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Directional Frames for Image Recovery: Multi-scale Discrete Gabor Frames

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Abstract

Sparsity-driven image recovery methods assume that images of interest can be sparsely approximated under some suitable system. As discontinuities of 2D images often show geometrical regularities along image edges with different orientations, an effective sparsifying system should have high orientation selectivity. There have been enduring efforts on constructing discrete frames and tight frames for improving the orientation selectivity of tensor product real-valued wavelet bases/frames. In this paper, we studied the general theory of discrete Gabor frames for finite signals, and constructed a class of discrete 2D Gabor frames with optimal orientation selectivity for sparse image approximation. Besides high orientation selectivity, the proposed multi-scale discrete 2D Gabor frames also allow us to simultaneously exploit sparsity prior of cartoon image regions in spatial domain and the sparsity prior of textural image regions in local frequency domain. Using a composite sparse image model, we showed the advantages of the proposed discrete Gabor frames over the existing wavelet frames in several image recovery experiments.

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  1. http://www3.math.tu-berlin.de/numerik/www.shearlab.org/software.

References

  1. Bölcskei, H., Hlawatsch, F.: Discrete Zak transforms, polyphase transforms, and applications. IEEE Trans. Signal Process. 45(4), 851–867 (1997)

    Article  Google Scholar 

  2. Cai, J., Chan, R., Shen, Z.: A framelet-based image inpainting algorithm. Appl. Comput. Harmon. Anal. 24(2), 131–149 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cai, J., Chan, R., Shen, Z.: Simultaneous cartoon and texture inpainting. Inverse Probl. Imaging 4(3), 379–395 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cai, J., Don, B., Shen, Z., Osher, S.: Image restoration: total variation; wavelet frames; and beyond. J. Am. Math. Soc. 25(4), 1033–1089 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cai, J., Ji, H., Liu, C., Shen, Z.: Framelet based blind image deblurring from a single image. IEEE Trans. Image Process. 21(2), 562–572 (2012)

    Article  MathSciNet  Google Scholar 

  6. Cai, J., Osher, S., Shen, Z.: Split Bregman methods and frame based image restoration. Multiscale Model. Simul. 8(2), 337–369 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Candes, E., Donoho, D.L.: New tight frames of curvelets and optimal representations of objects with piecewise-\(C^2\) singularities. Commun. Pure Appl. Math. 57, 219–266 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  8. Christensen, O.: An Introduction to Frames and Riesz Bases. Birkhäuser, Boston (2002)

    MATH  Google Scholar 

  9. Christensen, O.: Pairs of dual Gabor frame generators with compact support and desired frequency localization. Appl. Comput. Harmon. Anal. 20(3), 403–410 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  10. Coifman, R., Donoho, D.: Translation-invariant de-noising. In: Antoniadis, A., Oppenheim, G. (eds.) Wavelet and Statistics. Springer Lecture Notes in Statistics, vol. 103, pp. 125–150. Springer (1994)

  11. Cvetkovic, Z., Vetterli, M.: Tight Weyl–Heisenberg frame in \(\ell ^2(Z)\). IEEE Trans. Signal Process. 46(5), 1256–1260 (1998)

    Article  MathSciNet  Google Scholar 

  12. Daubechies, I.: Ten lectures on wavelets, vol. 61, pp. 198–202. SIAM, Philadelphia (1992)

  13. Daubechies, I., Han, B., Ron, A., Shen, Z.: Framelets: MRA-based constructions of wavelet frames. Appl. Comput. Harmon. Anal. 14, 1–46 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  14. Daugmann, J.G.: Two-dimensional spectral analysis of cortical receptive field profile. Vis. Res. 20, 847–856 (1980)

    Article  Google Scholar 

  15. De Boor, C.: B (asic)-Spline Basics. Mathematics Research Center, University of Wisconsin-Madison (1986)

  16. Elad, M., Milanfar, P., Rubinstein, R.: Analysis versus synthesis in signal priors. Inverse Probl. 23(3), 947 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. Elad, M., Starck, J.L., Querre, P., Donoho, D.L.: Simultaneous cartoon and texture image inpainting using morphological component analysis (mca). Appl. Comput. Harmon. Anal. 19(3), 340–358 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  18. Fan, Z., Heinecke, A., Shen, Z.: Duality for frames. J. Fourier Anal. Appl. 22(1), 71–136 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  19. Fan, Z., Ji, H., Shen, Z.: Dual gramian analysis: duality principle and unitary extension principle. Math. Comput. 85(297), 239–270 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  20. Feichtinger, H.G., Kozek, W., Luef, F.: Gabor analysis over finite Abelian groups. Appl. Comput. Harmon. Anal. 26(2), 230–248 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  21. Gabor, D.: Theory of communication. J. IEE 93(26), 429–457 (1946)

    Google Scholar 

  22. Guo, K., Labate, D., Lim, W., Wei, G., Wilson, E.: Wavelets with composite dilations. Electr. Res. Ann. AMS 10, 78–87 (2004)

