×

zbMATH — the first resource for mathematics

Wave atoms and sparsity of oscillatory patterns. (English) Zbl 1132.68068
Summary: We introduce “wave atoms” as a variant of 2D wavelet packets obeying the parabolic scaling wavelength \(\sim\)(diameter)\(^{2}\). We prove that warped oscillatory functions, a toy model for texture, have a significantly sparser expansion in wave atoms than in other fixed standard representations like wavelets, Gabor atoms, or curvelets. We propose a novel algorithm for a tight frame of wave atoms with redundancy two, directly in the frequency plane, by the “wrapping” technique. We also propose variants of the basic transform for applications in image processing, including an orthonormal basis, and a shift-invariant tight frame with redundancy four. Sparsity and denoising experiments on both seismic and fingerprint images demonstrate the potential of the tool introduced.

MSC:
68U10 Computing methodologies for image processing
Software:
DT-CWT
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Antoine, J.P.; Demanet, L.; Jacques, L.; Hochedez, J.F.; Terrier, R.; Verwichte, E., Applications of the 2-D wavelet transform to astrophysical images, Physicalia mag., 24, 93-116, (2002)
[2] Antoine, J.P.; Murenzi, R., Two-dimensional directional wavelets and the scale-angle representation, Signal process., 52, 259-281, (1996) · Zbl 0875.94074
[3] Brislawn, C., The FBI fingerprint image compression standard
[4] Candès, E.J., Harmonic analysis of neural networks, Appl. comput. harmon. anal., 6, 197-218, (1999) · Zbl 0931.68104
[5] Candès, E.J.; Demanet, L., Curvelets and Fourier integral operators, C. R. acad. sci. Paris, ser. I, 336, 395-398, (2003) · Zbl 1056.42025
[6] Candès, E.J.; Demanet, L., The curvelet representation of wave propagators is optimally sparse, Comm. pure appl. math., 58, 11, 1472-1528, (2005) · Zbl 1078.35007
[7] Candès, E.J.; Demanet, L.; Donoho, D.L.; Ying, L., Fast discrete curvelet transforms, SIAM multiscale model. simul., 5, 3, 861-899, (2006) · Zbl 1122.65134
[8] Candès, E.J.; Donoho, D.L., Curvelets—A surprisingly effective nonadaptive representation for objects with edges, (), 105-120
[9] Candès, E.J.; Donoho, D.L., New tight frames of curvelets and optimal representations of objects with piecewise-\(C^2\) singularities, Comm. pure appl. math., 57, 219-266, (2004) · Zbl 1038.94502
[10] Candès, E.J.; Romberg, J.; Tao, T., Stable signal recovery from incomplete and inaccurate measurements, Comm. pure appl. math., 59, 1207-1223, (2006) · Zbl 1098.94009
[11] C. Chaux, L. Duval, J.C. Pesquet, 2D dual-tree M-band wavelet decomposition, in: Proc. International Conference on Acoustics, Speech, and Signal Processing, ICASSP 05, Philadelphia, PA, March 18-23, 2005, pp. 537-540
[12] Chaux, C.; Duval, L.; Pesquet, J.C., Image analysis using a dual-tree M-band wavelet transform, IEEE trans. image process., 15, 8, 2397-2412, (2006)
[13] H. Choi, J. Romberg, R. Baraniuk, N. Kingsbury, Hidden Markov tree modeling of complex wavelet transforms, in: Proc. International Conference on Acoustics, Speech, and Signal Processing, ICASSP 00, Istanbul, Turkey, June 2000, pp. 133-136
[14] Córdoba, A.; Fefferman, C., Wave packets and Fourier integral operators, Comm. partial differential equations, 3, 11, 979-1005, (1978) · Zbl 0389.35046
