Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-06T12:25:02.420Z Has data issue: false hasContentIssue false
  • English
  • Français

3D Data Denoising Using Combined SparseDictionaries

Published online by Cambridge University Press:  28 January 2013

G. Easley ,
D. Labate  and
P. Negi
G. Easley
Affiliation:
System Planning Corporation, Arlington, VA 22201, USA
D. Labate*
Affiliation:
Department of Mathematics, University of Houston, Houston, Texas 77204, USA
P. Negi
Affiliation:
Department of Mathematics, University of Houston, Houston, Texas 77204, USA
*
∗ Corresponding author. E-mail: [email protected]
Get access

Abstract

Directional multiscale representations such as shearlets and curvelets have gainedincreasing recognition in recent years as superior methods for the sparse representationof data. Thanks to their ability to sparsely encode images and other multidimensionaldata, transform-domain denoising algorithms based on these representations are among thebest performing methods currently available. As already observed in the literature, theperformance of many sparsity-based data processing methods can be further improved byusing appropriate combinations of dictionaries. In this paper, we consider the problem of3D data denoising and introduce a denoising algorithm which uses combined sparsedictionaries. Our numerical demonstrations show that the realization of the algorithmwhich combines 3D shearlets and local Fourier bases provides highly competitive results ascompared to other 3D sparsity-based denosing algorithms based on both single and combineddictionaries.

Keywords

curveletsdenoisingnonlinear approximationspursuit algorithmsshearletssparse approximationswavelets
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 8 , Issue 1: Harmonic analysis , 2013 , pp. 60 - 74
Copyright
© EDP Sciences, 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Références

