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Hyperanalytic Wavelet Packets

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  1. Hyperanalytic Wavelet Packets Ioana Firoiu, Dorina Isar, Jean-Marc Boucher, Alexandru Isar WISP 2009, Budapest, Hungary

  2. Introduction Wavelet techniques based on the Discrete Wavelet Transform (DWT) • Advantages • Sparsity of coefficients • Disadvantages • Shift-sensitivity(input signal shift → unpredictable change in the output coefficients) • Poor directional selectivity WISP 2009, Budapest, Hungary 2

  3. Wavelet Packets 2D-DWT and 2D-DWPT implementations. WISP 2009, Budapest, Hungary 3

  4. Shift-Invariant Wavelet Packets Transforms • One-Dimensional DWPT (1D - DWPT) • Shift Invariant Wavelet Packets Transform (SIWPT) • Non-decimated DWPT (NDWPT) • Dual-Tree Complex Wavelet Packets Transform (DT-CWPT) • Analytical Wavelet Packets Transform (AWPT) WISP 2009, Budapest, Hungary 4

  5. Two-Dimensional DWT (2D - DWT) • 2D-SIWPT • 2D-NDWPT • Poor directional selectivity • 2D-DT-CWPT • Reduced flexibility in choosing the mother wavelets • Hyperanalytical Wavelet Packets Transform (HWPT) WISP 2009, Budapest, Hungary 5

  6. Advantages Quasi shift-invariant Good directional selectivity Disadvantages Low flexibility in choosing the mother wavelets Filters from the 2nd branch can be only approximated DT-CWPT Ilker Bayram and Ivan W. Selesnick, “On the Dual-Tree Complex Wavelet Packet and M-Band Transforms”, IEEE Trans. Signal Processing, 56(6) : 2298-2310, June 2008. WISP 2009, Budapest, Hungary 6

  7. AWT DWT at whose entry we apply the analytical signal defined as: xa=x+iH{x} where H{x} denotes the Hilbert transform of x. WISP 2009, Budapest, Hungary 7

  8. AWPT AWT AWPT WISP 2009, Budapest, Hungary 8

  9. input 1.2 1 0.8 0.6 0.4 0.2 0 -0.2 0 5 10 15 20 25 30 35 Simulation ResultsAWPT Best basis tree used DWPT AWPT WISP 2009, Budapest, Hungary 9

  10. HWT WISP 2009, Budapest, Hungary 10

  11. HWPT WISP 2009, Budapest, Hungary 11

  12. HWPT’s Shift-Invariance Deg2D-DWPT =0.3DegHWPT =0.81. WISP 2009, Budapest, Hungary 12

  13. DWPT’s Directional Selectivity WISP 2009, Budapest, Hungary 13

  14. HWPT’s Directional Selectivity WISP 2009, Budapest, Hungary 14

  15. Directional Selectivity Experiment WISP 2009, Budapest, Hungary 15

  16. Simulation Results. Comparison with the 2D-DWPT WISP 2009, Budapest, Hungary 16

  17. HWPT’s Direction Separation Capacity WISP 2009, Budapest, Hungary 17

  18. Conclusion The hyperanalytic wavelet packets have: • good frequency localization, • quasi shift-invariance, • quasi analyticity, • quasi rotational invariance.