Construction of Hilbert Transform Pairs of Wavelet Bases and Gabor-Like Transforms

被引:52
作者
Chaudhury, Kunal Narayan [1 ]
Unser, Michael [1 ]
机构
[1] Ecole Polytech Fed Lausanne, Biomed Imaging Grp, CH-1015 Lausanne, VD, Switzerland
基金
瑞士国家科学基金会;
关键词
Analytic signal; biorthogonal wavelet basis; B-spline multiresolution; directional Hilbert transform; dual-tree complex wavelet transform; Gabor function; Hilbert transform; time-frequency localization; SIGNALS;
D O I
10.1109/TSP.2009.2020767
中图分类号
TM [电工技术]; TN [电子技术、通信技术];
学科分类号
0808 ; 0809 ;
摘要
We propose a novel method for constructing Hilbert transform (HT) pairs of wavelet bases based on a fundamental approximation-theoretic characterization of scaling functions-the B-spline factorization theorem. In particular, starting from well-localized scaling functions, we construct HT pairs of biorthogonal wavelet bases of L-2(R) by relating the corresponding wavelet filters via a discrete form of the continuous HT filter. As a concrete application of this methodology, we identify HT pairs of spline wavelets of a specific flavor, which are then combined to realize a family of complex wavelets that resemble the optimally-localized Gabor function for sufficiently large orders. Analytic wavelets, derived from the complexification of HT wavelet pairs, exhibit a one-sided spectrum. Based on the tensor-product of such analytic wavelets, and, in effect, by appropriately combining four separable biorthogonal wavelet bases of L-2(R-2), we then discuss a methodology for constructing 2-D directional-selective complex wavelets. In particular, analogous to the HT correspondence between the components of the 1-D counterpart, we relate the real and imaginary components of these complex wavelets using a multidimensional extension of the HT-the directional HT. Next, we construct a family of complex spline wavelets that resemble the directional Gabor functions proposed by Daugman. Finally, we present an efficient fast Fourier transform (FFT)-based filterbank algorithm for implementing the associated complex wavelet transform.
引用
收藏
页码:3411 / 3425
页数:15
相关论文
共 40 条
[11]  
de Rivaz P, 2001, 2001 INTERNATIONAL CONFERENCE ON IMAGE PROCESSING, VOL II, PROCEEDINGS, P273, DOI 10.1109/ICIP.2001.958477
[12]   The monogenic signal [J].
Felsberg, M ;
Sommer, G .
IEEE TRANSACTIONS ON SIGNAL PROCESSING, 2001, 49 (12) :3136-3144
[13]   A new framework for complex wavelet transforms [J].
Fernandes, FCA ;
van Spaendonck, RLC ;
Burrus, CS .
IEEE TRANSACTIONS ON SIGNAL PROCESSING, 2003, 51 (07) :1825-1837
[14]  
Gabor D., 1946, J. Inst. Electr. Eng.-Part III: Radio Commun. Eng., V93, P429, DOI [DOI 10.1049/JI-3-2.1946.0074, 10.1049/ji-3-2.1946.0074]
[15]  
GRANLUND GH, 1995, SIGNAL PROCESSING CO, pCH4
[16]   MULTIDIMENSIONAL COMPLEX SIGNALS WITH SINGLE-ORTHANT SPECTRA [J].
HAHN, SL .
PROCEEDINGS OF THE IEEE, 1992, 80 (08) :1287-1300
[17]   Image texture description using complex wavelet transform [J].
Hatipoglu, S ;
Mitra, SK ;
Kingsbury, N .
2000 INTERNATIONAL CONFERENCE ON IMAGE PROCESSING, VOL II, PROCEEDINGS, 2000, :530-533
[18]   Shift invariant properties of the dual-tree complex wavelet transform [J].
Kingsbury, N .
ICASSP '99: 1999 IEEE INTERNATIONAL CONFERENCE ON ACOUSTICS, SPEECH, AND SIGNAL PROCESSING, PROCEEDINGS VOLS I-VI, 1999, :1221-1224
[19]   A dual-tree complex wavelet transform with improved orthogonality and symmetry properties [J].
Kingsbury, N .
2000 INTERNATIONAL CONFERENCE ON IMAGE PROCESSING, VOL II, PROCEEDINGS, 2000, :375-378
[20]   Complex wavelets for shift invariant analysis and filtering of signals [J].
Kingsbury, N .
APPLIED AND COMPUTATIONAL HARMONIC ANALYSIS, 2001, 10 (03) :234-253