Shearlets pp 283-325 | Cite as

Image Processing Using Shearlets

Chapter
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

Abstract

Since shearlets provide nearly optimally sparse representations for a large class of functions that are useful to model natural images, many image processing methods benefit from their use. In particular, the error rates of data estimation from noise are highly dependent on the sparsity properties of the representation, so that many successful applications of shearlets center around restoration tasks such as denoising and inverse problems. Other imaging problems, where also the application of the shearlet representation turns out to be very beneficial, include image enhancement, image separation, edge detection, and estimation of the geometric features of an object.

Key words

Curvelets Deconvolution Denoising Edge detection Geometric separation Image processing Shearlets Sparsity Wavelets Video denoising 

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Notes

Acknowledgements

D.L. acknowledges support from NSF grants DMS 1008900 and DMS (Career) 1005799.

References

  1. 1.
    J. Aelterman, H. Q. Luong, B. Goossens, A. Pizurica, W. Philips, Compass: a joint framework for Parallel Imaging and Compressive Sensing in MRI, Image Processing (ICIP), 17th IEEE International Conference on (2010), 1653–1656.Google Scholar
  2. 2.
    A. Averbuch, R. R. Coifman, D. L. Donoho, M. Israeli, and Y. Shkolnisky, A framework for discrete integral transformations I - the pseudo-polar Fourier transform, SIAM Journal on Scientific Computing 30(2) (2008), 764–784.MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    M. Bertero, Linear inverse and ill-posed problems, Advances in Electronics and Electron Physics (P.W. Hawkes, ed.), Academic Press, New York, 1989.Google Scholar
  4. 4.
    L. Blanc-Feraud, P. Charbonnier, G. Aubert, and M. Barlaud, Nonlinear image processing: modelling and fast algorithm for regularization with edge detection, Proc. IEEE ICIP-95, 1 (1995), 474–477.Google Scholar
  5. 5.
    P. J. Burt, E. H. Adelson, The Laplacian pyramid as a compact image code, IEEE Trans. Commun. 31 (4) (1983), 532–540.CrossRefGoogle Scholar
  6. 6.
    E. J. Candès, and D. L. Donoho, Recovering edges in ill-posed inverse problems: optimality of curvelet frames, Annals Stat. 30(3) (2002), 784–842.MATHCrossRefGoogle Scholar
  7. 7.
    E. J. Candès and F. Guo, New multiscale transforms, minimum total variation synthesis: applications to edge-preserving image reconstruction, Signal Proc. 82(11) (2002), 1519–1543.MATHCrossRefGoogle Scholar
  8. 8.
    F. J. Canny, A computational approach to edge detection, IEEE Trans. Pattern Anal. Machine Intell. 8(6) (1986), 679–698.CrossRefGoogle Scholar
  9. 9.
    H. Cao, W. Tian, C. Deng, Shearlet-based image denoising using bivariate model, Progress in Informatics and Computing (PIC), 2010 IEEE International Conference on 2 (2010), 818–821.Google Scholar
  10. 10.
    T. Chan, J. Shen, Image Processing And Analysis: Variational, PDE, Wavelet, And Stochastic Methods, SIAM, Philadelphia (2005).MATHGoogle Scholar
  11. 11.
    C. Chang, A. F. Laine, Coherence of Multiscale Features for Contrast Enhancement of Digital Mammograms, IEEE Trans. Info. Tech. in Biomedicine 3(1) (1999), 32–46.CrossRefGoogle Scholar
  12. 12.
    G. Chang, B. Yu and M. Vetterli, Adaptive Wavelet Thresholding for Image Denoising and Compression, IEEE Trans. Image Processing, 9 (2000), 1532–1546.MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    X. Chen, C. Deng, S. Wang, Shearlet-Based Adaptive Shrinkage Threshold for Image Denoising, E-Business and E-Government (ICEE), 2010 International Conference on (2010), 1616–1619.Google Scholar
  14. 14.
    X. Chen, H. Sun, C. Deng, Image Denoising Algorithm Using Adaptive Shrinkage Threshold Based on Shearlet Transform, Frontier of Computer Science and Technology, 2009, Fourth International Conference on (2009), 254–257.Google Scholar
  15. 15.
