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Microlocal analysis of edge flatness through directional multiscale representations

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Abstract

Edges and surface boundaries are often the most relevant features in images and multidimensional data. It is well known that multiscale methods including wavelets and their more sophisticated multidimensional siblings offer a powerful tool for the analysis and detection of such sets. Among such methods, the continuous shearlet transform has been especially successful. This method combines anisotropic scaling and directional sensitivity controlled by shear transformations in order to precisely identify not only the location of edges and boundary points but also edge orientation and corner points. In this paper, we show that this framework can be made even more flexible by controlling the scaling parameter of the anisotropic dilation matrix and considering non-parabolic scaling. We prove that, using ‘higher-than-parabolic’ scaling, the modified shearlet transform is also able to estimate the degree of local flatness of an edge or surface boundary, providing more detailed information about the geometry of edge and boundary points.

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References

  1. Bros, J., Iagolnitzer, D.: Support essentiel et structure analytique des distributions. Seminaire Goulaouic-Lions-Schwartz 19, 1975–1976 (1976)

    MATH  Google Scholar 

  2. Candès, E. J., Donoho, D. L.: Continuous curvelet transform: I. Resolution of the wavefront set. Appl. Comput. Harmon. Anal. 19, 162–197 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  3. Córdoba, A., Fefferman, C.: Wave packets and Fourier integral operators. Communications in Partial Differential Equations 3(11), 979–1005 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  4. Donoho, D., Kutyniok, G.: Microlocal analysis of the geometric separation problem. Comm. Pure Appl. Math. 66, 1–47 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  5. Duval-Poo, M. A., Odone, F., De Vito, E.: Edges and corners with shearlets. IEEE Trans. Image Process. 24(11), 3768–3780 (2015)

    Article  MathSciNet  Google Scholar 

  6. Easley, G., Labate, D., Lim, W.: Sparse directional image representations using the discrete shearlet transform. Appl. Comput. Harmon. Anal. 25, 25–46 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Genzel, M., Kutyniok, G.: Asymptotic analysis of inpainting via universal shearlet systems. SIAM J. Imaging Sci. 7, 2301–2339 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  8. Grohs, P.: Continuous shearlet frames and resolution of the wavefront set. Monatshefte för Mathematik 164(4), 393–426 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Guo, K., Labate, D.: Characterization and analysis of edges using the continuous shearlet transform. SIAM on Imaging Sciences 2, 959–986 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Guo, K., Labate, D.: Analysis and detection of surface discontinuities using the 3D continuous shearlet transform. Appl. Comput. Harmon. Anal. 30, 231–242 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. Guo, K., Labate, D.: Characterization of piecewise-smooth surfaces using the 3D continuous shearlet transform. J. Fourier Anal. Appl. 18, 488–516 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  12. Guo, Labate, D.: Geometric separation of singularities using combined multiscale dictionaries. J. Fourier Anal. Appl. 21(4), 667–693 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  13. Guo, K., Labate, D.: Characterization and analysis of edges in piecewise smooth functions. Appl. Comput. Harmon. Anal. 41(1), 139–163 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  14. Guo, K., Labate, D., Lim, W.: Edge analysis and identification using the continuous shearlet transform. Appl. Comput. Harmon. Anal. 27(1), 24–46 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Guo, K., Houska, R., Labate, D.: Microlocal analysis of singularities from directional multiscale representations. In: Proceedings from: Approximation Theory XIV: San Antonio 2013 (April 7-10, 2013, San Antonio, Texas). Springer Proceedings in Mathematics & Statistics, vol. 83, pp 173–196 (2014)

  16. Holschneider, M.: Wavelets. Analysis tool. Oxford University Press, Oxford (1995)

    MATH  Google Scholar 

  17. Jaffard, S.: Pointwise smoothness, two-microlocalization and wavelet coefficients. Publications Matematiques 35, 155–168 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  18. Jaffard, S., Meyer, Y.: Wavelet methods for pointwise regularity and local oscillations of functions. Memoirs of the AMS 123(587) (1996)

  19. King, E., Kutyniok, G., Zhuang, X.: Analysis of inpainting via clustered sparsity and microlocal analysis. J. Math. Imaging Vision 48(2), 205–234 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kutyniok, G., Labate, D.: Resolution of the wavefront set using continuous shearlets. Trans. Amer. Math. Soc. 361, 2719–2754 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kutyniok, G., Petersen, P.: Classification of edges using compactly supported shearlets. Appl. Comput. Harmon. Anal. (2015). doi:10.1016/j.acha.2015.08.006

    MATH  Google Scholar 

  22. Kutyniok, G., Shahram, M., Zhuang, X.: Shearlab: a rational design of a digital parabolic scaling algorithm. SIAM J. Imaging Sci. 5(4), 1291–1332 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  23. Meyer, Y.: Wavelets and Operators, Cambridge Stud. Adv Math, vol. 37. Cambridge University Press, Cambridge (1992)

  24. O’Leary, D. P., Schug, D. A., Easley, G. R.: Three-dimensional shearlet edge analysis. In: Proceedings of the SPIE 8058, Independent Component Analyses, Wavelets, Neural Networks, Biosystems, and Nanoengineering IX (2011)

  25. Ozcan, B., Negi, P., Laezza, F., Papadakis, M., Labate, D.: Automated detection of soma location and morphology in neuronal network cultures. PLos ONE 10(4) (2015)

  26. Reisenhofer, R., Kiefer, J., King, E. J.: Shearlet-based detection of flame fronts. Exp. Fluids 57(3), 41:1–41:14 (2016)

    Article  Google Scholar 

  27. Yi, S., Labate, D., Easley, G. R., Krim, H.: A shearlet approach to edge analysis and detection. IEEE Trans. Image Process. 18, 929–941 (2009)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Missouri State University, Springfield, MO, 65804, USA

    Kanghui Guo

  2. University of Houston, Houston, TX, 77204, USA

    Demetrio Labate

Authors
  1. Kanghui Guo
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  2. Demetrio Labate
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Correspondence to Demetrio Labate.

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Communicated by: Yang Wang

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Guo, K., Labate, D. Microlocal analysis of edge flatness through directional multiscale representations. Adv Comput Math 43, 295–318 (2017). https://doi.org/10.1007/s10444-016-9486-8

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  • DOI: https://doi.org/10.1007/s10444-016-9486-8

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