowei Shen*
Affiliation:
Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543
Yuhong Xu*
Affiliation:
Temasek Laboratories, National University of Singapore, 2 Science Drive 2, Singapore 117543
*
Corresponding author. Email: matjh@nus.edu.sg
Corresponding author. Email: matzuows@nus.edu.sg
Corresponding author. Email: tslxyh@nus.edu.sg
Get access

Abstract

This paper addresses the problem of how to restore degraded images where the pixels have been partly lost during transmission or damaged by impulsive noise. A wide range of image restoration tasks is covered in the mathematical model considered in this paper - e.g. image deblurring, image inpainting and super-resolution imaging. Based on the assumption that natural images are likely to have a sparse representation in a wavelet tight frame domain, we propose a regularization-based approach to recover degraded images, by enforcing the analysis-based sparsity prior of images in a tight frame domain. The resulting minimization problem can be solved efficiently by the split Bregman method. Numerical experiments on various image restoration tasks - simultaneously image deblurring and inpainting, super-resolution imaging and image deblurring under impulsive noise - demonstrated the effectiveness of our proposed algorithm. It proved robust to mis-detection errors of missing or damaged pixels, and compared favorably to existing algorithms.

Keywords

Image restorationimpulsive noisetight framesparse approximationsplit Bregman method
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 1 , Issue 2 , May 2011 , pp. 108 - 131
Copyright
Copyright © Global-Science Press 2011

