Advertisement

Shearlets pp 1-38 | Cite as

Introduction to Shearlets

  • Gitta KutyniokEmail author
  • Demetrio Labate
Chapter
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

Abstract

Shearlets emerged in recent years among the most successful frameworks for the efficient representation of multidimensional data. Indeed, after it was recognized that traditional multiscale methods are not very efficient at capturing edges and other anisotropic features which frequently dominate multidimensional phenomena, several methods were introduced to overcome their limitations. The shearlet representation stands out since it offers a unique combination of some highly desirable properties: it has a single or finite set of generating functions, it provides optimally sparse representations for a large class of multidimensional data, it is possible to use compactly supported analyzing functions, it has fast algorithmic implementations and it allows a unified treatment of the continuum and digital realms. In this chapter, we present a self-contained overview of the main results concerning the theory and applications of shearlets.

Key words

Affine systems Continuous wavelet transform Image processing Shearlets Sparsity Wavelets 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

G.K. acknowledges support by Deutsche Forschungsgemeinschaft (DFG) Grant KU 1446/14 and DFG Grant SPP-1324 KU 1446/13 and support by the Einstein Foundation Berlin. D.L. acknowledges support by Grant DMS 1008900 and DMS (Career) 1005799. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

References

  1. 1.
    J. P. Antoine, P. Carrette, R. Murenzi, and B. Piette, Image analysis with two-dimensional continuous wavelet transform, Signal Process. 31 (1993), 241–272.zbMATHCrossRefGoogle Scholar
  2. 2.
    R. H. Bamberger and M. J. T. Smith, A filter bank for the directional decomposition of images: theory and design, IEEE Trans. Signal Process. 40 (1992), 882–893.CrossRefGoogle Scholar
  3. 3.
    A. M. Bruckstein, D. L. Donoho, and M. Elad, From sparse solutions of systems of equations to sparse modeling of signals and images, SIAM Review 51 (2009), 34–81.MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    E. J. Candès and D. L. Donoho, New tight frames of curvelets and optimal representations of objects with piecewise C 2 singularities, Comm. Pure and Appl. Math. 56 (2004), 216–266.Google Scholar
  5. 5.
    P. G. Casazza and G. Kutyniok, Finite Frames: Theory and Applications, Birkhäuser, Boston, to appear.Google Scholar
  6. 6.
    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.MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    O. Christensen, An Introduction to Frames and Riesz Bases, Birkhäuser, Boston, 2003.zbMATHGoogle Scholar
  8. 8.
    S. Dahlke, G. Kutyniok, G. Steidl, and G. Teschke, Shearlet coorbit spaces and associated Banach frames, Appl. Comput. Harmon. Anal. 27 (2009), 195–214.MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    S. Dahlke, G. Kutyniok, P. Maass, C. Sagiv, H.-G. Stark, and G. Teschke, The uncertainty principle associated with the continuous shearlet transform, Int. J. Wavelets Multiresolut. Inf. Process. 6 (2008), 157–181.MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    S. Dahlke, G. Steidl, and G. Teschke, The continuous shearlet transform in arbitrary space dimensions, J. Fourier Anal. Appl. 16 (2010), 340–364.MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    S. Dahlke, G. Steidl and G. Teschke, Shearlet coorbit spaces: compactly supported analyzing shearlets, traces and embeddings, to appear in J. Fourier Anal. Appl. 17 (2011), 1232–1255.zbMATHCrossRefGoogle Scholar
  12. 12.