Bobin, J., Starck, J.-L., Fadili, M.J., Moudden, Y., Donoho, D.L.. Morphological component analysis : an adaptive thresholding strategy. IEEE Trans. Image Process. 16 (11) (2007), 2675–2681. CrossRefGoogle ScholarPubMed
Candès, E. J., Demanet, L., Donoho, D., Ying, L.. Fast discrete curvelet transforms. Multiscale Model. Simul. 5 (2006), 861–899. CrossRefGoogle Scholar
Candès, E. J., Donoho, D. L.. Ridgelets : the key to high dimensional intermittency? Philosophical Transactions of the Royal Society of London A, 357 (1999), 2495–2509. CrossRefGoogle Scholar
Candès, E. J., Donoho, D. L.. New tight frames of curvelets and optimal representations of objects with C2 singularities. Comm. Pure Appl. Math., 57 (2004), 219–266. CrossRefGoogle Scholar
Chen, S. S., Donoho, D. L., Saunders, M. A.. Atomic decomposition by basis pursuit. SIAM Rev. 43 (1) (2001), 129–159. CrossRefGoogle Scholar
Daubechies, I., Defrise, M., De Mol, C.. An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Comm. Pure Appl. Math. 57 (2004), 1413–1457. CrossRefGoogle Scholar
Donoho, D. L.. Denoising by soft thresholding. IEEE Trans. Inf. Theory, 41 (3) (1995), 613–627. CrossRefGoogle Scholar
Donoho, D. L.. Sparse components of images and optimal atomic decomposition. Constr. Approx. 17 (2001), 353–382. CrossRefGoogle Scholar
Donoho, D. L.. Wedgelets : nearly-minimax estimation of edges. Annals of Statistics, 27 (1999), 859–897. CrossRefGoogle Scholar
Donoho, D. L., Johnstone, I. M.. Ideal spatial adaptation by wavelet shrinkage. Biometrika, 81 (3) (1994), 425–455. CrossRefGoogle Scholar
Donoho, D. L., Johnstone, I. M.. Adapting to unknown smoothness via wavelet shrinkage. J. Amer. Statist. Assoc. 90 (1995), 1200–1224. CrossRefGoogle Scholar
Donoho, D. L., Johnstone, I. M., Kerkyacharian, G., Picard, D.. Wavelet shrinkage. Asymptopia. J. Roy. Statist. Soc. B, 57 (2) (1995), 301–337. Google Scholar
Easley, G. R., Labate, D., Colonna, F.. Shearlet-based total variation diffusion for denoising. IEEE Trans. Image Proc. 18 (2) (2009), 260–268. CrossRefGoogle ScholarPubMed
Easley, G. R., Labate, D., Lim, W.. Sparse directional image representations using the discrete shearlet transform. Appl. Comput. Harmon. Anal., 25 (1) (2008), 25–46. CrossRefGoogle Scholar
M. Elad. Sparse and Redundant Representations : From Theory to Applications in Signal and Image Processing. Springer, New York, NY, 2010.
Elad, M., Milanfar, P., Rubinstein, R.. Analysis Versus Synthesis in Signal Priors. Inverse Problems, 23 (3) (2007), 947–968. CrossRefGoogle Scholar
K. Guo, G. Kutyniok, 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.
Guo, K., Labate, D.. Optimally Sparse Multidimensional Representation using Shearlets. SIAM J. Math. Anal.. 9 (2007), 298–318. CrossRefGoogle Scholar
Guo, K., Labate, D.. Optimally sparse 3D approximations using shearlet representations. Electron. Res. Announc. Math. Sci. 17 (2010), 126–138. Google Scholar
Guo, K., Labate, D.. Optimally sparse representations of 3D Data with C2 surface singularities using Parseval frames of shearlets. SIAM J Math. Anal. 44 (2012), 851–886. CrossRefGoogle Scholar
Guo, K., Labate, D.. The Construction of Smooth Parseval Frames of Shearlets. Math. Model. Nat. Phenom. 8 (1) (2013), 3255. CrossRefGoogle Scholar
X. Huo. Sparse Image Representation Via Combined Transforms, Ph.D. Thesis, Stanford University, 1999.
G. Kutyniok. Clustered sparsity and separation of cartoon and texture, preprint (2012).
D. Labate, W.-Q Lim, G. Kutyniok, G. Weiss. Sparse multidimensional representation using shearlets, in Wavelets XI, edited by M. Papadakis, A. F. Laine, and M. A. Unser, SPIE Proc. 5914 (2005), SPIE, Bellingham, WA, 2005, 254–262.
Lu, Y., Do, M. N.. Multidimensional directional filter banks and surfacelets, IEEE Trans. Image Process., 16 (4) (2007), 918–931. CrossRefGoogle ScholarPubMed
Malgouyres, F.. Minimizing the total variation under a general convex constraint for image restoration. IEEE Trans. Signal Process. 11 (12) (2002), 1450–1456. Google Scholar
S. Mallat. A Wavelet Tour of Signal Processing.Third Edition : The Sparse Way, Academic Press, San Diego, CA, 2008.
Meyer, F. G., Averbuch, A. Z., Coifman, R.. Multi-layered image representation : Application to image compression, IEEE Trans. Image Process. 11(6) (1998), 1072–1080. CrossRefGoogle Scholar
Negi, P. S., Labate, D.. 3D discrete shearlet transform and video processing, IEEE Trans. Image Process. 21(6) (2012), 944–2954. CrossRefGoogle Scholar
V. M. Patel, G. R. Easley, R. Chellappa, Component-based restoration of speckled images, Proceedings 18th IEEE International Conference on Image Processing (ICIP), 2011.
Starck, J. L., Elad, M., Donoho, D.L.. Image decomposition via the combination of sparse representations and a variational approach, IEEE Trans. Image Process. 14(10) (2005), 1570–1582. CrossRefGoogle Scholar
Starck, J. L., Murtagh, F., Bijaoui, A.. Multiresolution support applied to image filtering and restoration, Graphic. Models Image Process. 57 (1995), 420–431. CrossRefGoogle Scholar
J. L. Starck, F. Murtagh, J. M. Fadili. Sparse Image and Signal Processing, Cambridge University Press, New York, NY, 2010.
Woiselle, A., Starck, J. L., Fadili, J. M.. 3-D Data denoising and inpainting with the Low-Redundancy Fast Curvelet Transform, J. Math. Imaging Vis. 39(2) (2011), 121–139. CrossRefGoogle Scholar