    R. R. Coifman and A. Sowa, Combining the calculus of variations and wavelets for image enhancement, Appl. Comput. Harmon. Anal., 9 (2000), 1–18.MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    F. Colonna, G. R. Easley, Generalized discrete Radon transforms and their use in the ridgelet transform, Journal of Mathematical Imaging and Vision, 23 (2005), 145–165.MathSciNetCrossRefGoogle Scholar
  17. 17.
    F. Colonna, G. R. Easley, K. Guo, and D. Labate, Radon Transform Inversion using the Shearlet Representation, Appl. Comput. Harmon. Anal. 29(2) (2010), 232–250.MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    C. Deng, H. Sun, X. Chen, Shearlet-Based Adaptive Bayesian Estimator for Image Denoising, Frontier of Computer Science and Technology, 2009, Fourth International Conference on (2009), 248–253.Google Scholar
  19. 19.
    C. Deng, S. Wang, X. Chen, Remote Sensing Images Fusion Algorithm Based on Shearlet Transform, Environmental Science and Information Application Technology, 2009, International Conference on 3 (2009), 451–454.Google Scholar
  20. 20.
    D. L. Donoho, Unconditional bases are optimal bases for data compression and for statistical estimation, Appl. Comput. Harmon. Anal. 1(1) (1993), 100–115.MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    D. L. Donoho, Nonlinear solution of linear inverse problems by wavelet-vaguelette decomposition, Appl. Comput. Harmon. Anal. 2 (1995), 101–126.MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    D. L. Donoho, De-noising by soft thresholding, IEEE Trans. Info. Theory 41 (1995), 613–627.MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    D. L. Donoho and I. M. Johnstone, Ideal spatial adaptation via wavelet shrinkage, Biometrika 81 (1994), 425–455.MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    D. L. Donoho and I. M. Johnstone, Adapting to unknown smoothness via wavelet shrinkage, J. Amer. Stat. Assoc. 90(432) (1995), 1200–1224.MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    D. L. Donoho and G. Kutyniok, Geometric Separation using a Wavelet-Shearlet Dictionary, SampTA-09 (Marseille, France, 2009), Proc., 2009.Google Scholar
  26. 26.
    D. L. Donoho and G. Kutyniok, Microlocal analysis of the geometric separation problem, Comm. Pure Appl. Math., to appear.Google Scholar
  27. 27.
    D. L. Donoho, M. Vetterli, R. A. DeVore, and I. Daubechies, Data compression and harmonic analysis, IEEE Trans. Inform. Theory, 44 (1998), 2435–2476.MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    S. Durand and J. Froment, Reconstruction of wavelet coefficients using total variation minimization, SIAM J. Sci. Comput., 24(5) (2003), 1754–1767.MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    G. R. Easley, F. Colonna, and D. Labate, Improved Radon Based Imaging using the Shearlet Transform, Proc. SPIE, Independent Component Analyses, Wavelets, Unsupervised Smart Sensors, Neural Networks, Biosystems, and Nanoengineering VII, 7343, Orlando, April 2009.Google Scholar
  30. 30.
    G. R. Easley, D. Labate, Critically sampled composite wavelets, Signals, Systems and Computers, 2009 Conference Record of the Forty-Third Asilomar Conference on (2009), 447–451.Google Scholar
  31. 31.
    G. R. Easley, D. Labate, and F. Colonna, Shearlet-Based Total Variation for Denoising, IEEE Trans. Image Processing, 18(2) (2009), 260–268.MathSciNetCrossRefGoogle Scholar
  32. 32.
    G. R. Easley, D. Labate, and W-Q Lim, Sparse Directional Image Representations using the Discrete Shearlet Transform, Appl. Comput. Harmon. Anal. 25(1) (2008), 25–46.MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    G. R. Easley, V. Patel, D. M. Healy, Jr., An M-channel Directional Filter Bank Compatible with the Contourlet and Shearlet Frequency Tiling, Wavelets XII, Proceedings of SPIE, San Diego, CA (2007), 26–30.Google Scholar
  34. 34.
    G. R. Easley, V. M. Patel, and D. M. Healy, Jr., Inverse halftoning using a shearlet representation, Proc. of SPIE Wavelets XIII, 7446, San Diego, August 2009.Google Scholar
  35. 35.
