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Complex Analysis and Operator Theory

, Volume 13, Issue 3, pp 985–1010 | Cite as

A Space-Based Method for the Generation of a Schwartz Function with Infinitely Many Vanishing Moments of Higher Order with Applications in Image Processing

  • Thomas Fink
  • Uwe KählerEmail author
Article
  • 37 Downloads

Abstract

In this article we construct a function with infinitely many vanishing (generalized) moments. This is motivated by an application to the Taylorlet transform which is based on the continuous shearlet transform. It can detect curvature and other higher order geometric information of singularities in addition to their position and the direction. For a robust detection of these features a function with higher order vanishing moments, \(\int _\mathbb {R}g(x^k)x^mdx=0\), is needed. We show that the presented construction produces an explicit formula of a function with \(\infty \) many vanishing moments of arbitrary order and thus allows for a robust detection of certain geometric features. The construction has an inherent connection to q-calculus, the Euler function and the partition function.

Keywords

Edge detection q-calculus Shearlets 

Mathematics Subject Classification

Primary 42C40 Secondary 65T60 

References

  1. 1.
    Berndt, B.C.: Ramanujan’s Notebooks, Part V. Springer, Berlin (2005)zbMATHGoogle Scholar
  2. 2.
    Bozejko, M., Speicher, R.: An example of a generalized brownian motion. Commun. Math. Phys. 137, 519–531 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Ernst, T.: The History of q-calculus and a New Method. Department of Mathematics, Uppsala University, Uppsala (2000)Google Scholar
  4. 4.
    Ernst, T.: A Comprehensive Treatment of q-calculus. Springer, Berlin (2012)CrossRefzbMATHGoogle Scholar
  5. 5.
    Euler, L.: Introductio in analysin infinitorum. apud Marcum-Michaelem Bousquet (1748)Google Scholar
  6. 6.
    Exton, H.: q-Hypergeometric Functions and Applications. Halsted Press, Horwood (1983)zbMATHGoogle Scholar
  7. 7.
    Fink, T.: Higher Order Analysis of the Geometry of Singularities Using the Taylorlet Transform. 03 (2017)Google Scholar
  8. 8.
    Grohs, P.: Continuous shearlet frames and resolution of the wavefront set. Monatshefte für Mathematik 164(4), 393–426 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Kac, V., Cheung, P.: Quantum Calculus. Springer, Berlin (2002)CrossRefzbMATHGoogle Scholar
  10. 10.
    Kutyniok, G., Labate, D.: Resolution of the wavefront set using continuous shearlets. Trans. Am. Math. Soc. 361(5), 2719–2754 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Mallat, S., Hwang, W.L.: Singularity detection and processing with wavelets. IEEE Trans. Inf. Theory 38(2), 617–643 (1992)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Fakultät für Informatik und MathematikUniversität PassauPassauGermany
  2. 2.Departamento de MatematicaUniversidade de AveiroAveiroPortugal

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