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The Construction of Smooth Parseval Frames ofShearlets

Published online by Cambridge University Press:  28 January 2013

K. Guo and
D. Labate
K. Guo
Affiliation:
Department of Mathematics, Missouri State University, Springfield, Missouri 65804, USA
D. Labate*
Affiliation:
Department of Mathematics, University of Houston, Houston, Texas 77204, USA
*
∗ Corresponding author. E-mail: dlabate@math.uh.edu

Abstract

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The shearlet representation has gained increasing recognition in recent years as aframework for the efficient representation of multidimensional data. This representationconsists of a countable collection of functions defined at various locations, scales andorientations, where the orientations are obtained through the use of shear matrices. Whileshear matrices offer the advantage of preserving the integer lattice and being moreappropriate than rotations for digital implementations, the drawback is that the action ofthe shear matrices is restricted to cone-shaped regions in the frequency domain. Hence, inthe standard construction, a Parseval frame of shearlets is obtained by combiningdifferent systems of cone-based shearlets which are projected onto certain subspaces ofL2(ℝD) with the consequence thatthe elements of the shearlet system corresponding to the boundary of the cone regions losetheir good spatial localization property. In this paper, we present a new constructionyielding smooth Parseval frame of shearlets forL2(ℝD). Specifically, allelements of the shearlet systems obtained from this construction are compactly supportedand C∞ in the frequency domain, hence ensuring that the systemhas also excellent spatial localization.

Keywords

affine systemsnonlinear approximationsshearletssparse approximationswavelets

Information

Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 8 , Issue 1: Harmonic analysis , 2013, pp. 82 - 105
Copyright
© EDP Sciences, 2013

