Daubechies Wavelets A first look Ref: Walker (Ch.2) Jyun-Ming Chen, Spring 2001.
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Transcript of Daubechies Wavelets A first look Ref: Walker (Ch.2) Jyun-Ming Chen, Spring 2001.
![Page 1: Daubechies Wavelets A first look Ref: Walker (Ch.2) Jyun-Ming Chen, Spring 2001.](https://reader033.fdocuments.us/reader033/viewer/2022052312/56649c775503460f9492bf9f/html5/thumbnails/1.jpg)
Daubechies Wavelets
A first look
Ref: Walker (Ch.2)
Jyun-Ming Chen, Spring 2001
![Page 2: Daubechies Wavelets A first look Ref: Walker (Ch.2) Jyun-Ming Chen, Spring 2001.](https://reader033.fdocuments.us/reader033/viewer/2022052312/56649c775503460f9492bf9f/html5/thumbnails/2.jpg)
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.
![Page 3: Daubechies Wavelets A first look Ref: Walker (Ch.2) Jyun-Ming Chen, Spring 2001.](https://reader033.fdocuments.us/reader033/viewer/2022052312/56649c775503460f9492bf9f/html5/thumbnails/3.jpg)
Haar Wavelets Revisited
• The elements in the synthesis and analysis matrices are
2
121
2
1,
2
121
2
1
2
1
2
2
1
2
1
2 Q ,P
![Page 4: Daubechies Wavelets A first look Ref: Walker (Ch.2) Jyun-Ming Chen, Spring 2001.](https://reader033.fdocuments.us/reader033/viewer/2022052312/56649c775503460f9492bf9f/html5/thumbnails/4.jpg)
Haar Revisited
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
1
1
1
1
1
1
1
1
SynthesisFilter P3
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1
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1
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1
2
1
2
1
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1
2
1
1
1
1
1
1
1
1
1
SynthesisFilter Q3
21V
21W
![Page 5: Daubechies Wavelets A first look Ref: Walker (Ch.2) Jyun-Ming Chen, Spring 2001.](https://reader033.fdocuments.us/reader033/viewer/2022052312/56649c775503460f9492bf9f/html5/thumbnails/5.jpg)
In Other Words
482
1
2
1
2
1
2
1
813
83
73
63
53
43
33
23
1412
42
32
22
1
VVVVVVVVVVVV
322
311
21 VVV
4,,1,322
3121
2 mVVV mmm
4,,1,322
3121
2 mVVW mmm
![Page 6: Daubechies Wavelets A first look Ref: Walker (Ch.2) Jyun-Ming Chen, Spring 2001.](https://reader033.fdocuments.us/reader033/viewer/2022052312/56649c775503460f9492bf9f/html5/thumbnails/6.jpg)
How we got the numbers
• Orthonormal; also lead to energy conservation
• Averaging
• Orthogonality
• Differencing
122
21
021
1221 ,
2
121
2
1,
2
121
122
21
221
22
22
2
then
if
21
2122112
1
21
fff
fffVf
fff
022
then
if
21
2122112
1
21
ff
fffWf
fff
022112
12
1 WV
![Page 7: Daubechies Wavelets A first look Ref: Walker (Ch.2) Jyun-Ming Chen, Spring 2001.](https://reader033.fdocuments.us/reader033/viewer/2022052312/56649c775503460f9492bf9f/html5/thumbnails/7.jpg)
How we got the numbers (cont)
8
7
6
5
4
3
2
1
21
21
21
21
21
21
21
21
4
4
3
3
2
2
1
1
f
f
f
f
f
f
f
f
d
c
d
c
d
c
d
c
fDy OR,
fDDffDDfyyddcc )()( TTTTT24
21
24
21
IDDffyy TTT re therefo,
:onConservatiEnergy
8
21
21
21
21
21
21
21
21
22
1
22
1
212
1
22
11
I
1 and 1
Hence22
21
22
21
![Page 8: Daubechies Wavelets A first look Ref: Walker (Ch.2) Jyun-Ming Chen, Spring 2001.](https://reader033.fdocuments.us/reader033/viewer/2022052312/56649c775503460f9492bf9f/html5/thumbnails/8.jpg)
Daubechies Wavelets• How they look like:
– Translated copy
– dilation
Scaling functions Wavelets
![Page 9: Daubechies Wavelets A first look Ref: Walker (Ch.2) Jyun-Ming Chen, Spring 2001.](https://reader033.fdocuments.us/reader033/viewer/2022052312/56649c775503460f9492bf9f/html5/thumbnails/9.jpg)
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
![Page 10: Daubechies Wavelets A first look Ref: Walker (Ch.2) Jyun-Ming Chen, Spring 2001.](https://reader033.fdocuments.us/reader033/viewer/2022052312/56649c775503460f9492bf9f/html5/thumbnails/10.jpg)
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
![Page 11: Daubechies Wavelets A first look Ref: Walker (Ch.2) Jyun-Ming Chen, Spring 2001.](https://reader033.fdocuments.us/reader033/viewer/2022052312/56649c775503460f9492bf9f/html5/thumbnails/11.jpg)
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
