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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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Published in: | IEEE transactions on signal processing 2003-07, Vol.51 (7), p.1825-1837 |
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container_title | IEEE transactions on signal processing |
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creator | Fernandes, F.C.A. van Spaendonck, R.L.C. Burrus, C.S. |
description | 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. |
doi_str_mv | 10.1109/TSP.2003.812841 |
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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. 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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.</description><subject>Applied sciences</subject><subject>Continuous wavelet transforms</subject><subject>Discrete transforms</subject><subject>Discrete wavelet transforms</subject><subject>Exact sciences and technology</subject><subject>Filter bank</subject><subject>Flexibility</subject><subject>Function space</subject><subject>Hilbert space</subject><subject>Image coding</subject><subject>Image processing</subject><subject>Information, signal and communications theory</subject><subject>Instruments</subject><subject>Mapping</subject><subject>Redundancy</subject><subject>Signal processing</subject><subject>Telecommunications and information theory</subject><subject>Transforms</subject><subject>Wavelet transforms</subject><issn>1053-587X</issn><issn>1941-0476</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2003</creationdate><recordtype>article</recordtype><recordid>eNp9kMtLw0AQhxdRsFbPHrwEQT0lnX0meyzFFxQUrOBt2Ww2kJpH3U1b_e_dkkLBg5eZgd83A_MhdIkhwRjkZPH2mhAAmmSYZAwfoRGWDMfAUnEcZuA05ln6cYrOvF8CYMakGKFkGrV2G5VON3bbuc-o7FxkumZV2-9oqze2tn3UO936EDT-HJ2Uuvb2Yt_H6P3hfjF7iucvj8-z6Tw2LBN9LHQB2kCaakmK3GBdYlKQPOVS5JLnBOcFSG2BCQFGc05kaZi2lKeCSyxLOkZ3w92V677W1veqqbyxda1b2629kpBKQajggbz9lwwuJCeMBvD6D7js1q4NX6gs-MpoqAGaDJBxnffOlmrlqka7H4VB7TSroFntNKtBc9i42Z_V3ug6iGxN5Q9rLJOUkx13NXCVtfYQExBCEvoLPpKD5Q</recordid><startdate>20030701</startdate><enddate>20030701</enddate><creator>Fernandes, F.C.A.</creator><creator>van Spaendonck, R.L.C.</creator><creator>Burrus, C.S.</creator><general>IEEE</general><general>Institute of Electrical and Electronics Engineers</general><general>The Institute of Electrical and Electronics Engineers, Inc. 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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. 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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 |
title | A new framework for complex wavelet transforms |
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