    S. Dahlke and G. Teschke, The continuous shearlet transform in higher dimensions: variations of a theme, in Group Theory: Classes, Representation and Connections, and Applications, edited by C. W. Danellis, Math. Res. Develop., Nova Publishers, 2010, 167–175.Google Scholar
  13. 13.
    I. Daubechies, Ten Lectures on Wavelets, SIAM, Philadelphia, 1992.zbMATHCrossRefGoogle Scholar
  14. 14.
    M. N. Do and M. Vetterli, The contourlet transform: an efficient directional multiresolution image representation, IEEE Trans. Image Process. 14 (2005), 2091–2106.MathSciNetCrossRefGoogle Scholar
  15. 15.
    D. L. Donoho, Sparse components of images and optimal atomic decomposition, Constr. Approx. 17 (2001), 353–382.MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    D. L. Donoho, Emerging applications of geometric multiscale analysis, Proceedings International Congress of Mathematicians Vol. I (2002), 209–233.Google Scholar
  17. 17.
    D. L. Donoho and G. Kutyniok, Geometric separation using a wavelet-shearlet dictionary, SampTA’09 (Marseille, France, 2009), Proc., 2009.Google Scholar
  18. 18.
    D. L. Donoho and G. Kutyniok, Microlocal analysis of the geometric separation problem, Comm. Pure Appl. Math., to appear.Google Scholar
  19. 19.
    D. L. Donoho, M. Vetterli, R. DeVore, and I. Daubechies, Data compression and harmonic analysis, IEEE Trans. Info. Theory 44 (1998), 2435–2476.MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    R.J. Duffin and A.C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc. 72 (1952), 341–366.MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    G. R. Easley, D. Labate, and F. Colonna, Shearlet based Total Variation for denoising, IEEE Trans. Image Process. 18(2) (2009), 260–268.MathSciNetCrossRefGoogle Scholar
  22. 22.
    G. Easley, D. Labate, and W.-Q Lim, Sparse directional image representations using the discrete shearlet transform, Appl. Comput. Harmon. Anal. 25 (2008), 25–46.Google Scholar
  23. 23.
    M. Elad, Sparse and Redundant Representations, Springer, New York, 2010.zbMATHCrossRefGoogle Scholar
  24. 24.
    C. Fefferman, A note on spherical summation multipliers, Israel J. Math. 15 (1973), 44–52.MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    G. Folland, Fourier Analysis and Its Applications, American Mathematical Society, Rhode Island, 2009.zbMATHGoogle Scholar
  26. 26.
    P. Grohs, Continuous shearlet frames and resolution of the wavefront set, Monatsh. Math. 164 (2011), 393–426.zbMATHCrossRefGoogle Scholar
  27. 27.
    P. Grohs, Continuous shearlet tight frames, J. Fourier Anal. Appl. 17 (2011), 506–518.MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    P. Grohs, Tree Approximation with anisotropic decompositions. Applied and Computational Harmonic Analysis (2011), to appear.Google Scholar
  29. 29.
    A. Grossmann, J. Morlet, and T. Paul, Transforms associated to square integrable group representations I: General Results, J. Math. Phys. 26 (1985), 2473–2479.MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    K. Guo, G. Kutyniok, and D. Labate, Sparse multidimensional representations using anisotropic dilation and shear operators, in Wavelets and Splines (Athens, GA, 2005), Nashboro Press, Nashville, TN, 2006, 189–201.Google Scholar
  31. 31.
    K. Guo and D. Labate, Optimally sparse multidimensional representation using shearlets, SIAM J. Math Anal. 39 (2007), 298–318.MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    K. Guo, and D. Labate, Characterization and analysis of edges using the Continuous Shearlet Transform, SIAM on Imaging Sciences 2 (2009), 959–986.MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    K. Guo, and D. Labate, Analysis and detection of surface discontinuities using the 3D continuous shearlet transform, Appl. Comput. Harmon. Anal. 30 (2011), 231–242.MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    K. Guo, and D. Labate, Optimally sparse 3D approximations using shearlet representations, Electronic Research Announcements in Mathematical Sciences 17 (2010), 126–138.MathSciNetGoogle Scholar
