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