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Compression and denoising using l 0-norm

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Abstract

In this paper, we deal with l 0-norm data fitting and total variation regularization for image compression and denoising. The l 0-norm data fitting is used for measuring the number of non-zero wavelet coefficients to be employed to represent an image. The regularization term given by the total variation is to recover image edges. Due to intensive numerical computation of using l 0-norm, it is usually approximated by other functions such as the l 1-norm in many image processing applications. The main goal of this paper is to develop a fast and effective algorithm to solve the l 0-norm data fitting and total variation minimization problem. Our idea is to apply an alternating minimization technique to solve this problem, and employ a graph-cuts algorithm to solve the subproblem related to the total variation minimization. Numerical examples in image compression and denoising are given to demonstrate the effectiveness of the proposed algorithm.

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References

  1. Antonini, M., Barlaud, M., Mathieu, P., Daubechies, I.: Image coding using wavelet transform. IEEE Trans. Image Process. 1, 205–220 (1992)

    Article  Google Scholar 

  2. Bae, E., Tai, X.-C.: Graph cuts for the multiphase Mumford-Shah model using piecewise constant level set methods. UCLA, Appl. Math., CAM-Report No. 08-36, June 2008

  3. Bae, E., Tai, X.-C.: Graph cut optimization for the piecewise constant level set method applied to multiphase image segmentation. In: Processing in Scale Space and Variational Methods in Computer Vision, 2009. Lecture Notes in Computer Science, vol. 5567, pp. 1–13. Springer, Berlin/Heidelberg (2009)

    Chapter  Google Scholar 

  4. Chambolle, A.: Total variation minimization and a class of binary mrf models. In: Energy Minimization Methods in Computer Vision and Pattern Recognition, pp. 136–152. Springer, Berlin/Heidelberg (2005)

    Chapter  Google Scholar 

  5. Chan, T., Zhou, H.M.: Optimal constructions of wavelet coefficients using total variation regularization in image compression. UCLA, Applied Mathematics, CAM Report No. 00–27, July 2000

  6. Chen, Y.W.: Vector quantization by principal component analysis. M.S. Thesis, National Tsing Hua University, June (1998)

  7. Chen, S.S., Donoho, D.L., Saunders, M.A.: Atomic decomposition by basis pursuit. SIAM J. Sci. Comput. 20(1), 33–61 (1998)

    Article  MathSciNet  Google Scholar 

  8. Darbon, J., Sigelle, M.: Image restoration with discrete constrained total variation. Part I. Fast and exact optimization. J. Math. Imaging Vis. 26(3), 261–276 (2006)

    Article  MathSciNet  Google Scholar 

  9. DeVore, R.A., Temlyakov, V.N.: Some remarks on greedy algorithm. Adv. Comput. Math. 12, 213–227 (1996)

    MathSciNet  Google Scholar 

  10. Donoho, D.L.: For most large undetermined systems of linear equations the minimal l 1-norm solution is also the sparsest solution. Technical Report, September 2004

  11. Durand, S., Froment, J.: Artifact free signal denoising with wavelets. In: Proceedings of ICASSP’01, vol. 6, pp. 3685–3688 (2001)

  12. Goldberg, A.V., Tarjan, R.E.: A new approach to the maximum-flow problem. J. Asoc. Comput. Mach. 35(4), 921–940 (1998)

    MathSciNet  Google Scholar 

  13. Huang, Y., Ng, M.K., Wen, Y.: A fast total variation minimization method for image restoration. Multiscale Model. Simul. 7(2), 774–795 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. Ishikawa, H.: Exact optimization for Markov random fields with convex priors. IEEE Trans. Pattern Anal. Mach. Intel. 25(10), 1333–1336 (2003)

    Article  Google Scholar 

  15. Ishikawa, H., Geiger, D.: Segmentation by grouping junctions. In: CVPR’ 98: Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition, p. 125. Washington, DC, USA, 1998

  16. Jacquin, A.E.: Image coding based on a fractal theory of iterated contractive image transformations. IEEE Trans. Image Process. 1, 18–30 (1992)