![Page 12: Daubechies Wavelets A first look Ref: Walker (Ch.2) Jyun-Ming Chen, Spring 2001.](https://reader033.fdocuments.us/reader033/viewer/2022052312/56649c775503460f9492bf9f/html5/thumbnails/12.jpg)
Numbers of Scaling Function and Wavelets (Daub4)
![Page 13: Daubechies Wavelets A first look Ref: Walker (Ch.2) Jyun-Ming Chen, Spring 2001.](https://reader033.fdocuments.us/reader033/viewer/2022052312/56649c775503460f9492bf9f/html5/thumbnails/13.jpg)
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 )()()(
![Page 14: Daubechies Wavelets A first look Ref: Walker (Ch.2) Jyun-Ming Chen, Spring 2001.](https://reader033.fdocuments.us/reader033/viewer/2022052312/56649c775503460f9492bf9f/html5/thumbnails/14.jpg)
Property of Daub4 (cont)
![Page 15: Daubechies Wavelets A first look Ref: Walker (Ch.2) Jyun-Ming Chen, Spring 2001.](https://reader033.fdocuments.us/reader033/viewer/2022052312/56649c775503460f9492bf9f/html5/thumbnails/15.jpg)
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
![Page 16: Daubechies Wavelets A first look Ref: Walker (Ch.2) Jyun-Ming Chen, Spring 2001.](https://reader033.fdocuments.us/reader033/viewer/2022052312/56649c775503460f9492bf9f/html5/thumbnails/16.jpg)
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
244
233
222
211
11 VVVVV
224
213
242
231
12 VVVVV
244
233
222
211
11 VVVVW
224
213
242
231
12 VVVVW
124
113
122
111
01 VVVVV
124
113
122
111
01 VVVVW
24
24
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23
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22
21
21
12
12
11
11
01
01
01
01
)()()()(
)()()()(
WWfWWfWWfWWf
WWfWWfWWfVVff
![Page 17: Daubechies Wavelets A first look Ref: Walker (Ch.2) Jyun-Ming Chen, Spring 2001.](https://reader033.fdocuments.us/reader033/viewer/2022052312/56649c775503460f9492bf9f/html5/thumbnails/17.jpg)
More on Scaling Functions (Daub4, N=8)
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P
Or,
1
1
1
1
1
1
1
1
VVVVVVVVVVVV
SynthesisFilter P3
![Page 18: Daubechies Wavelets A first look Ref: Walker (Ch.2) Jyun-Ming Chen, Spring 2001.](https://reader033.fdocuments.us/reader033/viewer/2022052312/56649c775503460f9492bf9f/html5/thumbnails/18.jpg)
Scaling Function (Daub4, N=16)
338
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4234
4133
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312343
22144234
21134133
123224
113123
2214
2113
12
11
P
Or,
VVVVVVVVVVVV
SynthesisFilter P3
![Page 19: Daubechies Wavelets A first look Ref: Walker (Ch.2) Jyun-Ming Chen, Spring 2001.](https://reader033.fdocuments.us/reader033/viewer/2022052312/56649c775503460f9492bf9f/html5/thumbnails/19.jpg)
Scaling Functions (Daub4)
24
13
42
31
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11
VVVVVV
SynthesisFilter P2
4
3
2
1
12
11
12
11
01
VVVVV
SynthesisFilter P1
![Page 20: Daubechies Wavelets A first look Ref: Walker (Ch.2) Jyun-Ming Chen, Spring 2001.](https://reader033.fdocuments.us/reader033/viewer/2022052312/56649c775503460f9492bf9f/html5/thumbnails/20.jpg)
More on Wavelets (Daub4)
24
13
24
13
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42
31
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31
24
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VVVVVVVVWWWW
24
13
42
31
24
23
22
21
12
11
VVVVWW
SynthesisFilter Q2
4
3
2
1
12
11
12
11
01
VVVVW
SynthesisFilter Q1
SynthesisFilter Q3
![Page 21: Daubechies Wavelets A first look Ref: Walker (Ch.2) Jyun-Ming Chen, Spring 2001.](https://reader033.fdocuments.us/reader033/viewer/2022052312/56649c775503460f9492bf9f/html5/thumbnails/21.jpg)
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
![Page 22: Daubechies Wavelets A first look Ref: Walker (Ch.2) Jyun-Ming Chen, Spring 2001.](https://reader033.fdocuments.us/reader033/viewer/2022052312/56649c775503460f9492bf9f/html5/thumbnails/22.jpg)
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
![Page 23: Daubechies Wavelets A first look Ref: Walker (Ch.2) Jyun-Ming Chen, Spring 2001.](https://reader033.fdocuments.us/reader033/viewer/2022052312/56649c775503460f9492bf9f/html5/thumbnails/23.jpg)
