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Advances in Computational Mathematics

, Volume 45, Issue 2, pp 675–705 | Cite as

Higher order analysis of the geometry of singularities using the Taylorlet transform

  • Thomas FinkEmail author
Article
  • 39 Downloads

Abstract

We consider an extension of the continuous shearlet transform which additionally uses higher order shears. This extension, called the Taylorlet transform, allows for a detection of the position, the orientation, the curvature, and other higher order geometric information of singularities. Employing the novel vanishing moment conditions of higher order, \({\int }_{\mathbb {R}} g(\pm t^{k})t^{m} dt= 0\) for \(k,m\in \mathbb {N}\), k ≥ 1, on the analyzing function \(g\in \mathcal {S}(\mathbb {R})\), we can show that the Taylorlet transform exhibits different decay rates for decreasing scales depending on the choice of the higher order shearing variables. This enables a faster detection of the geometric information of singularities in terms of the decay rate with respect to the dilation parameter. Furthermore, we present a construction that yields analyzing functions which fulfill vanishing moment conditions of different orders simultaneously.

Keywords

Shearlets Edge classification Curvature 

Mathematics Subject Classification (2010)

Primary 42C40 Secondary 65D18 

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Notes

Acknowledgements

The author thanks Johannes Nagler and Uwe Kähler for valuable discussions and Brigitte Forster for her continuing support. Moreover, the author expresses his gratitude to the anonymous referees for their helpful and precise advice.

Funding information

This work received support from the DFG project FO 792/2-1 “Splines of complex order, fractional operators and applications in signal and image processing,” awarded to Brigitte Forster.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Universität PassauPassauGermany

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