Optimally sparse 3D approximations using shearlet representations

Kanghui Guo; Demetrio Labate

doi:10.3934/era.2010.17.125

Article Contents

2010, Volume 17: 125-137. Doi: 10.3934/era.2010.17.125

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Optimally sparse 3D approximations using shearlet representations

Kanghui Guo^1, and
Demetrio Labate^2,

1.
Department of Mathematics, Missouri State University, Springfield, Missouri 65804
2.
Department of Mathematics, University of Houston, Houston, Texas 77204

Received: August 31, 2010

Published: December 2009

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Abstract

This paper introduces a new Parseval frame, based on the 3-D shearlet representation, which is especially designed to capture geometric features such as discontinuous boundaries with very high efficiency. We show that this approach exhibits essentially optimal approximation properties for 3-D functions $f$ which are smooth away from discontinuities along $C^2$ surfaces. In fact, the $N$ term approximation $f_N^S$ obtained by selecting the $N$ largest coefficients from the shearlet expansion of $f$ satisfies the asymptotic estimate

||$f-f_N^S$||$_2^2$ ≍ $N^{-1} (\log N)^2, as N \to \infty.$

Up to the logarithmic factor, this is the optimal behavior for functions in this class and significantly outperforms wavelet approximations, which only yields a $N^{-1/2}$ rate. Indeed, the wavelet approximation rate was the best published nonadaptive result so far and the result presented in this paper is the first nonadaptive construction which is provably optimal (up to a loglike factor) for this class of 3-D data.
Our estimate is consistent with the corresponding 2-D (essentially) optimally sparse approximation results obtained by the authors using 2-D shearlets and by Candès and Donoho using curvelets.

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Mathematics Subject Classification: 42C15, 42C40.

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References

[1]	E. J. Candès and D. L. Donoho, New tight frames of curvelets and optimal representations of objects with $C^2$ singularities, Comm. Pure Appl. Math., 57 (2004), 219-266.doi: doi:10.1002/cpa.10116.
[2]	F. Colonna, G. Easley, K. Guo and D. Labate, Radon transform inversion using the shearlet representation, Appl. Comput. Harmon. Anal., 29 (2010), 232-250.doi: doi:10.1016/j.acha.2009.10.005.
[3]	D. L. Donoho, Wedgelets: Nearly-minimax estimation of edges, Annals of Statistics, 27 (1999), 859-897.doi: doi:10.1214/aos/1018031261.
[4]	D. L. Donoho, Sparse components of images and optimal atomic decomposition, Constr. Approx., 17 (2001), 353-382.doi: doi:10.1007/s003650010032.
[5]	D. L. Donoho and G. Kutyniok., Microlocal analysis of the geometric separation problem, preprint, (2010).
[6]	D. L. Donoho, M. Vetterli, R. A. DeVore and I. Daubechies, Data compression and harmonic analysis, IEEE Trans. Inform. Th., 44 (1998), 2435-2476.doi: doi:10.1109/18.720544.
[7]	G. R. Easley, D. Labate and F. Colonna, Shearlet-based total variation diffusion for denoising, IEEE Trans. Image Proc., 18 (2009), 260-268.doi: doi:10.1109/TIP.2008.2008070.
[8]	G. R. Easley, D. Labate and W. Lim, Sparse directional image representations using the discrete shearlet transform, Appl. Comput. Harmon. Anal., 25 (2008), 25-46.doi: doi:10.1016/j.acha.2007.09.003.
[9]	K. Guo, G. Kutyniok and 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), 189-201.
[10]	K. Guo and D. Labate, Optimally sparse multidimensional representation using shearlets, SIAM J. Math. Anal., 9 (2007), 298-318.doi: doi:10.1137/060649781.
[11]	K. Guo and D. Labate, Characterization and analysis of edges using the continuous shearlet transform, SIAM J. Imag. Sci., 2 (2009), 959-986.doi: doi:10.1137/080741537.
[12]	K. Guo and D. Labate, "Optimally Sparse Representations of 3D Data with $C^2$ Surface Singularities Using Parseval Frames of Shearlets," Technical Report, University of Houston, 2010.
[13]	K. Guo, W.-Q Lim, D. Labate, G. Weiss and E. Wilson, Wavelets with composite dilations, Electron. Res. Announc. Amer. Math. Soc., 10 (2004), 78-87.doi: doi:10.1090/S1079-6762-04-00132-5.
[14]	K. Guo, W-Q. Lim, D. Labate, G. Weiss and E. Wilson, Wavelets with composite dilations and their MRA properties, Appl. Computat. Harmon. Anal., 20 (2006), 231-249.doi: doi:10.1007/0-8176-4504-7_11.
[15]	G. Kutyniok and W. Lim, Compactly supported shearlets are optimally sparse, preprint, (2010).
[16]	G. Kutyniok and T. Sauer., Adaptive directional subdivision schemes and shearlet multiresolution analysis, SIAM J. Math. Anal., 41 (2009), 1436-1471.doi: doi:10.1137/08072276X.
[17]	G. Kutyniok, M. Shahram and D. L. Donoho., Development of a digital shearlet transform based on pseudo-polar FFT, in "Wavelets XIII" (San Diego, CA, 2009), SPIE Proc. 7446, SPIE, Bellingham, WA, (2009), 74460B-1-74460B-13.
[18]	S. Mallat, "A Wavelet Tour of Signal Processing. The Sparse Way," Academic Press, San Diego, CA, 2009.
[19]	E. M. Stein and G. Weiss, "Introduction to Fourier Analysis on Euclidean Spaces," Princeton University Press, Princeton, NJ, 1970.
[20]	S. Yi, D. Labate, G. R. Easley and H. Krim, A Shearlet approach to edge analysis and detection, IEEE Trans. Image Process, 18 (2009), 929-941.doi: doi:10.1109/TIP.2009.2013082.