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Periodic spline-based frames for image restoration
1. | School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel, Israel |
2. | Dept. of Mathematical Information Technology, Faculty of Information Technology, Agora, P.O. Box 35, FI-40014, University of Jyväskylä |
References:
[1] |
O. Amrani, A. Averbuch, T. Cohen and V. A. Zheludev, Symmetric interpolatory framelets and their error correction properties,, International Journal of Wavelets, 5 (2007), 541.
doi: 10.1142/S0219691307001896. |
[2] |
A. Averbuch and V. Zheludev, Construction of biorthogonal discrete wavelet transforms using interpolatory splines,, Applied and Comp. Harmonic Analysis, 12 (2002), 25.
doi: 10.1006/acha.2001.0367. |
[3] |
A. Averbuch and V. Zheludev, Wavelet transforms generated by splines,, International Journal of Wavelets, 5 (2007), 257.
doi: 10.1142/S0219691307001756. |
[4] |
A. Z. Averbuch, A. B. Pevnyi and V. A. Zheludev, Biorthogonal Butterworth wavelets derived from discrete interpolatory splines,, IEEE Trans. on Sign. Proc., 49 (2001), 2682.
doi: 10.1109/78.960415. |
[5] |
A. Z. Averbuch, A. B. Pevnyi and V. A. Zheludev, Butterworth wavelet transforms derived from discrete interpolatory splines: Recursive implementation,, Signal Processing, 81 (2001), 2363. Google Scholar |
[6] |
A. Z. Averbuch, V. A. Zheludev and T. Cohen, Interpolatory frames in signal space,, IEEE Trans. Sign. Proc., 54 (2006), 2126.
doi: 10.1109/TSP.2006.870562. |
[7] |
A. Z. Averbuch, V. A. Zheludev and T. Cohen, Tight and sibling frames originated from discrete splines,, Sign. Proc. J., 86 (2006), 1632.
doi: 10.1016/j.sigpro.2005.09.007. |
[8] |
H. Bölcskei, F. Hlawatsch and H. G. Feichtinger, Frame-theoretic analysis of oversampled filter banks,, IEEE Transactions on Sign. Proc., 46 (1998), 3256. Google Scholar |
[9] |
C. K. Chui and W. He, Compactly supported tight frames associated with refinable functions,, Applied and Comp. Harmonic Analysis, 8 (2000), 293.
doi: 10.1006/acha.2000.0301. |
[10] |
C. K. Chui, W. He and J. Stöckler, Compactly supported tight and sibling frames with maximum vanishing moments,, Applied and Comp. Harmonic Analysis, 13 (2002), 224.
doi: 10.1016/S1063-5203(02)00510-9. |
[11] |
Z. Cvetković and M. Vetterli, Oversampled filter banks,, IEEE Transactions on Signal Processing, 46 (1998), 1245. Google Scholar |
[12] |
I. Daubechies, B. Han, A. Ron and Z. Shen, Framelets: Mra-based constructions of wavelet frames,, Applied and Computational Harmonic Analysis, 14 (2003), 1.
doi: 10.1016/S1063-5203(02)00511-0. |
[13] |
B. Dong and Z. Shen, Pseudo-splines, wavelets and framelets,, Applied and Computational Harmonic Analysis, 22 (2007), 78.
doi: 10.1016/j.acha.2006.04.008. |
[14] |
D. H. Foster, K. Amano, S. M. C. Nascimento and M. J. Foster, Frequency of metamerism in natural scenes,, Journal of the Optical Society of America A, 23 (2006), 2359. Google Scholar |
[15] |
T. Goldstein and S. Osher, The split {Bregman} method for l1-regularized problems,, SIAM J. Imaging Sciences, 2 (2009), 323.
