Image analysis using a dual-tree M-band wavelet transform

We propose a two-dimensional generalization to the M-band case of the dual-tree decomposition structure (initially proposed by Kingsbury and further investigated by Selesnick) based on a Hilbert pair of wavelets. We particularly address: 1) the construction of the dual basis and 2) the resulting directional analysis. We also revisit the necessary pre-processing stage in the M-band case. While several reconstructions are possible because of the redundancy of the representation, we propose a new optimal signal reconstruction technique, which minimizes potential estimation errors. The effectiveness of the proposed M-band decomposition is demonstrated via denoising comparisons on several image types (natural, texture, seismics), with various M-band wavelets and thresholding strategies. Significant improvements in terms of both overall noise reduction and direction preservation are observed.

[1]  Ramesh A. Gopinath,et al.  The phaselet transform-an integral redundancy nearly shift-invariant wavelet transform , 2003, IEEE Trans. Signal Process..

[2]  Ivan W. Selesnick,et al.  Video denoising using 2D and 3D dual-tree complex wavelet transforms , 2003, SPIE Optics + Photonics.

[3]  Stéphane Mallat,et al.  A Wavelet Tour of Signal Processing, 2nd Edition , 1999 .

[4]  Xiao-Ping Zhang,et al.  Orthogonal complex filter banks and wavelets: some properties and design , 1999, IEEE Trans. Signal Process..

[5]  B. Silverman,et al.  Incorporating Information on Neighboring Coefficients Into Wavelet Estimation , 2001 .

[6]  I. Selesnick,et al.  Bivariate shrinkage with local variance estimation , 2002, IEEE Signal Processing Letters.

[7]  I. Johnstone,et al.  Adapting to Unknown Smoothness via Wavelet Shrinkage , 1995 .

[8]  Stéphane Mallat,et al.  On denoising and best signal representation , 1999, IEEE Trans. Inf. Theory.

[9]  Ying-Jui Chen,et al.  M-channel lifting factorization of perfect reconstruction filter banks and reversible M-band wavelet transforms , 2003, IEEE Trans. Circuits Syst. II Express Briefs.

[10]  Caroline Chaux,et al.  Hilbert pairs of M-band orthonormal wavelet bases , 2004, 2004 12th European Signal Processing Conference.

[11]  P. Abry,et al.  Multiresolution transient detection , 1994, Proceedings of IEEE-SP International Symposium on Time- Frequency and Time-Scale Analysis.

[12]  Truong Q. Nguyen,et al.  Lattice structure for regular paraunitary linear-phase filterbanks and M-band orthogonal symmetric wavelets , 2001, IEEE Trans. Signal Process..

[13]  Sofia C. Olhede,et al.  'Analytic' wavelet thresholding , 2004 .

[14]  Josiane Zerubia,et al.  Satellite Image Deblurring Using Complex Wavelet Packets , 2003, International Journal of Computer Vision.

[15]  Thierry Blu,et al.  The fractional spline wavelet transform: definition end implementation , 2000, 2000 IEEE International Conference on Acoustics, Speech, and Signal Processing. Proceedings (Cat. No.00CH37100).

[16]  Olivier Rioul,et al.  Fast algorithms for discrete and continuous wavelet transforms , 1992, IEEE Trans. Inf. Theory.

[17]  C. Sidney Burrus,et al.  Complex wavelet transforms with allpass filters , 2003, Signal Process..

[18]  Truong Q. Nguyen,et al.  Linear-phase perfect reconstruction filter bank: lattice structure, design, and application in image coding , 2000, IEEE Trans. Signal Process..

[19]  Richard G. Baraniuk,et al.  Non-redundant, linear-phase, semi-orthogonal, directional complex wavelets [image/video processing applications] , 2004, 2004 IEEE International Conference on Acoustics, Speech, and Signal Processing.

[20]  Edward H. Adelson,et al.  Shiftable multiscale transforms , 1992, IEEE Trans. Inf. Theory.

[21]  Henrique S. Malvar,et al.  Signal processing with lapped transforms , 1992 .

[22]  Hüseyin Özkaramanli,et al.  On the phase condition and its solution for Hilbert transform pairs of wavelets bases , 2003, 2003 IEEE International Conference on Acoustics, Speech, and Signal Processing, 2003. Proceedings. (ICASSP '03)..

[23]  Caroline Chaux,et al.  2D dual-tree M-band wavelet decomposition , 2005, Proceedings. (ICASSP '05). IEEE International Conference on Acoustics, Speech, and Signal Processing, 2005..

[24]  Hüseyin Özkaramanli,et al.  On the phase condition and its solution for Hilbert transform pairs of wavelet bases , 2003, IEEE Trans. Signal Process..

[25]  I. Johnstone,et al.  Ideal spatial adaptation by wavelet shrinkage , 1994 .

[26]  William T. Freeman,et al.  Presented at: 2nd Annual IEEE International Conference on Image , 1995 .

[27]  Frédéric Truchetet,et al.  Discrete wavelet transform implementation in Fourier domain for multidimensional signal , 2002, J. Electronic Imaging.

[28]  Raghuveer M. Rao,et al.  Solution to the orthogonal M-channel bandlimited wavelet construction proposition , 2003, 2003 IEEE International Conference on Acoustics, Speech, and Signal Processing, 2003. Proceedings. (ICASSP '03)..

[29]  Thierry Blu,et al.  Orthogonal Hilbert transform filter banks and wavelets , 2003, 2003 IEEE International Conference on Acoustics, Speech, and Signal Processing, 2003. Proceedings. (ICASSP '03)..

[30]  B. Silverman,et al.  The Stationary Wavelet Transform and some Statistical Applications , 1995 .

[31]  N. Kingsbury.  Complex Wavelets for Shift Invariant Analysis and Filtering of Signals , 2001 .

[32]  Ivan W. Selesnick,et al.  The design of approximate Hilbert transform pairs of wavelet bases , 2002, IEEE Trans. Signal Process..

[33]  Peter N. Heller,et al.  Theory of regular M-band wavelet bases , 1993, IEEE Trans. Signal Process..

[34]  Hervé Carfantan,et al.  Time-invariant orthonormal wavelet representations , 1996, IEEE Trans. Signal Process..

[35]  I. Selesnick.  The Double-Density Dual-Tree Discrete Wavelet Transform , 2001 .

[36]  Oktay Alkin,et al.  Design of efficient M-band coders with linear-phase and perfect-reconstruction properties , 1995, IEEE Trans. Signal Process..

[37]  M. Palaniswami,et al.  Design of Approximate Hilbert Transform Pair of Wavelets with exact Symmetry. , 2004 .

[38]  Ingrid Daubechies,et al.  Ten Lectures on Wavelets , 1992 .

[39]  S. Mallat.  A wavelet tour of signal processing , 1998 .

[40]  Richard G. Baraniuk,et al.  Directional hypercomplex wavelets for multidimensional signal analysis and processing , 2004, 2004 IEEE International Conference on Acoustics, Speech, and Signal Processing.

[41]  Ivan W. Selesnick,et al.  The double-density dual-tree DWT , 2004, IEEE Transactions on Signal Processing.

[42]  I. Selesnick.  Hilbert transform pairs of wavelet bases , 2001, IEEE Signal Processing Letters.