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

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Wavelet Packets

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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)

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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)

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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.

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AWT

DWT at whose entry we apply the analytical signal defined as:

xa=x+iH{x}

where H{x} denotes theHilbert transform of x.

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AWPT

AWT AWPT

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Simulation ResultsAWPT

0 5 10 15 20 25 30 35-0.2

0

0.2

0.4

0.6

0.8

1

1.2 input

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Best basis tree used

DWPT AWPT

HWT

, , , , .aHWT f x y f x y x y

, ,

, ,

,

, , , , .

x

y x

a a

HWT f x y DWT f x y

iDWT f x y jDWT f x y

kDWT f x y

f x y x y DWT f x y

yH H

H H

, , ,

, ,

x

y x y

x y x y i x y

j x y k x y

a H

H H H2 2 2 1, and i j k ij ji k

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HWPT

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HWPT’s Shift-Invariance

Best basis EnergDWPT EnergHWPT

1 3.6916 1.2390e+005 1.0469e+006

2 3.94033 5.9904e+005 1.5056e+006

3 3.94033 5.9904e+005 1.5056e+006

4 3.6916 1.2390e+005 1.0469e+006

5 3.6916 1.2390e+005 1.0469e+006

6 3.94033 5.9904e+005 1.5056e+006

7 3.94033 5.9904e+005 1.5056e+006

8 3.6916 1.2390e+005 1.0469e+006

Deg=1- /sd m

Deg2D-DWPT =0.3 DegHWPT =0.81.

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DWPT’s Directional Selectivity

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HWPT’s Directional Selectivity

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Directional Selectivity Experiment

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Simulation Results. Comparison with the 2D-DWPT

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HWPT’s Direction Separation Capacity

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Conclusion

The hyperanalytic wavelet packets have:

• good frequency localization,

• quasi shift-invariance,

• quasi analyticity,

• quasi rotational invariance.