Abstract
We show that the cone-adapted shearlet coefficients can be computed by means of the limited angle horizontal and vertical (affine) Radon transforms and the one-dimensional wavelet transform. This yields formulas that open new perspectives for the inversion of the Radon transform.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
D. Labate, W.-Q. Lim, G. Kutyniok, G. Weiss. Sparse multidimensional representation using shearlets. Optics & Photonics 2005, International Society for Optics and Photonics (2005), 59140U-59140U.
E. Cordero, and A. Tabacco. Triangular subgroups of \(Sp(d,\mathbb R)\) and reproducing formulae. J. Fourier Anal. Appl.264 (2013), no. 9, 2034–2058.
E. Cordero, F. De Mari, K. Nowak, and A. Tabacco. Analytic features of reproducing groups for the metaplectic representation. J. Fourier Anal. Appl.12 (2006), no. 2, 157–180.
E. Cordero, F. De Mari, K. Nowak, and A. Tabacco. Dimensional upper bounds for admissible subgroups for the metaplectic representation. Math. Nachr.283 (2010), no. 7, 982–993.
F. Bartolucci, F. De Mari, E. De Vito, and F. Odone. The Radon transform intertwines wavelets and shearlets. Appl. Comput. Harmon. Anal.47 (2019), no. 3, 822–847 (available on line https://doi.org/10.1016/j.acha.2017.12.005).
G. Kutyniok and D. Labate. Resolution of the wavefront set using continuous shearlets. Trans. Amer. Math. Soc.361 (2009), no. 5, 2719–2754.
G. Kutyniok and D. Labate. Shearlets. Appl. Numer. Harmon. Anal. Birkhäuser/Springer, New York, 2012.
H. Führ. Continuous wavelet transforms with abelian dilation groups. J. Math. Phys.39 (1998), no. 8, 3974–3986.
H. Führ and R. R. Tousi. Simplified vanishing moment criteria for wavelets over general dilation groups, with applications to abelian and shearlet dilation groups. Appl. Comput. Harmon. Anal. (2016).
L. Borup and M. Nielsen. Frame decomposition of decomposition spaces. J. Fourier Anal. Appl.13 (2007), no. 1, 39–70.
P. Grohs. Continuous shearlet frames and resolution of the wavefront set. Monatsh. Math.164 (2011), no. 4, 393–426.
S. Dahlke, G. Kutyniok, P. Maass, C. Sagiv, H. Stark, and G. Teschke. The uncertainty principle associated with the continuous shearlet transform. Int. J. Wavelets Multiresolution Inf. Process.6 (2008), no. 2, 157–181.
S. Dahlke, G. Steidl, and G. Teschke. The continuous shearlet transform in arbitrary space dimensions. J. Fourier Anal. Appl.16 (2010), no. 3, 340–364.
S. Helgason. The Radon transform. Progress in Mathematics. Birkhäuser Boston, Inc., Boston, MA, 2nd edition, 1999.
Acknowledgements
F. Bartolucci, F. De Mari and E. De Vito are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Bartolucci, F., De Mari, F., De Vito, E. (2020). Cone-Adapted Shearlets and Radon Transforms. In: Boggiatto, P., et al. Advances in Microlocal and Time-Frequency Analysis. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-36138-9_4
Download citation
DOI: https://doi.org/10.1007/978-3-030-36138-9_4
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-030-36137-2
Online ISBN: 978-3-030-36138-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)