    M. J. Fadilli, J. L Starck, M. Elad, and D. L. Donoho, MCALab: reproducible research in signal and image decomposition and inpainting, IEEE Comput. Sci. Eng. Mag. 12(1) (2010), 44–63.Google Scholar
  36. 36.
    R. W. Floyd and L. Steinberg, An adaptive algorithm for spatial grayscale, Proc. Soc. Image Display 17(2) (1976), 75–77.Google Scholar
  37. 37.
    A. Foi, V. Katkovnik, K. Egiazarian, and J. Astola, Inverse halftoning based on the anisotropic LPA-ICI deconvolution, Proc. Int. TICSP Workshop Spectral Methods Multirate Signal Processing, (Vienna, Austria) (2004), 49–56.Google Scholar
  38. 38.
    D. Geman and C. Yang, Nonlinear image recovery with half-quadratic regularization, IEEE Trans. Image Proc. 4 (1995), 932–946.CrossRefGoogle Scholar
  39. 39.
    J. Geusebroek, A. W. M. Smeulders, and J. van de Weijer, Fast anisotropic Gauss filtering, IEEE Trans. Image Proc. 8 (2003), 938–943.CrossRefGoogle Scholar
  40. 40.
    G. Gilboa, Y. Y. Zeevi, and N. Sochen, Texture preserving variational denoising using an adaptive fidelity term, Proc. VLSM, Nice (2003), 137–144.Google Scholar
  41. 41.
    R. Gomathi and A. Kumar, An efficient GEM model for image inpainting using a new directional sparse representation: Discrete Shearlet Transform, Computational Intelligence and Computing Research (ICCIC), 2010 IEEE International Conference on (2010), 1–4.Google Scholar
  42. 42.
    B. Goossens, J. Aelterman, H. Luong, A. Pizurica, and W. Philips, Efficient design of a low redundant Discrete Shearlet Transform, Local and Non-Local Approximation in Image Processing, 2009, International Workshop on (2009), 112–124.Google Scholar
  43. 43.
    K. Guo and D. Labate, Optimally sparse multidimensional representation using shearlets, SIAM J. Math. Anal. 39 (2007), 298–318.MathSciNetMATHCrossRefGoogle Scholar
  44. 44.
    K. Guo and D. Labate, Characterization and analysis of edges using the continuous shearlet transform, SIAM J. Imaging Sciences 2 (2009), 959–986.MathSciNetMATHCrossRefGoogle Scholar
  45. 45.
    K. Guo, D. Labate and W. Lim, Edge analysis and identification using the continuous shearlet transform, Appl. Comput. Harmon. Anal. 27 (2009), 24–46.MathSciNetMATHCrossRefGoogle Scholar
  46. 46.
    Q. Guo, S. Yu, X. Chen, C. Liu, and W. Wei, Shearlet-based image denoising using bivariate shrinkage with intra-band and opposite orientation dependencies, Computational Sciences and Optimization, 2009, International Joint Conference on, 1 (2009), 863–866.Google Scholar
  47. 47.
    J. Jarvis, C. Judice, and W. Ninke, A survey of techniques for the display of continuous tone pictures on bilevel displays, Comput. Graph and Image Proc. 5 (1976), 13–40.CrossRefGoogle Scholar
  48. 48.
    T. D. Kite, B. L. Evans, and A. C. Bovik, Modeling and quality assessment of halftoning by error diffusion, IEEE Trans. Image Proc. 9 (2000), 909–922.CrossRefGoogle Scholar
  49. 49.
    G. Kutyniok and D. Labate, Resolution of the wavefront set using continuous shearlets, Trans. Amer. Math. Soc. 361 (2009), 2719–2754.MathSciNetMATHCrossRefGoogle Scholar
  50. 50.
    G. Kutyniok and W. Lim, Image separation using shearlets, in: Curves and Surfaces (Avignon, France, 2010), Lecture Notes in Computer Science 6920, Springer, 2012.Google Scholar
  51. 51.
    D. Labate and P. Negi, 3D Discrete shearlet transform and video denoising, Wavelets and Sparsity XIV (San Diego, CA, 2011), SPIE Proc. 8138, SPIE, Bellingham, WA, 2011.Google Scholar
  52. 52.
    A. F. Laine, S. Schuler, J. Fan, and W. Huda, Mammographic feature enhancement by multiscale analysis, IEEE Trans. Med. Imag. 13(4) (1994), 725–752.CrossRefGoogle Scholar
  53. 53.
    A. F. Laine and X. Zong, A multiscale sub-octave wavelet transform for de-noising and enhancement, Wavelet Applications, Proc. SPIE, Denver, CO, August 6-9, 1996, 2825, 238–249.Google Scholar
  54. 54.
    N. Lee and B J Lucier, Wavelets methods for inverting the Radon transform with noisy data, IEEE Trans. Image Proc. 10(1) (2001), 79–94.Google Scholar
