A Globally Optimal Bilinear Programming Approach to the Design of Approximate Hilbert Pairs of Orthonormal Wavelet Bases
It is understood that the Hilbert transform pairs of orthonormal wavelet bases can only be realized approximately by the scaling filters of conjugate quadrature filter (CQF) banks. In this paper, the approximate FIR realization of the Hilbert transform pairs is formulated as an optimization problem...
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| Vydané v: | IEEE transactions on signal processing Ročník 58; číslo 1; s. 233 - 241 |
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| Hlavní autori: | , |
| Médium: | Journal Article |
| Jazyk: | English |
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New York, NY
IEEE
01.01.2010
Institute of Electrical and Electronics Engineers |
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| ISSN: | 1053-587X, 1941-0476 |
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| Abstract | It is understood that the Hilbert transform pairs of orthonormal wavelet bases can only be realized approximately by the scaling filters of conjugate quadrature filter (CQF) banks. In this paper, the approximate FIR realization of the Hilbert transform pairs is formulated as an optimization problem in the sense of the lp ( p =1, 2, or infinite) norm minimization on the approximate error of the magnitude and phase conditions of the scaling filters. The orthogonality and regularity conditions of the CQF bank pairs are taken as the constraints of such an optimization problem. Whereafter the branch and bound technique is employed to obtain the globally optimal solution of the resulting bilinear program optimization problem. Since the orthogonality and regularity conditions are explicitly taken as the constraints of our optimization problem, the attained solution is an approximate Hilbert transform pair satisfying these conditions exactly. Some orthogonal wavelet bases designed herein demonstrate that our design scheme is superior to those that have been reported in the literature. Moreover, the designed orthogonal wavelet bases show that minimizing the l 1 norm of the approximate error should be advocated for obtaining better approximated Hilbert pairs. |
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| AbstractList | It is understood that the Hilbert transform pairs of orthonormal wavelet bases can only be realized approximately by the scaling filters of conjugate quadrature filter (CQF) banks. In this paper, the approximate FIR realization of the Hilbert transform pairs is formulated as an optimization problem in the sense of the lp ( p =1, 2, or infinite) norm minimization on the approximate error of the magnitude and phase conditions of the scaling filters. The orthogonality and regularity conditions of the CQF bank pairs are taken as the constraints of such an optimization problem. Whereafter the branch and bound technique is employed to obtain the globally optimal solution of the resulting bilinear program optimization problem. Since the orthogonality and regularity conditions are explicitly taken as the constraints of our optimization problem, the attained solution is an approximate Hilbert transform pair satisfying these conditions exactly. Some orthogonal wavelet bases designed herein demonstrate that our design scheme is superior to those that have been reported in the literature. Moreover, the designed orthogonal wavelet bases show that minimizing the l 1 norm of the approximate error should be advocated for obtaining better approximated Hilbert pairs. |
| Author | Jian Qiu Zhang Jiang Wang |
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| CitedBy_id | crossref_primary_10_1016_j_sigpro_2013_08_016 crossref_primary_10_1016_j_sigpro_2013_11_040 crossref_primary_10_1016_j_sigpro_2011_08_004 crossref_primary_10_1109_LSP_2010_2080674 crossref_primary_10_1109_TSP_2020_3001906 crossref_primary_10_1109_LSP_2012_2196692 |
| Cites_doi | 10.1109/TPAMI.2008.79 10.1109/TSP.2007.913156 10.1007/BF01580367 10.1109/97.923042 10.1098/rsta.1999.0447 10.1002/cpa.3160410705 10.1109/MSP.2005.1550194 10.1109/TSP.2006.872542 10.1109/LSP.2006.874453 10.1007/BF01580892 10.1017/CBO9780511804441 10.1109/78.995070 10.1109/18.119725 10.1287/opre.14.4.699 10.1109/LSP.2007.913609 10.1109/TCSI.2005.856048 10.1109/CVPR.2008.4587846 10.1109/TSP.2008.924134 10.1109/TSP.2008.917408 10.1109/34.799911 10.1109/LSP.2007.910318 |
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| Keywords | dual-tree complex wavelet transform (DTWT) conjugate quadrature filter (CQF) Branch and bound method L1 approximation Programming Bilinear programming Orthogonality Hilbert transformation Constrained optimization Hilbert transform Wavelet transformation Optimal solution Wavelet base Signal processing Filter bank orthonormal wavelet bases |
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| References | ref13 ref12 ref14 ref11 yang (ref22) 2008 ref2 ref1 ref17 ref16 ref19 ref18 mallat (ref15) 1999 shi (ref8) 2008; 56 lu (ref10) 1999 ref24 ref25 ref20 ref21 ref7 ref9 ref4 ref3 ref6 ref5 tibshirani (ref23) 1996; 58 |
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| SubjectTerms | Applied sciences Bilinear programming conjugate quadrature filter (CQF) Constraint optimization Delay Detection, estimation, filtering, equalization, prediction Discrete transforms Discrete wavelet transforms dual-tree complex wavelet transform (DTWT) Exact sciences and technology Filter bank Finite impulse response filter Frequency Hilbert transform Information, signal and communications theory Miscellaneous orthonormal wavelet bases Quadratic programming Signal and communications theory Signal processing Signal, noise Telecommunications and information theory Wavelet transforms |
| Title | A Globally Optimal Bilinear Programming Approach to the Design of Approximate Hilbert Pairs of Orthonormal Wavelet Bases |
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