doi: 10.1137/080725891. |
[16] |
M. Golub, M. Nathan, A. Averbuch, A. Kagan, V. A. Zheludev and R. Malinsky, Snapshot spectral imaging based on digital cameras,, US Patent Application Publication Golub et al., (2013). Google Scholar |
[17] |
V. K. Goyal, J. Kovacevic and J. A. Kelner, Quantized frame expansions with erasures,, Appl. and Comput. Harmonic Analysis, 10 (2001), 203.
doi: 10.1006/acha.2000.0340. |
[18] |
V. K. Goyal, M. Vetterli and N. T. Thao, Quantized overcomplete expansions in $\mathbbR^n$: Analysis, synthesis and algorithms,, IEEE Trans. on Information Theory, 44 (1998), 16.
doi: 10.1109/18.650985. |
[19] |
B. Han, S. Song Goh and Z. Shen, Tight periodic wavelet frames and approximation orders,, Applied and Computational Harmonic Analysis, 31 (2011), 228.
doi: 10.1016/j.acha.2010.12.001. |
[20] |
C. Herley and M. Vetterli, Wavelets and recursive filter banks,, IEEE Trans. Signal Proc., 41 (1993), 2536.
doi: 10.1109/78.229887. |
[21] |
J. Kovacevic, P. L. Dragotti and V. K. Goyal, Filter bank frame expansions with erasures,, IEEE Trans. Inform, 48 (2002), 1439.
doi: 10.1109/TIT.2002.1003832. |
[22] |
A. V. Oppenheim and R. W. Schafer, Discrete-time Signal Processing,, New York, (2010). Google Scholar |
[23] |
S. Osher, J. Cai, B. Dong and Z. Shen, Image restoration: Total variation, wavelet frames, and beyond,, Journal of the American Mathematical Society, 25 (2012), 1033.
doi: 10.1090/S0894-0347-2012-00740-1. |
[24] |
G. Polya and G. Szegö, Aufgaben und Lehrsätze aus der Analysis, Vol. II,, Springer, (1971).
|
[25] |
J. Romberg, E. Candes and T. Tao, Stable signal recovery from incomplete and inaccurate measurements,, Communications on Pure and Applied Mathematics, 59 (2006), 1207.
doi: 10.1002/cpa.20124. |
[26] |
A. Ron and Z. Shen, Compactly supported tight affine spline frames in$ l^2(\mathbbR)$,, Mathematics of Computation, 67 (1998), 191.
doi: 10.1090/S0025-5718-98-00898-9. |
[27] |
I. J. Schoenberg, Contribution to the problem of approximation of equidistant data by analytic functions,, Quart. Appl. Math., 4 (1946), 45. Google Scholar |
[28] |
Z. Shen, Wavelet frames and image restorations,, in Proc. Int. Congress of Mathematicians. Vol. IV (eds. Rajendra Bhatia), (2010), 2834.
|
[29] |
Z. Shen, J. Cai and S. Osher, Split Bregman methods and frame based image restoration,, Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal, 8 (2009), 337.
doi: 10.1137/090753504. |
[30] |
Z. Shen H. Ji and Y. Xu, Wavelet frame based image restoration with missing/damaged pixels,, East Asia Journal on Applied Mathematics, 1 (2011), 108. Google Scholar |
[31] |
G. Strang and G. Fix, A Fourier analysis of the finite element variational method,, Construct. Asp. Funct. Anal., 57 (2011), 793.
doi: 10.1007/978-3-642-10984-3_7. |
[32] |
V. A. Zheludev, V. N. Malozemov and A. B. Pevnyi, Filter banks and frames in the discrete periodic case,, in Proceedings of the St. Petersburg Mathematical Society. Vol. XIV, (2009), 1.
|
[33] |
V. A. Zheludev, Local spline approximation on a uniform grid,, Zh. Vychisl. Mat. i Mat. Fiz., 27 (1987), 1296.
|
[34] |
V. A. Zheludev, Periodic splines and the fast Fourier transform,, Comput. Math. & Math Phys., 32 (1992), 149.
|
[35] |
V. A. Zheludev, Interpolatory subdivision schemes with infinite masks originated from splines,, Advances in Comp. Math., 25 (2006), 475.