  55. 55.
    W. Q. Lim, The discrete shearlet transform: a new directional transform and compactly supported shearlet frames, Image Proc. IEEE Transactions on 19(5) (2010), 1166–1180.CrossRefGoogle Scholar
  56. 56.
    J. Lu and D. M. Healy, Jr., Contrast enhancement via multi-scale gradient transformation, Wavelet Applications, Proc. SPIE, Orlando, FL, April 5-8, 1994.Google Scholar
  57. 57.
    L. Lü , J. Zhao, and H. Sun, Multi-focus image fusion based on shearlet and local energy, Signal Processing Systems (ICSPS), 2010 2nd International Conference on, 1 (2010), V1-632–V1-635.Google Scholar
  58. 58.
    J. Ma and M. Fenn, Combined complex ridgelet shrinkage and total variation minimization, SIAM J. Sci. Comput., 28(3) (2006), 984–1000.MathSciNetMATHCrossRefGoogle Scholar
  59. 59.
    S. Mallat, A Wavelet Tour of Signal Processing, Academic Press, San Diego, 1998.MATHGoogle Scholar
  60. 60.
    S. Mallat and W. L. Hwang, Singularity detection and processing with wavelets, IEEE Trans. Inf. Theory 38(2) (1992), 617–643.MathSciNetMATHCrossRefGoogle Scholar
  61. 61.
    S. Mallat and S. Zhong, Characterization of signals from multiscale edges, IEEE Trans. Pattern Anal. Mach. Intell. 14(7) (1992), 710–732.CrossRefGoogle Scholar
  62. 62.
    Y. Meyer, Oscillating Patterns in Image Processing and Nonlinear Evolution Equations, AMS, Providence, 2001.MATHGoogle Scholar
  63. 63.
    F. Natterer, The Mathematics of Computerized Tomography, Wiley, New York, 1986.MATHGoogle Scholar
  64. 64.
    F. Natterer and F. Wübbeling, Mathematical Methods in Image Reconstruction, SIAM Monographs on Mathematical Modeling and Computation, Philadelphia, 2001.MATHCrossRefGoogle Scholar
  65. 65.
    R. Neelamani, H. Choi, and R. G. Baraniuk, ForWaRD: Fourier-wavelet regularized deconvolution for ill-conditioned systems, IEEE Trans. Image Proc. 52(2) (2004), 418–433.MathSciNetGoogle Scholar
  66. 66.
    R. Neelamani, R. Nowak, and R. Baraniuk, Model-based inverse halftoning with Wavelet-Vaguelette Deconvolution, Proc. IEEE Int. Conf. Image Proc. (2000), 973–976.Google Scholar
  67. 67.
    P. Negi and D. Labate, 3D discrete shearlet transform and video processing, IEEE Trans. Image Proc., in press 2012.Google Scholar
  68. 68.
    V. M. Patel, G. R. Easley, and R. Chellappa, Multiscale directional filtering of noisy InSAR phase images, Proc. SPIE, Independent Component Analyses, Wavelets, Neural Networks, Biosystems, and Nanoengineering VII 7703, Orlando, April 2010.Google Scholar
  69. 69.
    V. M. Patel, G. R. Easley, and D. M. Healy, Jr., A new multiresolution generalized directional filter bank design and application in image enhancement, Proc. IEEE International Conference on Image Proc., San Diego, October 2008, 2816–2819.Google Scholar
  70. 70.
    V. M. Patel, G. R. Easley, and D. M. Healy, Jr., Shearlet-based deconvolution, IEEE Trans. Image Proc. 18(12) (2009), 2673–2685.MathSciNetCrossRefGoogle Scholar
  71. 71.
    P. Perona, Steerable-scalable kernels for edge detection and junction analysis, Image Vis. Comput. 10 (1992), 663–672.CrossRefGoogle Scholar
  72. 72.
    P. Perona and J. Malik, Scale-space and edge detection using anisotropic diffusion, IEEE Trans. Pattern Anal. Mach. Intel. 12 (1990), 629–639.CrossRefGoogle Scholar
  73. 73.
    L. Rudin, S. Oscher, and E. Fatemi, Nonlinear total variation based noise removal algorithms, Phys. D 60 (1992), 259–268.MATHCrossRefGoogle Scholar
  74. 74.
    O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier and F. Lenzen, Variational Methods in Imaging, Springer, Applied Mathematical Sciences 167, 2009.Google Scholar
  75. 75.
    O. Scherzer and J. Weickert, Relations between regularization and diffusion filtering Journal of Mathematical Imaging and Vision 12(1) (2000), 43–63.MathSciNetMATHCrossRefGoogle Scholar
  76. 76.
    D. A. Schug and G. R. Easley, Three dimensional Bayesian state estimation using shearlet edge analysis and detection, Communications, Control and Signal Processing (ISCCSP), 2010 4th International Symposium on (2010), 1–4.Google Scholar
