Original Paper
Anisotropic shearlet transforms for L2
Wojciech Czaja,
Emily J. King,
Corresponding Author
Wojciech Czaja
Department of Mathematics, University of Maryland, College Park, MD, 20742 U.S.A.
Corresponding author: e-mail: [email protected], Phone: +1 301 405 5106Search for more papers by this authorEmily J. King
- [email protected]
- +49 (0) 30 314 25181
Department of Mathematics, University of Maryland, College Park, MD, 20742 U.S.A.
Department of Mathematics, Technical University Berlin, Marchstrasse 6, 10587 Berlin, Germany
Search for more papers by this authorWojciech Czaja,
Emily J. King,
Corresponding Author
Wojciech Czaja
Department of Mathematics, University of Maryland, College Park, MD, 20742 U.S.A.
Corresponding author: e-mail: [email protected], Phone: +1 301 405 5106Search for more papers by this authorEmily J. King
- [email protected]
- +49 (0) 30 314 25181
Department of Mathematics, University of Maryland, College Park, MD, 20742 U.S.A.
Department of Mathematics, Technical University Berlin, Marchstrasse 6, 10587 Berlin, Germany
Search for more papers by this authorAbstract
In this paper, we present a new anisotropic generalization of the continuous shearlet transformation. This is achieved by means of an explicit construction of a family of reproducing Lie subgroups of the symplectic group. We study the properties of this new family of anisotropic shearlet transformations. In particular, we provide an analog of the Calderón admissibility condition for anisotropic shearlet reproducing functions.
References
- 1I. Daubechies, Ten Lectures on Wavelets (SIAM, 1992).
10.1137/1.9781611970104 Google Scholar
- 2S. Mallat, Wavelet Tour of Signal Processing (Academic Press, 1999).
- 3A. Chambolle, R. A. DeVore, N. Lee, and B. J. Lucier, Nonlinear wavelet image processing: Variational problems, compression, and noise removal through wavelet shrinkage, IEEE Transactions on Image Processing 7(3), 319–333 (1998).
- 4R. R. Coifman and D. L. Donoho, Translation-invariant de-noising, Lecture Notes in Statistics: Wavelets and Statistics (Springer, New York, NY, 1995).
10.1007/978-1-4612-2544-7_9 Google Scholar
- 5M. N. Do and M. Vetterli, Contourlets, in: Beyond Wavelets, edited by G. V. Welland, Studies in Computational Mathematics Vol. 10 (Academic Press/Elsevier Science, San Diego, CA, 2003), pp. 1–27.
- 6E. J. Candès and F. Guo, New multiscale transforms, minimum total variation synthesis: applications to edge-preserving image reconstruction, Signal Processing 82(11), 1519–1543 (2002).
- 7E. L. Pennec and S. Mallat, Bandelet image approximation and compression, SIAM Journal of Multiscale Modeling and Simulation 4, 2005 (2005).
- 8D. L. Donoho, Wedgelets: nearly minimax estimation of edges, Ann. Statist. 27(3), 859–897 (1999).
- 9K. Guo, G. Kutyniok, and D. Labate, Sparse multidimensional representations using anisotropic dilation and shear operators, in: Wavelets and splines: Athens 2005, Mod. Methods Math. (Nashboro Press, Brentwood, TN, 2006), pp. 189–201.
- 10D. Labate, W. Q. Lim, G. Kutyniok, and G. Weiss, Sparse multidimensional representation using shearlets, SPIE Proc. 5914, SPIE, Bellingham pp. 254–262 (2005).
- 11W. Czaja and E. J. King, Isotropic shearlet analogs for
, Numerical Functional Analysis and Optimization 33, 872–905 (2012).
- 12S. Dahlke, G. Steidl, and G. Teschke, The continuous shearlet transform in arbitrary space dimensions, in: Structured Decompositions and Efficient Algorithms, edited by S. Dahlke, I. Daubechies, M. Elad, G. Kutyniok, and G. Teschke, No. 08492 in Dagstuhl Seminar Proceedings (Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, Germany, Dagstuhl, Germany, 2009).
- 13S. Dahlke, G. Steidl, and G. Teschke, The continuous shearlet transform in arbitrary space dimensions, J. Fourier Anal. Appl. 16(3), 340–364 (2010).
- 14S. Dahlke and G. Teschke, The continuous shearlet transform in higher dimensions: variations of a theme, in: Group Theory: Classes, Representation and Connections, and Applications, edited by C. W. Danellis, Mathematics Research Developments (Nova Publishers, 2010).
- 15K. Guo, D. Labate, W. Q. Lim, G. Weiss, and E. Wilson, Wavelets with composite dilations, Electron. Res. Announc. Amer. Math. Soc. 10, 78–87 (2004) (electronic).