[15] Daubechies, I.; Jaffard, S.; Journ, J.L., A simple Wilson orthonormal basis with exponential decay, SIAM J. math. anal., 22, 554-572, (1991) · Zbl 0754.46016
[16] L. Demanet, Curvelets, wave atoms and wave equations, Ph.D. thesis, California Institute of Technology, 2006
[17] Do, M.; Vetterli, M., The contourlet transform: an efficient directional multiresolution image representation, IEEE trans. image process., 14, 12, 2091-2106, (2005)
[18] Elad, M.; Starck, J.-L.; Donoho, D.L.; Querre, P., Simultaneous cartoon and texture image inpainting using morphological component analysis (MCA), Appl. comput. harmon. anal., 19, 340-358, (2005) · Zbl 1081.68732
[19] F. Fernandes, M. Wakin, R. Baraniuk, Non-redundant, linear-phase, semi-orthogonal, directional complex wavelets, in: Proc. IEEE ICASSP, Montreal, Canada, 2004
[20] Hennenfent, G.; Herrmann, F.J., Seismic denoising with unstructured curvelets, Comput. sci. eng., 8, 3, 16-25, (2006)
[21] F.J. Herrmann, P.P. Moghaddam, C.C. Stolk, Sparsity- and continuity-promoting seismic image recovery with curvelet frames, in preparation · Zbl 1135.68057
[22] Kingsbury, N., Image processing with complex wavelets, Philos. trans. R. soc. A, 357, 1760, 2543-2560, (1999) · Zbl 0976.68527
[23] E. Kolaczyk, Wavelet methods for the inversion of certain homogeneous linear operators in the presence of noisy data, Ph.D. thesis, Stanford University, 1994
[24] D. Labate, W.-Q. Lim, G. Kutyniok, G. Weiss, Sparse multidimensional representation using shearlets, in: Proc. SPIE Wavelets XI, San Diego, 2005, pp. 254-262
[25] Lintner, S.; Malgouyres, F., Solving a variational image restoration model which involves \(\ell_\infty\) constraints, Inverse problems, 20, 815-831, (2004) · Zbl 1063.94007
[26] Mallat, S., A wavelet tour of signal processing, (1999), Academic Press Orlando/San Diego · Zbl 0998.94510
[27] Meyer, F.G.; Coifman, R.R., Brushlets: A tool for directional image analysis and image compression, Appl. comput. harmon. anal., 4, 147-187, (1997) · Zbl 0879.68117
[28] Peyré, G.; Mallat, S., Surface compression with geometric bandelets, Proc. SIGGRAPH 2005, ACM trans. graphics, 24, 3, 601-608, (2005)
[29] Selesnick, I.; Baraniuk, R.G.; Kingsbury, N.G., The dual-tree complex wavelet transform, IEEE signal process. mag., 22, 6, 123-151, (2005)
[30] Seré, E., Localisation fréquentielle des paquets d’ondelettes, Rev. mat. iberoamericana, 11, 2, 334-354, (1995) · Zbl 0829.42024
[31] Starck, J.L.; Candès, E.J.; Donoho, D.L., The curvelet transform for image denoising, IEEE trans. image process., 11, 6, 670-684, (2002) · Zbl 1288.94011
[32] Starck, J.-L.; Elad, M.; Donoho, D.L., Image decomposition via the combination of sparse representation and a variational approach, IEEE trans. image process., 14, 10, 1570-1582, (2005) · Zbl 1288.94012
[33] Vandergheynst, P.; Gobbers, J.F., Directional dyadic wavelet transforms: design and algorithms, IEEE trans. image process., 11, 4, 363-372, (2002)
[34] Villemoes, L., Wavelet packets with uniform time – frequency localization, C. R. math., 335, 10, 793-796, (2002) · Zbl 1015.42026
[35] Wakin, M.; Romberg, J.; Choi, H.; Baraniuk, R., Wavelet-domain approximation and compression of piecewise smooth images, IEEE trans. image process., 15, 5, 1071-1087, (2006)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.