References

Références

Candès, E. J., Demanet, L.. The curvelet representation of wave propagators is optimally sparse. Comm. Pure Appl. Math. 58 (2005), 1472–1528. CrossRefGoogle Scholar
Candès, E. J., Demanet, L., Donoho, D., Ying, L.. Fast Discrete Curvelet Transforms. Multiscale Model. Simul. 5 (2006), 861–899. CrossRefGoogle Scholar
Candès, E. J., Donoho, D. L.. Ridgelets : the key to high dimensional intermittency?. Philosophical Transactions of the Royal Society of London A 357 (1999), 2495–2509. CrossRefGoogle Scholar
Candès, E. J., Donoho, D. L.. New tight frames of curvelets and optimal representations of objects with C2 singularities. Comm. Pure Appl. Math. 57 (2004), 219–266. CrossRefGoogle Scholar
Colonna, F., Easley, G., Guo, K., Labate, D.. Radon Transform Inversion using the Shearlet Representation. Appl. Comput. Harmon. Anal. 29 (2) (2010), 232–250. CrossRefGoogle Scholar
Dahlke, S., Kutyniok, G., Maass, P., Sagiv, C., Stark, H.-G., Teschke, G.. The uncertainty principle associated with the continuous shearlet transform. Int. J. Wavelets Multiresolut. Inf. Process. 6 (2008), 157–181. CrossRefGoogle Scholar
Do, M. N., Vetterli, M.. The contourlet transform : an efficient directional multiresolution image representation. IEEE Trans. Image Process. 14 (2005), 2091–2106. CrossRefGoogle Scholar
Donoho, D. L.. Wedgelets : Nearly-minimax estimation of edges. Annals of Statistics, 27 (1999), 859–897. CrossRefGoogle Scholar
Easley, G. R., Labate, D., Colonna, F.. Shearlet-Based Total Variation Diffusion for Denoising. IEEE Trans. Image Proc. 18 (2) (2009), 260–268. CrossRefGoogle ScholarPubMed
Easley, G. R., Labate, D., Lim, W.. Sparse Directional Image Representations using the Discrete Shearlet Transform. Appl. Comput. Harmon. Anal. 25 (1) (2008), 25–46. CrossRefGoogle Scholar
Grohs, P.. Tree Approximation with anisotropic decompositions. Appl. Comput. Harmon. Anal. 33(1) (2012), 44–57. CrossRefGoogle Scholar
P. Grohs. Bandlimited Shearlet Frames with nice Duals. SAM Report 2011-55, ETH Zurich, July 2011.
K. Guo, G. Kutyniok, D. Labate. Sparse Multidimensional Representations using Anisotropic Dilation and Shear Operators in : Wavelets and Splines, G. Chen and M. Lai (eds.), Nashboro Press, Nashville, TN (2006), pp. 189–201.
Guo, K., Labate, D.. Optimally Sparse Multidimensional Representation using Shearlets. SIAM J. Math. Anal. 9 (2007), 298–318 CrossRefGoogle Scholar
Guo, K., Labate, D.. Representation of Fourier Integral Operators using Shearlets. J. Fourier Anal. Appl. 14 (2008), 327–371 CrossRefGoogle Scholar
Guo, K., Labate, D.. Characterization and analysis of edges using the continuous shearlet transform. SIAM J. Imag. Sci. 2 (2009), 959–986. CrossRefGoogle Scholar
Guo, K., Labate, D.. Optimally sparse 3D approximations using shearlet representations. Electron. Res. Announc. Math. Sci. 17 (2010), 126–138. Google Scholar
Guo, K., Labate, D.. Optimally sparse representations of 3D Data with C2 surface singularities using Parseval frames of shearlets. SIAM J Math. Anal. 44 (2012), 851–886. CrossRefGoogle Scholar
Guo, K., Labate, D., Lim, W.-Q.. Edge analysis and identification using the Continuous Shearlet Transform. Appl. Comput. Harmon. Anal. 27 (2009), 24–46. CrossRefGoogle Scholar
Guo, K., Labate, D., Lim, W.-Q, Weiss, G., Wilson, E.. Wavelets with composite dilations. Electron. Res. Announc. Amer. Math. Soc. 10 (2004), 78–87. CrossRefGoogle Scholar
K. Guo, D. Labate, W-Q. Lim, G. Weiss, E. Wilson. The theory of wavelets with composite dilations. in : Harmonic Analysis and Applications, C. Heil (ed.), Birkhäuser, Boston, MA, 2006.
Guo, K., Lim, W-Q., Labate, D., Weiss, G., Wilson, E.. Wavelets with composite dilations and their MRA properties. Appl. Computat. Harmon. Anal. 20 (2006), 231–249. CrossRefGoogle Scholar
Han, B.. Pairs of frequency-based nonhomogeneous dual wavelet frames in the distribution space. Appl. Comput. Harmon. Anal. 29 (2010), 330–353. CrossRefGoogle Scholar
Han, B.. Nonhomogeneous wavelet systems in high dimensions. Appl. Comput. Harmon. Anal. 32 (2012), 169–196. CrossRefGoogle Scholar
Houska, R.. The nonexistence of shearlet scaling functions. Appl. Comput Harmon. Anal. 32 (1) (2012), 28–44. CrossRefGoogle Scholar
P. Kittipoom, G. Kutyniok, W.-Q Lim. Construction of compactly supported shearlet frames. Constr. Approx., to appear (2012).
G. Kutyniok. Sparsity Equivalence of Anisotropic Decompositions. preprint (2012).
Kutyniok, G., Labate, D.. Resolution of the wavefront set using continuous shearlets. Trans. Amer. Math. Soc. 361 (2009), 2719–2754. CrossRefGoogle Scholar
Kutyniok, G., Lim, W.-Q.. Compactly supported shearlets are optimally sparse. J. Approx. Theory 163 (2011), 1564–1589. CrossRefGoogle Scholar
Kutyniok, G., Sauer, T.. Adaptive Directional Subdivision Schemes and Shearlet Multiresolution Analysis. SIAM J. Math. Anal. 41 (2009), 1436–1471. CrossRefGoogle Scholar
D. Labate, W.-Q Lim, G. Kutyniok, G. Weiss. Sparse multidimensional representation using shearlets. in Wavelets XI, edited by M. Papadakis, A. F. Laine, and M. A. Unser, SPIE Proc. 5914 (2005), SPIE, Bellingham, WA, 2005, 254–262.
Y. Meyer, R. Coifman. Wavelets, Calderón-Zygmund Operators and Multilinear Operators. Cambridge Univ. Press, Cambridge, 1997.
Negi, P. S., Labate, D.. 3D Discrete Shearlet Transform and Video Processing. IEEE Trans. Image Process. 21 (6) (2012), 2944–2954. CrossRefGoogle Scholar
Patel, V.M., Easley, G., Healy, D. M.. Shearlet-based deconvolution. IEEE Trans. Image Process. 18 (12) (2009), 2673-2685 CrossRefGoogle ScholarPubMed
Yi, S., Labate, D., Easley, G. R., Krim, H.. A Shearlet approach to Edge Analysis and Detection. IEEE Trans. Image Process 18 (5) (2009), 929–941. Google ScholarPubMed