  77. 77.
    D. A. Schug, G. R. Easley, and D. P. O’Leary, Three-dimensional shearlet edge analysis, Proc. SPIE, Independent Component Analyses, Wavelets, Neural Networks, Biosystems, and Nanoengineering IX, Orlando, April 2011.Google Scholar
  78. 78.
    J. L. Starck, E. J. Candès, and D. L. Donoho, The curvelet transform for image denoising, IEEE Trans. Im. Proc. 11 (2002), 670–684.CrossRefGoogle Scholar
  79. 79.
    J. L Starck, M. Elad, and D. L. Donoho, Image decomposition via the combination of sparse representation and a variational approach, IEEE Trans. Image Proc. 14 (2005), 1570–1582.Google Scholar
  80. 80.
    J. L. Starck, F. Murtagh, E. J. Candès, and D. L. Donoho, Gray and color image contrast enhancement by the curvelet transform, IEEE Trans. Imag. Proc. 12(6) (2003), 706–717.CrossRefGoogle Scholar
  81. 81.
    G. Steidl, J. Weickert, T. Brox, P. Mrázek, and M. Welk, On the equivalence of soft wavelet shrinkage, total variation diffusion, total variation regularization, and SIDEs, SIAM J. Numer. Anal. 42 (2004), 686–713.MathSciNetMATHCrossRefGoogle Scholar
  82. 82.
    R. N. Strickland and H. I. Hahn, Wavelet Transforms for Detecting Microcalcifications in Mammograms, IEEE Trans. on Med. Imag. 15(2) (1996), 218–229.CrossRefGoogle Scholar
  83. 83.
    H. Sun and J. Zhao, Shearlet Threshold Denoising Method Based on Two Sub-swarm Exchange Particle Swarm Optimization, Granular Computing (GrC), 2010 IEEE International Conference on (2010), 449–452.Google Scholar
  84. 84.
    S. Teboul, L. Blanc-Feraud, G. Aubert, and M. Barlaud, Variational approach for edge-preserving regularization using coupled PDEs, IEEE Trans. Image Proc. 7 (1998), 387–397.CrossRefGoogle Scholar
  85. 85.
    W. Tian, H. Cao, and C. Deng, Shearlet-based adaptive MMSE estimator for image denoising, Intelligent Computing and Intelligent Systems (ICIS), 2010 IEEE International Conference on 2 (2010), 689–692.Google Scholar
  86. 86.
    A. N. Tikhonov, Solution of incorrectly formulated problems and the regularization method, Soviet Math. Doklady 4 (1963), 1035–1039.Google Scholar
  87. 87.
    R. Ulichney, Digital Halftoning, MIT Press, Cambridge, MA, 1987.Google Scholar
  88. 88.
    J. Weickert, Foundations and applications of nonlinear anisotropic diffusion filtering, Z. Angew. Math. Mechan. 76 (1996), 283–286.MATHGoogle Scholar
  89. 89.
    J. Weickert, Anisotropic Diffusion in Image Processing, Teubner, Stuttgart, 1998.MATHGoogle Scholar
  90. 90.
    M. Welk, G. Steidl and J. Weickert, Locally analytic schemes: A link between diffuusion filtering and wavelet shrinkage, Appl. and Comput. Harmon. Anal. 24 (2008), 195–224.MathSciNetMATHCrossRefGoogle Scholar
  91. 91.
    S. Yi, D. Labate, G. R. Easley, and H. Krim, Edge detection and processing using shearlets, Proc. IEEE Int. Conference on Image Proc., San Diego, October 12-15, 2008.Google Scholar
  92. 92.
    S. Yi, D. Labate, G. R. Easley, and H. Krim, A Shearlet approach to edge analysis and detection, IEEE Trans. Image Proc. 18(5) (2009), 929–941CrossRefGoogle Scholar
  93. 93.
    X. Zhang, X. Sun, L. Jiao, and J. Chen, A Non-Local Means Filter with Translating Invariant Shearlet Feature Descriptors, Wireless Communications Networking and Mobile Computing (WiCOM), 2010 6th International Conference on (2010), 1–4.Google Scholar
  94. 94.
    X. Zhang, Q. Zhang, and L. Jiao, Image Denoising with Non-Local Means in the Shearlet Domain, Multi-Platform/Multi-Sensor Remote Sensing and Mapping (M2RSM), 2011 International Workshop on (2011), 1–5.Google Scholar
  95. 95.
    D. Ziou and S. Tabbone, Edge Detection Techniques An Overview, Internat. J. Pattern Recognition and Image Anal. 8(4) (1998), 537–559.Google Scholar

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© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.System Planning CorporationArlingtonUSA
  2. 2.Department of MathematicsUniversity of HoustonHoustonUSA

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