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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2 Q ,P

Haar Revisited

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SynthesisFilter P3

2

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SynthesisFilter Q3

21V

21W

In Other Words

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813

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VVVVVVVVVVVV

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4,,1,322

3121

2 mVVV mmm

4,,1,322

3121

2 mVVW mmm

How we got the numbers

• Orthonormal; also lead to energy conservation

• Averaging

• Orthogonality

• Differencing

122

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021

1221 ,

2

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1,

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221

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then

if

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fff

fffVf

fff

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then

if

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ff

fffWf

fff

022112

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1 WV

How we got the numbers (cont)

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:onConservatiEnergy

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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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11 VVVVV

224

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11 VVVVW

224

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)()()()(

)()()()(

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More on Scaling Functions (Daub4, N=8)

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Or,

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SynthesisFilter P3

Scaling Function (Daub4, N=16)

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Or,

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SynthesisFilter P3

Scaling Functions (Daub4)

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VVVVVV

SynthesisFilter P2

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VVVVV

SynthesisFilter P1

More on Wavelets (Daub4)

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SynthesisFilter Q2

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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