A new framework for complex wavelet transforms

Although the discrete wavelet transform (DWT) is a powerful tool for signal and image processing, it has three serious disadvantages: shift sensitivity, poor directionality, and lack of phase information. To overcome these disadvantages, we introduce two-stage mapping-based complex wavelet transform...

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Bibliographic Details
Published in:IEEE transactions on signal processing 2003-07, Vol.51 (7), p.1825-1837
Main Authors: Fernandes, F.C.A., van Spaendonck, R.L.C., Burrus, C.S.
Format: Article
Language:English
Subjects:
Applied sciences
Continuous wavelet transforms
Discrete transforms
Discrete wavelet transforms
Exact sciences and technology
Filter bank
Flexibility
Function space
Hilbert space
Image coding
Image processing
Information, signal and communications theory
Instruments
Mapping
Redundancy
Signal processing
Telecommunications and information theory
Transforms
Wavelet transforms
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Summary:Although the discrete wavelet transform (DWT) is a powerful tool for signal and image processing, it has three serious disadvantages: shift sensitivity, poor directionality, and lack of phase information. To overcome these disadvantages, we introduce two-stage mapping-based complex wavelet transforms that consist of a mapping onto a complex function space followed by a DWT of the complex mapping. Unlike other popular transforms that also mitigate DWT shortcomings, the decoupled implementation of our transforms has two important advantages. First, the controllable redundancy of the mapping stage offers a balance between degree of shift sensitivity and transform redundancy. This allows us to create a directional, non-redundant, complex wavelet transform with potential benefits for image coding systems. To the best of our knowledge, no other complex wavelet transform is simultaneously directional and non-redundant. The second advantage of our approach is the flexibility to use any DWT in the transform implementation. As an example, we can exploit this flexibility to create the complex double-density DWT (CDDWT): a shift-insensitive, directional, complex wavelet transform with a low redundancy of (3/sup m/-1/2/sup m/-1) in m dimensions. To the best of our knowledge, no other transform achieves all these properties at a lower redundancy.
ISSN:1053-587X
1941-0476
DOI:10.1109/TSP.2003.812841