    Article  Google Scholar 

  17. Lewis, A.S., Knowles, K.: Image compression using 2D wavelet transform. IEEE Trans. Image Process. 1, 244–250 (1992)

    Article  Google Scholar 

  18. Linde, Y., Buzo, A., Gray, R.M.: An algorithm for vector quantizer design. IEEE Trans. Commun. 36, 84–95 (1980)

    Article  Google Scholar 

  19. Mohimani, G.H., Babaie-Zadeh, M., Jutten, C.: A fast approach for overcomplete sparse decomposition based on smoothed L 0 norm. IEEE Trans. Signal Process. 57(1), 289–301 (2009)

    Article  MathSciNet  Google Scholar 

  20. Mancera, L., Portilla, J.: L 0-norm-based sparse representation through alternate projections. In: IEEE International Conference on Image Processing, pp. 2089–2092, Atlanta, 2006

  21. Pennebaker, W.B., Mitchell, J.: JPEG Still Image Compression Standard. Van Nostrand Reinhold, New York (1993)

    Google Scholar 

  22. Ranchin, F., Chambolle, A., Dibos, F.: Total variation minimization and graph cuts for moving objects segmentation: scale space and variational methods in computer vision, pp. 743–753. Springer, Berlin (2006)

    Google Scholar 

  23. Rudin, L., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60, 259–268 (1992)

    MATH  Google Scholar 

  24. Said, A., Pearlman, W.A.: A new, fast, and efficient image codec based on set partitioning in hierarchical trees. IEEE Trans. Circuits Syst. Video Technol. 6, 243–250 (1996)

    Article  Google Scholar 

  25. Starck, J.-L., Elad, M., Donoho, D.L.: Image decomposition via the combination of sparse representations and a variational approach. IEEE Trans. Image Process. 14(10), 1570–1582 (2005)

    Article  MathSciNet  Google Scholar 

  26. Shapiro, J.M.: Embedded image coding using zerotree of wavelet coefficients. IEEE Trans. Signal Process. 41, 3445–3462 (1993)

    Article  MATH  Google Scholar 

  27. Wang, Y., Zhou, H.: Total variation wavelet-based medical image denoising. Int. J. Biomed. Imaging 2006, 1–6 (2006)

    Google Scholar 

  28. Yau, A.C., Tai, X.C., Ng, M.K.: L 0-norm and total variation for wavelet inpainting. In: Processing in Scale Space and Variational Methods in Computer Vision, 2009. Lecture Notes in Computer Science, vol. 5567, pp. 539–551. Springer, Berlin/Heidelberg (2009)

    Chapter  Google Scholar 

  29. Wallace, G.K.: The JPEG still picture compression standard. Commun. ACM 34, 31–44 (1991)

    Article  Google Scholar 

  30. Wipf, D.P., Rao, B.D.: Sparse Bayesian learning for basis selection. IEEE Trans. Signal Process. 52(8), 2153–2164 (2004)

    Article  MathSciNet  Google Scholar 

  31. Wang, Y., Yang, J., Yin, W., Zhang, Y.: A new alternating minimization algorithm for total variation image reconstruction. SIAM J. Imaging Sci. 1(3), 248–272 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Division of Mathematical Science, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore, Singapore

    Andy C. Yau & Xuecheng Tai

  2. Center for Mathematical Imaging and Vision and Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong

    Michael K. Ng

Authors
  1. Andy C. Yau
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  2. Xuecheng Tai
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  3. Michael K. Ng
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Corresponding author

Correspondence to Michael K. Ng.

Additional information

The paper is dedicated to Professor Liqun Qi in celebration of his 65th birthday. Michael Ng would like to thank Professor Liqun Qi for his guidance in Michael’s research career.

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Yau, A.C., Tai, X. & Ng, M.K. Compression and denoising using l 0-norm. Comput Optim Appl 50, 425–444 (2011). https://doi.org/10.1007/s10589-010-9352-4

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  • DOI: https://doi.org/10.1007/s10589-010-9352-4

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