doi: 10.1007/s10444-004-4149-6. |
show all references
References:
[1] |
O. Amrani, A. Averbuch, T. Cohen and V. A. Zheludev, Symmetric interpolatory framelets and their error correction properties,, International Journal of Wavelets, 5 (2007), 541.
doi: 10.1142/S0219691307001896. |
[2] |
A. Averbuch and V. Zheludev, Construction of biorthogonal discrete wavelet transforms using interpolatory splines,, Applied and Comp. Harmonic Analysis, 12 (2002), 25.
doi: 10.1006/acha.2001.0367. |
[3] |
A. Averbuch and V. Zheludev, Wavelet transforms generated by splines,, International Journal of Wavelets, 5 (2007), 257.
doi: 10.1142/S0219691307001756. |
[4] |
A. Z. Averbuch, A. B. Pevnyi and V. A. Zheludev, Biorthogonal Butterworth wavelets derived from discrete interpolatory splines,, IEEE Trans. on Sign. Proc., 49 (2001), 2682.
doi: 10.1109/78.960415. |
[5] |
A. Z. Averbuch, A. B. Pevnyi and V. A. Zheludev, Butterworth wavelet transforms derived from discrete interpolatory splines: Recursive implementation,, Signal Processing, 81 (2001), 2363. Google Scholar |
[6] |
A. Z. Averbuch, V. A. Zheludev and T. Cohen, Interpolatory frames in signal space,, IEEE Trans. Sign. Proc., 54 (2006), 2126.
doi: 10.1109/TSP.2006.870562. |
[7] |
A. Z. Averbuch, V. A. Zheludev and T. Cohen, Tight and sibling frames originated from discrete splines,, Sign. Proc. J., 86 (2006), 1632.
doi: 10.1016/j.sigpro.2005.09.007. |
[8] |
H. Bölcskei, F. Hlawatsch and H. G. Feichtinger, Frame-theoretic analysis of oversampled filter banks,, IEEE Transactions on Sign. Proc., 46 (1998), 3256. Google Scholar |
[9] |
C. K. Chui and W. He, Compactly supported tight frames associated with refinable functions,, Applied and Comp. Harmonic Analysis, 8 (2000), 293.
doi: 10.1006/acha.2000.0301. |
[10] |
C. K. Chui, W. He and J. Stöckler, Compactly supported tight and sibling frames with maximum vanishing moments,, Applied and Comp. Harmonic Analysis, 13 (2002), 224.
doi: 10.1016/S1063-5203(02)00510-9. |
[11] |
Z. Cvetković and M. Vetterli, Oversampled filter banks,, IEEE Transactions on Signal Processing, 46 (1998), 1245. Google Scholar |
[12] |
I. Daubechies, B. Han, A. Ron and Z. Shen, Framelets: Mra-based constructions of wavelet frames,, Applied and Computational Harmonic Analysis, 14 (2003), 1.
doi: 10.1016/S1063-5203(02)00511-0. |
[13] |
B. Dong and Z. Shen, Pseudo-splines, wavelets and framelets,, Applied and Computational Harmonic Analysis, 22 (2007), 78.
doi: 10.1016/j.acha.2006.04.008. |
[14] |
D. H. Foster, K. Amano, S. M. C. Nascimento and M. J. Foster, Frequency of metamerism in natural scenes,, Journal of the Optical Society of America A, 23 (2006), 2359. Google Scholar |
[15] |
T. Goldstein and S. Osher, The split {Bregman} method for l1-regularized problems,, SIAM J. Imaging Sciences, 2 (2009), 323.
doi: 10.1137/080725891. |
[16] |
M. Golub, M. Nathan, A. Averbuch, A. Kagan, V. A. Zheludev and R. Malinsky, Snapshot spectral imaging based on digital cameras,, US Patent Application Publication Golub et al., (2013). Google Scholar |
[17] |
V. K. Goyal, J. Kovacevic and J. A. Kelner, Quantized frame expansions with erasures,, Appl. and Comput. Harmonic Analysis, 10 (2001), 203.
