Construction of Compactly Supported Shearlet Frames
- 363 Downloads
Shearlet tight frames have been extensively studied in recent years due to their optimal approximation properties of cartoon-like images and their unified treatment of the continuum and digital settings. However, these studies only concerned shearlet tight frames generated by a band-limited shearlet, whereas for practical purposes compact support in spatial domain is crucial.
In this paper, we focus on cone-adapted shearlet systems that—accounting for stability questions—are associated with a general irregular set of parameters. We first derive sufficient conditions for such cone-adapted irregular shearlet systems to form a frame and provide explicit estimates for their frame bounds. Secondly, exploring these results and using specifically designed wavelet scaling functions and filters, we construct a family of cone-adapted shearlet frames consisting of compactly supported shearlets. For this family, we derive estimates for the ratio of their frame bounds and prove that they provide optimally sparse approximations of cartoon-like images. Finally, we discuss an implementation strategy for the shearlet transform based on compactly supported shearlets. We further show that an optimally sparse N-term approximation of an M×M cartoon-like image can be derived with asymptotic computational cost O(M 2(1+τ)) for some positive constant τ depending on M and N.
KeywordsCurvilinear discontinuities Edges Filters Nonlinear approximation Optimal sparsity Scaling functions Shearlets Wavelets
Mathematics Subject Classification (2000)42C40 42C15 65T60 65T99 94A08
Unable to display preview. Download preview PDF.
- 1.Candés, E.J., Donoho, D.L.: New tight frames of curvelets and optimal representations of objects with piecewise C 2 singularities. Commun. Pure Appl. Math. 56, 216–266 (2004) Google Scholar
- 9.Donoho, D.L., Kutyniok, G.: Geometric separation using a wavelet-shearlet dictionary. Proc., SampTA’09, Marseille, France (2009) Google Scholar
- 10.Donoho, D.L., Kutyniok, G.: Microlocal Analysis of the Geometric Separation Problems. Preprint Google Scholar
- 13.Grohs, P.: Continuous shearlet frames and resolution of the wavefront set. Preprint Google Scholar
- 14.Guo, K., Kutyniok, G., Labate, D.: Sparse Multidimensional Representations Using Anisotropic Dilation and Shear Operators, Wavelets and Splines (Athens, GA, 2005), pp. 189–201. Nashboro Press, Nashville (2006) Google Scholar
- 18.Kutyniok, G., Labate, D.: Construction of regular and irregular shearlets. J. Wavelet Theory Appl. 1, 1–10 (2007) Google Scholar
- 21.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) Google Scholar
- 22.Labate, D., Lim, W.-Q, Kutyniok, G., Weiss, G.: Sparse multidimensional representation using shearlets. Wavelets XI, San Diego, CA, 2005. SPIE Proc., vol. 5914, pp. 254–262. SPIE, Bellingham (2005) Google Scholar
- 24.Lu, Y., Do, M.N.: CRISP-contourlets: a critically sampled directional multiresolution image representation. In: Papadakis, M., Laine, A.F., Unser, M.A. (eds.) Wavelets X, San Diego, CA, 2003. SPIE Proc. SPIE, Bellingham (2003) Google Scholar
- 25.Smith, H.F.: A hardy space for Fourier integral operators. J. Geom. Anal. 8, 629–653 Google Scholar