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

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

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

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

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

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) : , June WISP 2009, Budapest, Hungary 6

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

AWPT AWT AWPT WISP 2009, Budapest, Hungary 8

Simulation Results AWPT input WISP 2009, Budapest, Hungary 9 Best basis tree used DWPT AWPT

HWT WISP 2009, Budapest, Hungary 10

HWPT WISP 2009, Budapest, Hungary 11

HWPT’s Shift-Invariance Best basisEnergDWPTEnergHWPT e e e e e e e e e e e e e e e e+006 Deg 2D-DWPT =0.3 Deg HWPT =0.81. WISP 2009, Budapest, Hungary 12

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

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

Directional Selectivity Experiment WISP 2009, Budapest, Hungary 15

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

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

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