doi: 10.1006/acha.2000.0340. |
[18] |
V. K. Goyal, M. Vetterli and N. T. Thao, Quantized overcomplete expansions in $\mathbbR^n$: Analysis, synthesis and algorithms,, IEEE Trans. on Information Theory, 44 (1998), 16.
doi: 10.1109/18.650985. |
[19] |
B. Han, S. Song Goh and Z. Shen, Tight periodic wavelet frames and approximation orders,, Applied and Computational Harmonic Analysis, 31 (2011), 228.
doi: 10.1016/j.acha.2010.12.001. |
[20] |
C. Herley and M. Vetterli, Wavelets and recursive filter banks,, IEEE Trans. Signal Proc., 41 (1993), 2536.
doi: 10.1109/78.229887. |
[21] |
J. Kovacevic, P. L. Dragotti and V. K. Goyal, Filter bank frame expansions with erasures,, IEEE Trans. Inform, 48 (2002), 1439.
doi: 10.1109/TIT.2002.1003832. |
[22] |
A. V. Oppenheim and R. W. Schafer, Discrete-time Signal Processing,, New York, (2010). Google Scholar |
[23] |
S. Osher, J. Cai, B. Dong and Z. Shen, Image restoration: Total variation, wavelet frames, and beyond,, Journal of the American Mathematical Society, 25 (2012), 1033.
doi: 10.1090/S0894-0347-2012-00740-1. |
[24] |
G. Polya and G. Szegö, Aufgaben und Lehrsätze aus der Analysis, Vol. II,, Springer, (1971).
|
[25] |
J. Romberg, E. Candes and T. Tao, Stable signal recovery from incomplete and inaccurate measurements,, Communications on Pure and Applied Mathematics, 59 (2006), 1207.
doi: 10.1002/cpa.20124. |
[26] |
A. Ron and Z. Shen, Compactly supported tight affine spline frames in$ l^2(\mathbbR)$,, Mathematics of Computation, 67 (1998), 191.
doi: 10.1090/S0025-5718-98-00898-9. |
[27] |
I. J. Schoenberg, Contribution to the problem of approximation of equidistant data by analytic functions,, Quart. Appl. Math., 4 (1946), 45. Google Scholar |
[28] |
Z. Shen, Wavelet frames and image restorations,, in Proc. Int. Congress of Mathematicians. Vol. IV (eds. Rajendra Bhatia), (2010), 2834.
|
[29] |
Z. Shen, J. Cai and S. Osher, Split Bregman methods and frame based image restoration,, Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal, 8 (2009), 337.
doi: 10.1137/090753504. |
[30] |
Z. Shen H. Ji and Y. Xu, Wavelet frame based image restoration with missing/damaged pixels,, East Asia Journal on Applied Mathematics, 1 (2011), 108. Google Scholar |
[31] |
G. Strang and G. Fix, A Fourier analysis of the finite element variational method,, Construct. Asp. Funct. Anal., 57 (2011), 793.
doi: 10.1007/978-3-642-10984-3_7. |
[32] |
V. A. Zheludev, V. N. Malozemov and A. B. Pevnyi, Filter banks and frames in the discrete periodic case,, in Proceedings of the St. Petersburg Mathematical Society. Vol. XIV, (2009), 1.
|
[33] |
V. A. Zheludev, Local spline approximation on a uniform grid,, Zh. Vychisl. Mat. i Mat. Fiz., 27 (1987), 1296.
|
[34] |
V. A. Zheludev, Periodic splines and the fast Fourier transform,, Comput. Math. & Math Phys., 32 (1992), 149.
|
[35] |
V. A. Zheludev, Interpolatory subdivision schemes with infinite masks originated from splines,, Advances in Comp. Math., 25 (2006), 475.
doi: 10.1007/s10444-004-4149-6. |
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