- 16S. Dahlke, G. Kutyniok, G. Steidl, and G. Teschke, Shearlet coorbit spaces and associated Banach frames, Appl. Comput. Harmon. Anal. 27(2), 195–214 (2009).
- 17D. Labate and G. Weiss, Wavelets associated with composite dilations, in: Matemáticas: investigación y educación. Un homenaje a Miguel de Guzmán (Universidad Complutense de Madrid, 2005).
- 18G. Kutyniok and T. Sauer, Adaptive directional subdivision schemes and shearlet multiresolution analysis, SIAM J. Math. Anal. 41, 1436–1471 (2009).
- 19G. Kutyniok and T. Sauer, From wavelets to shearlets and back again, in: Approximation Theory XII: San Antonio, 2007, edited by M. Neamtu and L. Schumaker (Nashboro Press, 2008).
- 20J. D. Blanchard, Minimally supported frequency composite dilation wavelets, J. Fourier Anal. Appl. 15(6), 796–815 (2009).
- 21S. Yi, D. Labate, G. R. Easley, and H. Krim, A shearlet approach to edge analysis and detection, IEEE Transactions on Image Processing 18(5), 929–941 (2009).
- 22F. Colonna, G. R. Easley, K. Guo, and D. Labate, Radon transform inversion using the shearlet representation, Appl. Comput. Harmon. Anal. 29(2), 232–250 (2010).
- 23G. Kutyniok and W. Lim, Compactly supported shearlets are optimally sparse, J. Approx. Theory 163, 1564–1589 (2011).
- 24G. Easley, D. Labate, and W. Q. Lim, Sparse directional image representations using the discrete shearlet transform, Appl. Comput. Harmon. Anal. 25(1), 25–46 (2008).
- 25B. Han, G. Kutyniok, and Z. Shen, Adaptive multiresolution analysis structures and shearlet systems, SIAM Journal on Numerical Analysis 49, 1921–1946 (2011).
- 26P. Grohs and G. Kutyniok, Parabolic Molecules, Found. Comput. Math., to appear.
- 27G. Kutyniok and D. Labate, Shearlets: Multiscale Analysis for Multivariate Data (Birkhäuser, Boston, MA, 2012).
- 28E. Cordero, F. De Mari, K. Nowak, and A. Tabacco, Dimensional upper bounds for admissible subgroups for the metaplectic representation, Math. Nach. 283(7), 982–993 (2010).
- 29E. Cordero, F. De Mari, K. Nowak, and A. Tabacco, Reproducing groups for the metaplectic representation, in: Pseudo-differential operators and related topics, Oper. Theory Adv. Appl., Vol. 164 (Birkhäuser, Basel, 2006), pp. 227–244.
10.1007/3-7643-7514-0_16 Google Scholar
- 30E. Cordero, F. De Mari, K. Nowak, and A. Tabacco, Analytic features of reproducing groups for the metaplectic representation, J. Fourier Anal. Appl. 12(2), 157–180 (2006).
- 31F. De Mari and K. Nowak, Analysis of the affine transformations of the time-frequency plane, Bull. Austral. Math. Soc. 63(2), 195–218 (2001).
- 32F. De Mari and K. Nowak, Canonical subgroups of
, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 5(2), 405–430 (2002).
- 33K. Gröchenig, Foundations of Time-Frequency Analysis, Applied and Numerical Harmonic Analysis (Birkhäuser Boston Inc., Boston, MA, 2001).
- 34G. B. Folland, Harmonic analysis in phase space, Annals of Mathematics Studies, Vol. 122 (Princeton University Press, Princeton, NJ, 1989).
- 35G. Weiss and E. N. Wilson, The mathematical theory of wavelets. Twentieth century harmonic analysis—a celebration (Il Ciocco, 2000), vol. 33 of NATO Sci. Ser. II Math. Phys. Chem. 329–366 (2001).
- 36Z. Rzeszotnik, Calderón's condition and wavelets, Collect. Math. 5(2), 181–191 (2001).
- 37F. De Mari and E. De Vito, Admissible vectors for mock metaplectic representations, Appl. Comput. Harmon. Anal. 34(2), 163–200 (2013).
- 38W. Czaja, J. Dobrosotskaya, and B. Manning, Composite wavelet representations for reconstruction of missing data, SPIE Proceedings Vol. 8750 Independent Component Analyses, Compressive Sampling, Wavelets, Neural Net, Biosystems, and Nanoengineering XI, 875003 (2013).
- 39E. H. Bosch, A. Castrodad, J. S. Cooper, W. Czaja, and J. Dobrosotskaya, Tight frames for multiscale and multidirectional image analysis, SPIE Proceedings Vol. 8750 Independent Component Analyses, Compressive Sampling, Wavelets, Neural Net, Biosystems, and Nanoengineering XI, 875004 (2013).
