Irregular Shearlet Frames: Geometry and Approximation Properties

Journal of Fourier Analysis and Applications

Volumn 17, Issue 4, 2011, Pages 604-639

  Kittipoom, Pisamai ^a   Kutyniok, Gitta ^b   Lim, Wang Q ^b  

^a PRINCE OF SONGKLA UNIVERSITY   (Thailand)

^b UNIVERSITY OF OSNABRÜCK   (Germany)

Author keywords

Amalgam spaces; Density; Frame; Geometry; Homogeneous approximation property; Shearlet; Shearlet group

Indexed keywords

EID: 79960201109     PISSN: 10695869     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00041-010-9163-0     Document Type: Article

References (33)

1. Balan, R., Casazza, P. G., Heil, C., Landau, Z.: Density, overcompleteness, and localization of frames, I. Theory. J. Fourier Anal. Appl. 12, 105-143 (2006).
2. Czaja, W., Kutyniok, G., Speegle, D.: The geometry of sets of parameters of wave packet frames. Appl. Comput. Harmon. Anal. 20, 108-125 (2006).
3. 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, 157-181 (2008).
4. Dahlke, S., Kutyniok, G., Steidl, G., Teschke, G.: Shearlet coorbit spaces and associated Banach frames. Appl. Comput. Harmon. Anal. 27, 195-214 (2009).
5. Dahlke, S., Steidl, G., Teschke, G.: The continuous shearlet transform in arbitrary space dimensions. J. Fourier Anal. Appl. 16, 340-364 (2010).
6. Dahlke, S., Steidl, G., Teschke, G.: Shearlet coorbit spaces: compactly supported analyzing shearlets, traces and embeddings, preprint.
7. Dahlke, S., Teschke, G.: The continuous shearlet transform in higher dimensions: variations of a theme. In: Danellis, C. W. (ed.) Group Theory: Classes, Representation and Connections, and Applications. Mathematics Research Developments. Nova Publishers, New York (2010).
8. Easley, G., Labate, D., Lim, W.: Sparse directional image representations using the discrete shearlet transform. Appl. Comput. Harmon. Anal. 25, 25-46 (2008).
9. Feichtinger, H. G.: Banach convolution algebras of Wiener type. In: Functions, Series, Operators, Proc. Conf. Budapest. Colloq. Math. Soc. János Bolyai, vol. 38, pp. 509-524 (1980).
10. Gressman, P., Labate, D., Weiss, G., Wilson, E. N.: Affine, quasi-affine and co-affine wavelets. In: Beyond Wavelets, Studies in Computational Mathematics, vol. 10, pp. 215-224. Elsevier, Amsterdam (2003).
11. Gröchenig, K.: The homogeneous approximation property and the comparison theorem for coherent frames. Sampl. Theory Signal Image Process. 3, 311-319 (2008).
12. Gröchenig, K., Kutyniok, G., Seip, K.: Landau's necessary density conditions for LCA groups. J. Funct. Anal. 255, 1831-1850 (2008).
13. Grohs, P.: Continuous Shearlet frames and resolution of the Wavefront Set. Monatsh. Math. doi: 10. 1007/s00605-010-0264-2.
14. Grohs, P.: Continuous shearlet tight frames. J. Fourier Anal. Appl. doi: 10. 1007/s00041-010-9149-y.
15. Guo, K., Kutyniok, G., Labate, D.: Sparse multidimensional representations using anisotropic dilation and shear operators. In: Wavelets and Splines, Athens, GA, 2005, pp. 189-201. Nashboro Press, Nashville (2006).
16. Guo, K., Labate, D.: Optimally sparse multidimensional representation using shearlets. SIAM J. Math. Anal. 39, 298-318 (2007).
17. Heil, C.: An introduction to weighted Wiener amalgams. In: Proc. International Conference on Wavelets and Their Applications, Chennai, January 2002, pp. 183-216. Allied Publishers, New Delhi (2003).
18. Heil, C.: History and evolution of the density theorem for Gabor frames. J. Fourier Anal. Appl. 13, 113-166 (2007).
19. Heil, C., Kutyniok, G.: Density of weighted wavelet frames. J. Geom. Anal. 13, 479-493 (2003).
20. Heil, C., Kutyniok, G.: The homogeneous approximation property for wavelet frames. J. Approx. Theory 147, 28-46 (2007).
21. Hernandez, E., Labate, D., Weiss, G., Wilson, E.: Oversampling, quasi-affine frames and wave packets. Appl. Comput. Harmon. Anal. 16, 111-147 (2004).
22. Kittipoom, P., Kutyniok, G., Lim, W.-Q: Construction of compactly supported shearlets, preprint.
23. Kutyniok, G.: Affine Density in Wavelet Analysis. Springer, Berlin (2007).
24. Kutyniok, G.: Affine density, frame bounds, and the admissibility condition for wavelet frames. Constr. Approx. 25, 239-253 (2007).
25. Kutyniok, G., Labate, D.: Construction of regular and irregular shearlet frames. J. Wavelet Theory Appl. 1, 1-10 (2007).
26. Kutyniok, G., Labate, D.: Resolution of the wavefront set using continuous shearlets. Trans. Amer. Math. Soc. 361, 2719-2754 (2009).
27. Kutyniok, G., Lim, W.: Compactly supported shearlets are optimally sparse, preprint.
28. Kutyniok, G., Shahram, M., Donoho, D. L.: Development of a digital shearlet transform based on pseudo-polar FFT. In: Van De Ville, D., Goyal, V. K., Papadakis, M. (eds.) Wavelets XIII, San Diego, CA, 2009. SPIE Proc., vol. 7446, pp. 74460B-1-74460B-13. SPIE, Bellingham (2009).
29. Landau, H.: On the density of phase-space expansions. IEEE Trans. Inform. Theory 39, 1152-1156 (1993).
30. Lim, W.-Q: The discrete shearlet transform: a new directional transform and compactly supported shearlet frame. IEEE Trans. Image Proc. 19, 1166-1180 (2010).
31. Ramanathan, J., Steger, T.: Incompleteness of sparse coherent states. Appl. Comput. Harmon. Anal. 2, 148-153 (1995).
32. Sun, W., Zhou, X.: Density and stability of wavelet frames. Appl. Comput. Harmon. Anal. 15, 117-133 (2003).
33. Sun, W., Zhou, X.: Density of irregular wavelet frames. Proc. Am. Math. Soc. 132, 2377-2387 (2004).