Hyperanalytic Wavelet Packets Ioana Firoiu, Dorina Isar, Jean-Marc Boucher, Alexandru Isar WISP 2009, Budapest, Hungary
Jan 04, 2016
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) : 2298-2310, June 2008.
WISP 2009, Budapest, Hungary 6
AWT
DWT at whose entry we apply the analytical signal defined as:
xa=x+iH{x}
where H{x} denotes theHilbert transform of x.
WISP 2009, Budapest, Hungary 7
AWPT
AWT AWPT
WISP 2009, Budapest, Hungary 8
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
WISP 2009, Budapest, Hungary 9
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
WISP 2009, Budapest, Hungary 10
HWPT
WISP 2009, Budapest, Hungary 11
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.
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.