Continuous Shearlet Tight Frames

Philipp Grohs

doi:10.1007/s00041-010-9149-y

Continuous Shearlet Tight Frames

Philipp Grohs

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19 Scopus citations

Abstract

Based on the shearlet transform we present a general construction of continuous tight frames for L2(ℝ2) from any sufficiently smooth function with anisotropic moments. This includes for example compactly supported systems, piecewise polynomial systems, or both. From our earlier results in Grohs (Technical report, KAUST, 2009) it follows that these systems enjoy the same desirable approximation properties for directional data as the previous bandlimited and very specific constructions due to Kutyniok and Labate (Trans. Am. Math. Soc. 361:2719-2754, 2009). We also show that the representation formulas we derive are in a sense optimal for the shearlet transform. © 2010 Springer Science+Business Media, LLC.

Original language	English (US)
Pages (from-to)	506-518
Number of pages	13
Journal	Journal of Fourier Analysis and Applications
Volume	17
Issue number	3
DOIs	https://doi.org/10.1007/s00041-010-9149-y
State	Published - Oct 22 2010
Externally published	Yes

Bibliographical note

KAUST Repository Item: Exported on 2020-10-01
Acknowledgements: The research for this paper has been carried out while the author was working atthe Center for Geometric Modeling and Scientific Visualization at KAUST, Saudi Arabia.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.

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abstract = "Based on the shearlet transform we present a general construction of continuous tight frames for L2(ℝ2) from any sufficiently smooth function with anisotropic moments. This includes for example compactly supported systems, piecewise polynomial systems, or both. From our earlier results in Grohs (Technical report, KAUST, 2009) it follows that these systems enjoy the same desirable approximation properties for directional data as the previous bandlimited and very specific constructions due to Kutyniok and Labate (Trans. Am. Math. Soc. 361:2719-2754, 2009). We also show that the representation formulas we derive are in a sense optimal for the shearlet transform. {\textcopyright} 2010 Springer Science+Business Media, LLC.",

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N2 - Based on the shearlet transform we present a general construction of continuous tight frames for L2(ℝ2) from any sufficiently smooth function with anisotropic moments. This includes for example compactly supported systems, piecewise polynomial systems, or both. From our earlier results in Grohs (Technical report, KAUST, 2009) it follows that these systems enjoy the same desirable approximation properties for directional data as the previous bandlimited and very specific constructions due to Kutyniok and Labate (Trans. Am. Math. Soc. 361:2719-2754, 2009). We also show that the representation formulas we derive are in a sense optimal for the shearlet transform. © 2010 Springer Science+Business Media, LLC.

AB - Based on the shearlet transform we present a general construction of continuous tight frames for L2(ℝ2) from any sufficiently smooth function with anisotropic moments. This includes for example compactly supported systems, piecewise polynomial systems, or both. From our earlier results in Grohs (Technical report, KAUST, 2009) it follows that these systems enjoy the same desirable approximation properties for directional data as the previous bandlimited and very specific constructions due to Kutyniok and Labate (Trans. Am. Math. Soc. 361:2719-2754, 2009). We also show that the representation formulas we derive are in a sense optimal for the shearlet transform. © 2010 Springer Science+Business Media, LLC.

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JO - Journal of Fourier Analysis and Applications

JF - Journal of Fourier Analysis and Applications

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ER -

Continuous Shearlet Tight Frames

Abstract

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