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

Optimally sparse 3D approximations using shearlet representations

1. 

Department of Mathematics, Missouri State University, Springfield, Missouri 65804, United States

2. 

Department of Mathematics, University of Houston, Houston, Texas 77204, United States

Received  September 2010 Published  October 2010

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.

Citation: Kanghui Guo, Demetrio Labate. Optimally sparse 3D approximations using shearlet representations. Electronic Research Announcements, 2010, 17: 125-137. doi: 10.3934/era.2010.17.125
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.  doi: doi:10.1002/cpa.10116.  Google Scholar

[2]

F. Colonna, G. Easley, K. Guo and D. Labate, Radon transform inversion using the shearlet representation,, Appl. Comput. Harmon. Anal., 29 (2010), 232.  doi: doi:10.1016/j.acha.2009.10.005.  Google Scholar

[3]

D. L. Donoho, Wedgelets: Nearly-minimax estimation of edges,, Annals of Statistics, 27 (1999), 859.  doi: doi:10.1214/aos/1018031261.  Google Scholar

[4]

D. L. Donoho, Sparse components of images and optimal atomic decomposition,, Constr. Approx., 17 (2001), 353.  doi: doi:10.1007/s003650010032.  Google Scholar

[5]

D. L. Donoho and G. Kutyniok., Microlocal analysis of the geometric separation problem,, preprint, (2010).   Google Scholar

[6]

D. L. Donoho, M. Vetterli, R. A. DeVore and I. Daubechies, Data compression and harmonic analysis,, IEEE Trans. Inform. Th., 44 (1998), 2435.  doi: doi:10.1109/18.720544.  Google Scholar

[7]

G. R. Easley, D. Labate and F. Colonna, Shearlet-based total variation diffusion for denoising,, IEEE Trans. Image Proc., 18 (2009), 260.  doi: doi:10.1109/TIP.2008.2008070.  Google Scholar

[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.  doi: doi:10.1016/j.acha.2007.09.003.  Google Scholar

[9]

K. Guo, G. Kutyniok and D. Labate, Sparse multidimensional representations using anisotropic dilation and shear operators,, in, (2006), 189.   Google Scholar

[10]

K. Guo and D. Labate, Optimally sparse multidimensional representation using shearlets,, SIAM J. Math. Anal., 9 (2007), 298.  doi: doi:10.1137/060649781.  Google Scholar

[11]

K. Guo and D. Labate, Characterization and analysis of edges using the continuous shearlet transform,, SIAM J. Imag. Sci., 2 (2009), 959.  doi: doi:10.1137/080741537.  Google Scholar

[12]

K. Guo and D. Labate, "Optimally Sparse Representations of 3D Data with $C^2$ Surface Singularities Using Parseval Frames of Shearlets,", Technical Report, (2010).   Google Scholar

[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.  doi: doi:10.1090/S1079-6762-04-00132-5.  Google Scholar

[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.  doi: doi:10.1007/0-8176-4504-7_11.  Google Scholar

[15]

G. Kutyniok and W. Lim, Compactly supported shearlets are optimally sparse,, preprint, (2010).   Google Scholar

[16]

G. Kutyniok and T. Sauer., Adaptive directional subdivision schemes and shearlet multiresolution analysis,, SIAM J. Math. Anal., 41 (2009), 1436.  doi: doi:10.1137/08072276X.  Google Scholar

[17]

G. Kutyniok, M. Shahram and D. L. Donoho., Development of a digital shearlet transform based on pseudo-polar FFT,, in, 7446 (2009), 1.   Google Scholar

[18]

S. Mallat, "A Wavelet Tour of Signal Processing. The Sparse Way,", Academic Press, (2009).   Google Scholar

[19]

E. M. Stein and G. Weiss, "Introduction to Fourier Analysis on Euclidean Spaces,", Princeton University Press, (1970).   Google Scholar

[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.  doi: doi:10.1109/TIP.2009.2013082.  Google Scholar

show all references

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.  doi: doi:10.1002/cpa.10116.  Google Scholar

[2]

F. Colonna, G. Easley, K. Guo and D. Labate, Radon transform inversion using the shearlet representation,, Appl. Comput. Harmon. Anal., 29 (2010), 232.  doi: doi:10.1016/j.acha.2009.10.005.  Google Scholar

[3]

D. L. Donoho, Wedgelets: Nearly-minimax estimation of edges,, Annals of Statistics, 27 (1999), 859.  doi: doi:10.1214/aos/1018031261.  Google Scholar

[4]

D. L. Donoho, Sparse components of images and optimal atomic decomposition,, Constr. Approx., 17 (2001), 353.  doi: doi:10.1007/s003650010032.  Google Scholar

[5]

D. L. Donoho and G. Kutyniok., Microlocal analysis of the geometric separation problem,, preprint, (2010).   Google Scholar

[6]

D. L. Donoho, M. Vetterli, R. A. DeVore and I. Daubechies, Data compression and harmonic analysis,, IEEE Trans. Inform. Th., 44 (1998), 2435.  doi: doi:10.1109/18.720544.  Google Scholar

[7]

G. R. Easley, D. Labate and F. Colonna, Shearlet-based total variation diffusion for denoising,, IEEE Trans. Image Proc., 18 (2009), 260.  doi: doi:10.1109/TIP.2008.2008070.  Google Scholar

[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.  doi: doi:10.1016/j.acha.2007.09.003.  Google Scholar

[9]

K. Guo, G. Kutyniok and D. Labate, Sparse multidimensional representations using anisotropic dilation and shear operators,, in, (2006), 189.   Google Scholar

[10]