    MathSciNet  MATH  Google Scholar 

  23. Janssen, A.: Duality and biorthogonality for Weyl–Heisenberg frames. J. Fourier Anal. Appl. 1(4), 403–436 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  24. Janssen, A.: From continuous to discrete Weyl–Heisenberg frames through sampling. J. Fourier Anal. Appl. 3(5), 583–596 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  25. Kutyniok, G., Labate, D.: Construction of regular and irregular shearlet frames. J. Wavelet Theory Appl. 1, 1–10 (2007)

    Google Scholar 

  26. Li, S.: Discrete multi-Gabor expansions. IEEE Trans. Inf. Theory 45(6), 1954–1958 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  27. Lim, W.Q.: The discrete shearlet transform: a new directional transform and compactly supported shearlet frames. IEEE Trans. Image Process. 19(5), 1166–1180 (2010)

    Article  MathSciNet  Google Scholar 

  28. Lopez, J., Han, D.: Discrete Gabor frames in \(\ell ^2(Z^d)\). Proc. Am. Thorac. Soc. 141(11), 3839–3851 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  29. Lu, Y., Morris, J.M.: Some results on discrete gabor transforms for finite periodic sequences. IEEE Trans. Signal Process. 46(6), 1703–1708 (1998)

    Article  MATH  Google Scholar 

  30. Mallat, S., Lepennec, E.: Sparse geometric image representation with bandelets. IEEE Trans. Image Process. 14, 423–438 (2005)

    Article  MathSciNet  Google Scholar 

  31. Meyer, F.G., Coifman, R.R.: Brushlets: a tool for directional image analysis and image compression. Appl. Comput. Harmon. Anal. 4(2), 147–187 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  32. Morris, J.M., Lu, Y.: Discrete gabor expansion of discrete-time signals in \({\ell }^2(Z)\) via frame theory. Signal Process. 40, 155–181 (1994)

    Article  MATH  Google Scholar 

  33. Orr, R.S.: Derivation of the finite discrete Gabor transform by periodization and sampling. Signal Process. 34(1), 85–97 (1993)

    Article  MATH  Google Scholar 

  34. Qian, S., Chen, D.: Discrete Gabor transform. IEEE Trans. Signal Process. 41(7), 2429–2438 (1993)

    Article  MATH  Google Scholar 

  35. Qiu, S.: Discrete Gabor transforms: the Gabor–Gram matrix approach. J. Fourier Anal. Appl. 4(1), 1–17 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  36. Ron, A., Shen, Z.: Frames and stable bases for subspaces of \(L^2(R^d)\): the duality principle of Weyl-Heisenberg sets. In: Proceedings of the Lanczos Centenary Conference Raleigh, pp. 422–425. SIAM Pub (1993)

  37. Ron, A., Shen, Z.: Affine systems in \(L_2(R^d)\): the analysis of the analysis operator. J. Funct. Anal. 148, 408–447 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  38. Ron, A., Shen, Z.: Weyl–Heisenberg frames and Riesz bases in \(L^2(R^d)\). Duke Math. J. 89, 237–282 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  39. Selesnick, I.W., Baraniuk, R.G., Kingsbury, N.C.: The dual-tree complex wavelet transform. IEEE Signal Process. Mag. 22(6), 123–151 (2005)

    Article  Google Scholar 

  40. Shen, Z.: Wavelet frames and image restorations. In: Proceedings of the International congress of Mathematicians, vol. 4, pp. 2834–2863. Hindustan Book Agency (2010)

  41. Søndergaard, P.L.: Gabor frames by sampling and periodization. Adv. Comput. Math. 27(4), 355–373 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  42. Starck, J., Candès, E.: The curvelet transform for image denoising. IEEE Trans. Image Process. 11(6), 670–684 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  43. Starck, J., Nguyen, M., Murtagh, F.: Wavelets and curvelets for image deconvolution: a combined approach. Signal Process. 83(10), 2279–2283 (2003)

    Article  MATH  Google Scholar 

  44. 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, 1570–1582 (2005)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  46. Wexler, J., Raz, S.: Discrete gabor expansions. Signal Process. 21(3), 207–220 (1990)

    Article  Google Scholar 

Download references

Acknowledgments

The authors would like to thank the editor and the reviewers for their helpful comments and suggestions. This work was partially supported by Singapore MOE AcRF Research Grant MOE2012-T3-1-008 and R-146-000-165-112.

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Authors and Affiliations

  1. Department of Mathematics, National University of Singapore, Lower Kent Ridge Road, Singapore, 119076, Singapore

    Hui Ji, Zuowei Shen & Yufei Zhao

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  1. Hui Ji
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  2. Zuowei Shen
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  3. Yufei Zhao
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Correspondence to Hui Ji.

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Communicated by Akram Aldroubi.

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Ji, H., Shen, Z. & Zhao, Y. Directional Frames for Image Recovery: Multi-scale Discrete Gabor Frames. J Fourier Anal Appl 23, 729–757 (2017). https://doi.org/10.1007/s00041-016-9487-5

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