  35. 35.
    K. Guo, and D. Labate, Optimally sparse shearlet approximations of 3D data Proc. of SPIE Defense, Security, and Sensing (2011).Google Scholar
  36. 36.
    K. Guo, and D. Labate, Optimally sparse representations of 3D Data with C 2 surface singularities using Parseval frames of shearlets, SIAM J. Math Anal., to appear (2012).Google Scholar
  37. 37.
    K. Guo, and D. Labate, The Construction of Smooth Parseval Frames of Shearlets, preprint (2011).Google Scholar
  38. 38.
    K. Guo, D. Labate and W.-Q Lim, Edge analysis and identification using the Continuous Shearlet Transform, Appl. Comput. Harmon. Anal. 27 (2009), 24–46.Google Scholar
  39. 39.
    K. Guo, D. Labate, W.-Q Lim, G. Weiss, and E. Wilson, Wavelets with composite dilations, Electron. Res. Announc. Amer. Math. Soc. 10 (2004), 78–87.Google Scholar
  40. 40.
    K. Guo, D. Labate, W.-Q Lim, G. Weiss, and E. Wilson, The theory of wavelets with composite dilations, Harmonic analysis and applications, Appl. Numer. Harmon. Anal., Birkhäuser Boston, Boston, MA, 2006, 231–250.Google Scholar
  41. 41.
    K. Guo, W.-Q Lim, D. Labate, G. Weiss, and E. Wilson, Wavelets with composite dilations and their MRA properties, Appl. Comput. Harmon. Anal. 20 (2006), 220–236.Google Scholar
  42. 42.
    E. Hewitt and K.A. Ross, Abstract Harmonic Analysis I, II, Springer-Verlag, Berlin/ Heidelberg/New York, 1963.Google Scholar
  43. 43.
    M. Holschneider, Wavelets. Analysis Tool, Oxford University Press, Oxford, 1995.Google Scholar
  44. 44.
    N. Kingsbury, Image processing with complex wavelets, Phil. Trans. Royal Society London A, 357 (1999), 2543–2560.zbMATHCrossRefGoogle Scholar
  45. 45.
    N. Kingsbury, Complex wavelets for shift invariant analysis and filtering of signals, Appl. Computat. Harmon. Anal. 10 (2001), 234–253.MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    B. Han, G. Kutyniok, and Z. Shen. Adaptive multiresolution analysis structures and shearlet systems, SIAM J. Numer. Anal. 49 (2011), 1921–1946.MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    E. Hernandez and G. Weiss, A First Course on Wavelets, CRC, Boca Raton, FL, 1996zbMATHCrossRefGoogle Scholar
  48. 48.
    R. Houska, The nonexistence of shearlet scaling functions, to appear in Appl. Comput Harmon. Anal. 32 (2012), 28–44.zbMATHCrossRefGoogle Scholar
  49. 49.
    P. Kittipoom, G. Kutyniok, and W.-Q Lim, Construction of compactly supported shearlet frames, Constr. Approx. 35 (2012), 21–72.Google Scholar
  50. 50.
    P. Kittipoom, G. Kutyniok, and W.-Q Lim, Irregular shearlet frames: geometry and approximation properties J. Fourier Anal. Appl. 17 (2011), 604–639.Google Scholar
  51. 51.
    G. Kutyniok and D. Labate, Construction of regular and irregular shearlets, J. Wavelet Theory and Appl. 1 (2007), 1–10.Google Scholar
  52. 52.
    G. Kutyniok and D. Labate, Resolution of the wavefront set using continuous shearlets, Trans. Amer. Math. Soc. 361 (2009), 2719–2754.MathSciNetzbMATHCrossRefGoogle Scholar
  53. 53.
    G. Kutyniok, J. Lemvig, and W.-Q Lim, Compactly supported shearlets, in Approximation Theory XIII (San Antonio, TX, 2010), Springer Proc. Math. 13, 163–186, Springer, 2012.Google Scholar
  54. 54.
    G. Kutyniok, J. Lemvig, and W.-Q Lim, Compactly supported shearlet frames and optimally sparse approximations of functions in L 2 (ℝ 3 ) with piecewise C 2 singularities, preprint.Google Scholar
  55. 55.