K. Guo and D. Labate, Optimally sparse multidimensional representation using shearlets,, SIAM J. Math. Anal., 9 (2007), 298.  doi: doi:10.1137/060649781.  Google Scholar

[11]

K. Guo and D. Labate, Characterization and analysis of edges using the continuous shearlet transform,, SIAM J. Imag. Sci., 2 (2009), 959.  doi: doi:10.1137/080741537.  Google Scholar

[12]

K. Guo and D. Labate, "Optimally Sparse Representations of 3D Data with $C^2$ Surface Singularities Using Parseval Frames of Shearlets,", Technical Report, (2010).   Google Scholar

[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.  doi: doi:10.1090/S1079-6762-04-00132-5.  Google Scholar

[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.  doi: doi:10.1007/0-8176-4504-7_11.  Google Scholar

[15]

G. Kutyniok and W. Lim, Compactly supported shearlets are optimally sparse,, preprint, (2010).   Google Scholar

[16]

G. Kutyniok and T. Sauer., Adaptive directional subdivision schemes and shearlet multiresolution analysis,, SIAM J. Math. Anal., 41 (2009), 1436.  doi: doi:10.1137/08072276X.  Google Scholar

[17]

G. Kutyniok, M. Shahram and D. L. Donoho., Development of a digital shearlet transform based on pseudo-polar FFT,, in, 7446 (2009), 1.   Google Scholar

[18]

S. Mallat, "A Wavelet Tour of Signal Processing. The Sparse Way,", Academic Press, (2009).   Google Scholar

[19]

E. M. Stein and G. Weiss, "Introduction to Fourier Analysis on Euclidean Spaces,", Princeton University Press, (1970).   Google Scholar

[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.  doi: doi:10.1109/TIP.2009.2013082.  Google Scholar

[1]

Lingju Kong, Roger Nichols. On principal eigenvalues of biharmonic systems. Communications on Pure & Applied Analysis, 2021, 20 (1) : 1-15. doi: 10.3934/cpaa.2020254

[2]

Peizhao Yu, Guoshan Zhang, Yi Zhang. Decoupling of cubic polynomial matrix systems. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 13-26. doi: 10.3934/naco.2020012

[3]

Ilyasse Lamrani, Imad El Harraki, Ali Boutoulout, Fatima-Zahrae El Alaoui. Feedback stabilization of bilinear coupled hyperbolic systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020434

[4]

Felix Finster, Jürg Fröhlich, Marco Oppio, Claudio F. Paganini. Causal fermion systems and the ETH approach to quantum theory. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020451

[5]

Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020461

[6]

Simon Hochgerner. Symmetry actuated closed-loop Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 641-669. doi: 10.3934/jgm.2020030

[7]

Javier Fernández, Cora Tori, Marcela Zuccalli. Lagrangian reduction of nonholonomic discrete mechanical systems by stages. Journal of Geometric Mechanics, 2020, 12 (4) : 607-639. doi: 10.3934/jgm.2020029

[8]

Lingwei Ma, Zhenqiu Zhang. Monotonicity for fractional Laplacian systems in unbounded Lipschitz domains. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 537-552. doi: 10.3934/dcds.2020268

[9]

Peter H. van der Kamp, D. I. McLaren, G. R. W. Quispel. Homogeneous darboux polynomials and generalising integrable ODE systems. Journal of Computational Dynamics, 2021, 8 (1) : 1-8. doi: 10.3934/jcd.2021001

[10]

Yuri Fedorov, Božidar Jovanović. Continuous and discrete Neumann systems on Stiefel varieties as matrix generalizations of the Jacobi–Mumford systems. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020375

[11]

João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138

[12]

Awais Younus, Zoubia Dastgeer, Nudrat Ishaq, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Devendra Kumar. On the observability of conformable linear time-invariant control systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020444

[13]

Sergey Rashkovskiy. Hamilton-Jacobi theory for Hamiltonian and non-Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 563-583. doi: 10.3934/jgm.2020024

[14]

Meilan Cai, Maoan Han. Limit cycle bifurcations in a class of piecewise smooth cubic systems with multiple parameters. Communications on Pure & Applied Analysis, 2021, 20 (1) : 55-75. doi: 10.3934/cpaa.2020257

[15]

Touria Karite, Ali Boutoulout. Global and regional constrained controllability for distributed parabolic linear systems: RHUM approach. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020055

[16]

Shiqi Ma. On recent progress of single-realization recoveries of random Schrödinger systems. Electronic Research Archive, , () : -. doi: 10.3934/era.2020121

[17]

Jan Bouwe van den Berg, Elena Queirolo. A general framework for validated continuation of periodic orbits in systems of polynomial ODEs. Journal of Computational Dynamics, 2021, 8 (1) : 59-97. doi: 10.3934/jcd.2021004

[18]

Chongyang Liu, Meijia Han, Zhaohua Gong, Kok Lay Teo. Robust parameter estimation for constrained time-delay systems with inexact measurements. Journal of Industrial & Management Optimization, 2021, 17 (1) : 317-337. doi: 10.3934/jimo.2019113

[19]

Jian Zhang, Tony T. Lee, Tong Ye, Liang Huang. An approximate mean queue length formula for queueing systems with varying service rate. Journal of Industrial & Management Optimization, 2021, 17 (1) : 185-204. doi: 10.3934/jimo.2019106

[20]

Maoding Zhen, Binlin Zhang, Vicenţiu D. Rădulescu. Normalized solutions for nonlinear coupled fractional systems: Low and high perturbations in the attractive case. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020379

2019 Impact Factor: 0.5

Metrics

  • PDF downloads (34)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]