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]Bar, Leah, Kiryati, Nahum, and Sochen, Nir. Image deblurring in the presence of impulse noise. Int. J. Computer Vision, 70(3):279–298, 2006.Google Scholar
[2]Bar, Leah, Sochen, Nir, and Kiryati, Nahum. Image deblurring in the presence of salt-and- pepper noise. In Lecture Notes in Computer Science, Scale Space and PDE Methods in Computer Vision, LNCS), volume 3459, pages 107–118, 2005.Google Scholar
[3]Beck, A. and Teboulle, M.. A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci., 2(1):183–202, 2009.Google Scholar
[4]Bertalmio, Marcelo, Sapiro, Guillermo, Ballester, Coloma, and Caselles, Vicent. Image inpainting. In Computer Graphics, SIGGRAPH 2000, pages 417–424, 2000.Google Scholar
[5]Borup, Lasse, Gribonval, Rémi, and Nielsen, Morten. Bi-framelet systems with few vanishing moments characterize Besov spaces. Appl. Comput. Harmon. Anal., 17(1):3–28, 2004.Google Scholar
[6]Bose, N.K. and Boo, K.J.. High-resolution image reconstruction with multisensors. J. Imaging Syst. Technol., 9(4):294–304, 1998.Google Scholar
[7]Bovik, Alan C.. Handbook of Image and Video Processing. Academic Press Inc., San Diego, CA, 2000.Google Scholar
[8]Cai, Jian-Feng, Chan, Raymond H., and Nikolova, Mila. Two-phase approach for deblurring images corrupted by impulse plus Gaussian noise. Inverse Probl. Imaging, 2(2):187–204, 2008.Google Scholar
[9]Cai, Jian-Feng, Chan, Raymond H., and Nikolova, Mila. Fast two-phase image deblurring under impulse noises. J. Math. Imaging Vis., 36(1):46–53, 2010.Google Scholar
[10]Cai, Jian-Feng, Chan, Raymond H., Shen, Lixin, and Shen, Zuowei. Restoration of chopped and nodded images by framelets. SIAM J. Sci. Comput., 30(3):1205–1227, 2008.Google Scholar
[11]Cai, Jian-Feng, Chan, Raymond H., Shen, Lixin, and Shen, Zuowei. Simultaneouslyinpainting in image and transformed domains. Numer. Math., 112(4):509–533, 2009.Google Scholar
[12]Cai, Jian-Feng, Chan, Raymond H., and Shen, Zuowei. A framelet-based image inpainting algorithm. Appl. Comput. Harmon. Anal., 24(2):131–149, 2008.Google Scholar
[13]Cai, Jian-Feng, Ji, Hui, Liu, Chaoqiang, and Shen, Zuowei. Blind motion deblurring from a single image using sparse approximation. In IEEE Conference on Computer Vision and Pattern Recognition, Miami, 2009.Google Scholar
[14]Cai, Jian-Feng, Ji, Hui, Liu, Chaoqiang, and Shen, Zuowei. Blind motion deblurring using multiple images. J. Comput. Phys., 228(14):5057–5071, 2009.Google Scholar
[15]Cai, Jian-Feng, Osher, Stanley, and Shen, Zuowei. Linearized Bregman iterations for frame-based image deblurring. SIAM J. Imaging Sci., 2(1):226–252, 2009.Google Scholar
[16]Cai, Jian-Feng, Osher, Stanley, and Shen, Zuowei. Split bregman methods and frame based image restoration. Multiscale Model. Simul., 8(14):5057–5071, 2009.Google Scholar
[17]Chai, Anwei and Shen, Zuowei. Deconvolution: a wavelet frame approach. Numer. Math., 106(4):529–587, 2007.Google Scholar
[18]Chan, Raymond H., Chan, Tony F., Shen, Lixin, and Shen, Zuowei. Wavelet algorithms for high-resolution image reconstruction. SIAM J. Sci. Comput., 24(4):1408–1432, 2003.Google Scholar
[19]Chan, Raymond H., Riemenschneider, Sherman D., Shen, Lixin, and Shen, Zuowei. Tight frame: an efficient way for high-resolution image reconstruction. Appl. Comput. Harmon. Anal., 17(1):91–115, 2004.Google Scholar
[20]Chan, Tony F. and Shen, Jianhong. Image processing and analysis: Variational, PDE, wavelet, and stochastic methods. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2005.Google Scholar
[21]Chen, Tao, Ma, Kai-Kuang, and Chen, Li-Hui. Tri-state median filter for image denoising. IEEE Trans. Image Processing, 8(12):1834–1838, 1999.CrossRefGoogle ScholarPubMed
[22]Chen, Tao and Wu, Hong Ren. Adaptive impulse dectection using center-weighted median filters. IEEE Signal Processing Letters, 8(1):1–3, 2001.Google Scholar
[23]Criminisi, A., Perez, P., and Toyama, K.. Region filling and object removal by exemplar-based image inpainting. IEEE Trans. Image Processing, 13(9):1200–1212, 2004.Google Scholar
[24]Daubechies, Ingrid. Ten lectures on wavelets. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992.Google Scholar
[25]Daubechies, Ingrid, Han, Bin, Ron, Amos, and Shen, Zuowei. Framelets: MRA-based constructions of wavelet frames. Appl. Comput. Harmon. Anal., 14(1):1–46, 2003.CrossRefGoogle Scholar
[26]Esser, Ernie. Applications of lagrangian-based alternating direction methods and connections to split Bregman. Technical report, CAM report, UCLA, 2009.Google Scholar
[27]Goldstein, Tom, Bresson, Xavier, and Osher, Stanley. Geometric applications of the split breg-man method: Segmentation and surface reconstruction. Technical report, CAM report, UCLA, 2009.Google Scholar
[28]Goldstein, Tom and Osher, Stanley. The split Bregman method for L1-regularized problems. SIAM J. Imaging Sci., 2(2):323–343, 2009.Google Scholar
[29]Han, Bin and Shen, Zuowei. Dual wavelet frames and Riesz bases in Sobolev spaces. Constr. Approx., 29(3):369–406, 2009.CrossRefGoogle Scholar
[30]Hwang, H. and Haddad, R.A.. Adaptive median filters: new algorithms and results. IEEE Trans. Image Processing, 4(4):499–502, 1995.Google Scholar
[31]Ji, Hui, Shen, Zuowei, and Xu, Yuhong. Wavelet frame based scene reconstruction from range data. J. Comput. Phys. 229(6):2093–2108, 2010.CrossRefGoogle Scholar
[32]Ko, S.-J. and Lee, Y.H.. Center weighted median filters and their applications to image enhancement. IEEE Trans. Circuits and Systems, 38(9):984–993, 1991.Google Scholar
[33]Mumford, David and Shah, Jayant. Optimal approximations by piecewise smooth functions and associated variational problems. Comm. Pure Appl. Math., 42(5):577–685, 1989.Google Scholar
[34]Nikolova, Mila. Minimizers of cost-functions involving nonsmooth data-fidelity terms. Application to the processing of outliers. SIAM J. Numer. Anal., 40(3):965–994 (electronic), 2002.Google Scholar
[35]Nikolova, Mila. A variational approach to remove outliers and impulse noise. J. Math. Imaging Vis., 20:99–120, 2004.Google Scholar
[36]Osher, Stanley, Burger, Martin, Goldfarb, Donald, Xu, Jinjun, and Yin, Wotao. An iterative regularization method for total variation-based image restoration. SIAM Multiscale Model. Simul., 4:460–489, 2005.Google Scholar
[37]Ron, Amos and Shen, Zuowei. Affine systems in L 2(ℝd): the analysis of the analysis operator. J. Funct. Anal., 148(2):408–447, 1997.Google Scholar
[38]Rudin, Leonid, Osher, Stanley, and Fatemi, Emad. Nonlinear total variation based noise removal algorithms. Physica D, 60:259–268, 1992.Google Scholar
[39]Setzer, Simon. Split bregman algorithm, douglas-rachford splitting and frame shrinkage. In Proceedings ofthe Second International Conference on Scale Space Methods and Variational Methods in Computer Vision 2009, 2009.Google Scholar
[40]Shen, Zuowei, Toh, Kim-Chuan, and Yun, Sangwoon. An accelerated proximal gradient algorithm for image restoration. preprint, 2009.Google Scholar
[41]Tikhonov, AN and Arsenin, VA. Solution of Ill-posed Problems. Winston & Sons, Washington, 1977.Google Scholar
[42]Yang, Junfeng, Zhang, Yin, and Yin, Wotao. An efficient TVL1 algorithm for deblurring multichannel images corrupted by impulsive noise. SIAM J. Sci. Comput., 31(4):2842–2865, 2009.Google Scholar