    G. Kutyniok and W.-Q Lim, Compactly supported shearlets are optimally sparse, J. Approx. Theory 163 (2011), 1564–1589.Google Scholar
  56. 56.
    G. Kutyniok and W.-Q Lim, Image separation using wavelets and shearlets, Curves and Surfaces (Avignon, France, 2010), Lecture Notes in Computer Science 6920, Springer, 2012.Google Scholar
  57. 57.
    G. Kutyniok and W.-Q Lim, Shearlets on bounded domains, in Approximation Theory XIII (San Antonio, TX, 2010), Springer Proc. Math. 13, 187–206, Springer, 2012.Google Scholar
  58. 58.
    G. Kutyniok and T. Sauer, Adaptive directional subdivision schemes and shearlet multiresolution analysis, SIAM J. Math. Anal. 41 (2009), 1436–1471.MathSciNetzbMATHCrossRefGoogle Scholar
  59. 59.
    G. Kutyniok, M. Shahram, and D. L. Donoho, Development of a digital shearlet transform based on pseudo-polar FFT, in Wavelets XIII, edited by V. K. Goyal, M. Papadakis, D. Van De Ville, SPIE Proc. 7446 (2008), SPIE, Bellingham, WA, 2009, 7446-12.Google Scholar
  60. 60.
    G. Kutyniok, M. Shahram, and X. Zhuang, ShearLab: a rational design of a digital parabolic scaling algorithm, preprint.Google Scholar
  61. 61.
    D. Labate, W.-Q Lim, G. Kutyniok, and G. Weiss. Sparse multidimensional representation using shearlets, in Wavelets XI, edited by M. Papadakis, A. F. Laine, and M. A. Unser, SPIE Proc. 5914 (2005), SPIE, Bellingham, WA, 2005, 254–262.Google Scholar
  62. 62.
    D. Labate and P. S. Negi, 3D Discrete Shearlet Transform and video denoising, Wavelets XIV (San Diego, CA, 2011), SPIE Proc. (2011).Google Scholar
  63. 63.
    Laugesen, R. S., N. Weaver, G. Weiss, and E. Wilson, A characterization of the higher dimensional groups associated with continuous wavelets, J. Geom. Anal. 12 (2001), 89–102.MathSciNetCrossRefGoogle Scholar
  64. 64.
    W.-Q Lim, The discrete shearlet transform: a new directional transform and compactly supported shearlet frames, IEEE Trans. Image Process. 19 (2010), 1166–1180.Google Scholar
  65. 65.
    Mallat, S., A Wavelet Tour of Signal Processing, Academic Press, San Diego 1998.zbMATHGoogle Scholar
  66. 66.
    S. Mallat, Geometrical Grouplets, Appl. Comput. Harmon. Anal. 26 (2) (2009), 161–180.MathSciNetzbMATHCrossRefGoogle Scholar
  67. 67.
    P. S. Negi and D. Labate, Video denoising and enhancement using the 3D Discrete Shearlet Transform, to appear IEEE Trans. Image Process. (2012)Google Scholar
  68. 68.
    B. A. Olshausen, and D. J. Field, Natural image statistics and efficient coding, Network: Computation in Neural Systems 7 (1996), 333–339CrossRefGoogle Scholar
  69. 69.
    V.M. Patel, G. Easley, D. M. Healy, Shearlet-based deconvolution IEEE Trans. Image Process. 18(12) (2009), 2673-2685MathSciNetCrossRefGoogle Scholar
  70. 70.
    E. L. Pennec and S. Mallat, Sparse geometric image representations with bandelets, IEEE Trans. Image Process. 14 (2005), 423–438.MathSciNetCrossRefGoogle Scholar
  71. 71.
    E. P. Simoncelli, W. T. Freeman, E. H. Adelson, D. J. Heeger, Shiftable multiscale transforms, IEEE Trans. Inform. Theory 38 (1992), 587–607.MathSciNetCrossRefGoogle Scholar
  72. 72.
    H. F. Smith, A Hardy space for Fourier integral operators, J. Geom. Anal. 8 (1998), 629–653.MathSciNetzbMATHCrossRefGoogle Scholar
  73. 73.
    Stein, E., Harmonic Analysis: Real–Variable Mathods, Orthogonality and Oscillatory Integrals, Princeton University Press, Princeton, 1993.Google Scholar
  74. 74.
    S. Yi, D. Labate, G. R. Easley, and H. Krim, A shearlet approach to edge analysis and detection, IEEE Trans. Image Process. 18 (2009), 929–941.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Institut für MathematikTechnische Universität BerlinBerlinGermany
  2. 2.Department of MathematicsUniversity of HoustonHoustonUSA

Personalised recommendations