Analysis and Synthesis
332
332
cBd
cAc
1d2d0d
0c2c 1cf
221
221
cBd
cAc
110
110
cBd
cAc
![Page 24: Daubechies Wavelets A first look Ref: Walker (Ch.2) Jyun-Ming Chen, Spring 2001.](https://reader033.fdocuments.us/reader033/viewer/2022052312/56649c775503460f9492bf9f/html5/thumbnails/24.jpg)
MRA (Daub4)0c1c2c3c4c
5c6c7c8c
)(xf
![Page 25: Daubechies Wavelets A first look Ref: Walker (Ch.2) Jyun-Ming Chen, Spring 2001.](https://reader033.fdocuments.us/reader033/viewer/2022052312/56649c775503460f9492bf9f/html5/thumbnails/25.jpg)
Energy Compaction (Haar vs. Daub4)
![Page 26: Daubechies Wavelets A first look Ref: Walker (Ch.2) Jyun-Ming Chen, Spring 2001.](https://reader033.fdocuments.us/reader033/viewer/2022052312/56649c775503460f9492bf9f/html5/thumbnails/26.jpg)
How we got the numbers
• Orthonormal; also lead to energy conservation
• Orthogonality
• Averaging
• Differencing– Constant
– Linear
04231 4 unknowns; 4 eqns
![Page 27: Daubechies Wavelets A first look Ref: Walker (Ch.2) Jyun-Ming Chen, Spring 2001.](https://reader033.fdocuments.us/reader033/viewer/2022052312/56649c775503460f9492bf9f/html5/thumbnails/27.jpg)
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
![Page 28: Daubechies Wavelets A first look Ref: Walker (Ch.2) Jyun-Ming Chen, Spring 2001.](https://reader033.fdocuments.us/reader033/viewer/2022052312/56649c775503460f9492bf9f/html5/thumbnails/28.jpg)
Conservation of Energy
• Define
• Therefore (Exercise: verify)
![Page 29: Daubechies Wavelets A first look Ref: Walker (Ch.2) Jyun-Ming Chen, Spring 2001.](https://reader033.fdocuments.us/reader033/viewer/2022052312/56649c775503460f9492bf9f/html5/thumbnails/29.jpg)
Energy Conservation
• By definition:cc c
c c
c c
![Page 30: Daubechies Wavelets A first look Ref: Walker (Ch.2) Jyun-Ming Chen, Spring 2001.](https://reader033.fdocuments.us/reader033/viewer/2022052312/56649c775503460f9492bf9f/html5/thumbnails/30.jpg)
Orthogonal Wavelets
• By construction • Haar is also orthogonal
• Not all wavelets are orthogonal!– Semiorthogonal, Biorth
ogonal
![Page 31: Daubechies Wavelets A first look Ref: Walker (Ch.2) Jyun-Ming Chen, Spring 2001.](https://reader033.fdocuments.us/reader033/viewer/2022052312/56649c775503460f9492bf9f/html5/thumbnails/31.jpg)
Other Wavelets (Daub6)
nN 2
1n
1n
1n
1n
![Page 32: Daubechies Wavelets A first look Ref: Walker (Ch.2) Jyun-Ming Chen, Spring 2001.](https://reader033.fdocuments.us/reader033/viewer/2022052312/56649c775503460f9492bf9f/html5/thumbnails/32.jpg)
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.
![Page 33: Daubechies Wavelets A first look Ref: Walker (Ch.2) Jyun-Ming Chen, Spring 2001.](https://reader033.fdocuments.us/reader033/viewer/2022052312/56649c775503460f9492bf9f/html5/thumbnails/33.jpg)
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.
![Page 34: Daubechies Wavelets A first look Ref: Walker (Ch.2) Jyun-Ming Chen, Spring 2001.](https://reader033.fdocuments.us/reader033/viewer/2022052312/56649c775503460f9492bf9f/html5/thumbnails/34.jpg)
Comparison (Daub20)
0c1c2c3c4c
5c6c7c8c
)(xf
![Page 35: Daubechies Wavelets A first look Ref: Walker (Ch.2) Jyun-Ming Chen, Spring 2001.](https://reader033.fdocuments.us/reader033/viewer/2022052312/56649c775503460f9492bf9f/html5/thumbnails/35.jpg)
Supplemental on Daubechies Wavelets
![Page 36: Daubechies Wavelets A first look Ref: Walker (Ch.2) Jyun-Ming Chen, Spring 2001.](https://reader033.fdocuments.us/reader033/viewer/2022052312/56649c775503460f9492bf9f/html5/thumbnails/36.jpg)
![Page 37: Daubechies Wavelets A first look Ref: Walker (Ch.2) Jyun-Ming Chen, Spring 2001.](https://reader033.fdocuments.us/reader033/viewer/2022052312/56649c775503460f9492bf9f/html5/thumbnails/37.jpg)
Coiflets
• Designed for maintaining a close match between the trend value and the original signal
• Named after the inventor: R. R. Coifman
![Page 38: Daubechies Wavelets A first look Ref: Walker (Ch.2) Jyun-Ming Chen, Spring 2001.](https://reader033.fdocuments.us/reader033/viewer/2022052312/56649c775503460f9492bf9f/html5/thumbnails/38.jpg)
Ex: Coif6