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

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. DT-CWPT Advantages –Quasi shift- invariant –Good directional selectivity Disadvantages –Low flexibility in choosing the mother wavelets –Filters from the 2nd branch can be only approximated 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: x a =x+i H {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. Simulation Results AWPT 05101520253035 -0.2 0 0. 2 0.4 0.6 0.8 1 1.2 input WISP 2009, Budapest, Hungary 9 Best basis tree used DWPT AWPT

10. HWT WISP 2009, Budapest, Hungary 10

11. HWPT WISP 2009, Budapest, Hungary 11

12. HWPT’s Shift-Invariance Best basisEnergDWPTEnergHWPT 13.69161.2390e+0051.0469e+006 23.940335.9904e+0051.5056e+006 33.940335.9904e+0051.5056e+006 43.69161.2390e+0051.0469e+006 53.69161.2390e+0051.0469e+006 63.940335.9904e+0051.5056e+006 73.940335.9904e+0051.5056e+006 83.69161.2390e+0051.0469e+006 Deg 2D-DWPT =0.3 Deg HWPT =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.