Daubechies Wavelets
A first look
Ref: Walker (Ch.2)
Jyun-Ming Chen, Spring 2001
Introduction• A family of wavelet transforms disc
overed by Ingrid Daubechies
• Concepts similar to Haar (trend and fluctuation)
• Differs in how scaling functions and wavelets are defined– longer supports
Wavelets are building blocks that can quickly decorrelate data.
Haar Wavelets Revisited
• The elements in the synthesis and analysis matrices are
2
121
2
1,
2
121
2
1
2
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1
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2 Q ,P
Haar Revisited
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SynthesisFilter P3
2
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SynthesisFilter Q3
21V
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In Other Words
482
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322
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4,,1,322
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2 mVVV mmm
4,,1,322
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2 mVVW mmm
How we got the numbers
• Orthonormal; also lead to energy conservation
• Averaging
• Orthogonality
• Differencing
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1221 ,
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1,
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221
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then
if
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fff
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then
if
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1 WV
How we got the numbers (cont)
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1 and 1
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Daubechies Wavelets• How they look like:
– Translated copy
– dilation
Scaling functions Wavelets
1n n n n n
nk
nNk VV :around-Wrap
Daub4 Scaling Functions (n-1 level)
• Obtained from natural basis
• (n-1) level Scaling functions– wrap around at end due to
periodicity
• Each (n-1) level function– Support: 4
– Translation: 2
• Trend: average of 4 values
1n
1n
1n
1n
1n
nN 2
c j
Daub4 Scaling Function (n-2 level)
• Obtained from n-1 level scaling functions
• Each (n-2) scaling function– Support: 10
– Translation: 4
• Trend: average of 10 values
• This extends to lower levels
2n 1n 1n1n 1n
112/ VV :around-Wrap
nk
nNk
1j j j j j
jk
j
k j VV :around-Wrap2
Daub4 Wavelets
• Similar “wrap-around”• Obtained from natural
basis• Support/translation:
– Same as scaling functions
• Extends to lower-levels
1n
nN 2
1n
1n
1n
1n
1j j j j j
jk
j
k j VV :around-Wrap2
Numbers of Scaling Function and Wavelets (Daub4)
Property of Daub4
• If a signal f is (approximately) linear over the support of a Daub4 wavelet, then the corresponding fluctuation value is (approximately) zero.
• True for functions that have a continuous 2nd derivative
xconstxfconstxf )()()(
Property of Daub4 (cont)
MRA
)(d)(c)(c
)(d)(cf112
22
xxx
xx
)(c1 x 1 1 1 1 1 1
1 1 1 1 1 1)(d1 x
nn- Nxxx 2 where)(d)(d)(cf 100
Example (Daub4) 887654321 Nfffffffff
000043212
1 V
0000 43212
2 V
43212
3 0000 V
21432
4 0000 V
000043212
1 W
0000 43212
2 W
43212
3 0000 W
21432
4 0000 W
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224
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12 VVVVV
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11 VVVVW
224
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12 VVVVW
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122
111
01 VVVVV
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01 VVVVW
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)()()()(
)()()()(
WWfWWfWWfWWf
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More on Scaling Functions (Daub4, N=8)
338
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35
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32
31
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24
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P
Or,
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VVVVVVVVVVVV
SynthesisFilter P3
Scaling Function (Daub4, N=16)
338
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36
35
34
33
32
31
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23
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P
Or,
VVVVVVVVVVVV
SynthesisFilter P3
Scaling Functions (Daub4)
24
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31
24
23
22
21
12
11
VVVVVV
SynthesisFilter P2
4
3
2
1
12
11
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11
01
VVVVV
SynthesisFilter P1
More on Wavelets (Daub4)
24
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13
42
31
38
37
36
35
34
33
32
31
24
23
22
21
VVVVVVVVWWWW
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VVVVWW
SynthesisFilter Q2
4
3
2
1
12
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01
VVVVW
SynthesisFilter Q1
SynthesisFilter Q3
Summary
Daub4 (N=32)
j=5 j=4 j=3 j=2In
general
N=2n
support 1 4 10 22 ?
translation 1 2 4 8 ?
jjj PVV 1 jjj QVW 1
Analysis and Synthesis
• There is another set of matrices that are related to the computation of analysis/decomposition coefficient
• In the Daubechies case, they are also the transpose of each other
• Later we’ll show that this is a property unique to orthogonal wavelets
Analysis and Synthesis
332
332
cBd
cAc
1d2d0d
0c2c 1cf
221
221
cBd
cAc
110
110
cBd
cAc
MRA (Daub4)0c1c2c3c4c
5c6c7c8c
)(xf
Energy Compaction (Haar vs. Daub4)
How we got the numbers
• Orthonormal; also lead to energy conservation
• Orthogonality
• Averaging
• Differencing– Constant
– Linear
04231 4 unknowns; 4 eqns
Supplemental
22average
then
if
4321443322112
1
4321
f
fffffVf
fffff
202ncorrelatioconst
then
if
4321443322112
1
4321
fffffWf
fffff
202ncorrelatiolinear
3210
then
3,2,, if
43214321
443322112
1
4321
sk
ffffWf
skfskfskfkf
Conservation of Energy
• Define
• Therefore (Exercise: verify)
Energy Conservation
• By definition:cc c
c c
c c
Orthogonal Wavelets
• By construction • Haar is also orthogonal
• Not all wavelets are orthogonal!– Semiorthogonal, Biorth
ogonal
Other Wavelets (Daub6)
nN 2
1n
1n
1n
1n
Daub6 (cont)
• Constraints
• If a signal f is (approximately) quadratic over the support of a Daub6 wavelet, then the corresponding fluctuation value is (approximately) zero.
DaubJ• Constraints
• If a signal f is (approximately) equal to a polynomial of degree less than J/2 over the support of a DaubJ wavelet, then the corresponding fluctuation value is (approximately) zero.
Comparison (Daub20)
0c1c2c3c4c
5c6c7c8c
)(xf
Supplemental on Daubechies Wavelets
Coiflets
• Designed for maintaining a close match between the trend value and the original signal
• Named after the inventor: R. R. Coifman
Ex: Coif6