Haar Wavelets

A first look

Ref: Walker (ch1)

Jyun-Ming Chen, Spring 2001

Introduction

• Simplest; hand calculation suffice

• A prototype for studying more sophisticated wavelets

• Related to Haar transform, a mathematical operation

Haar Transform

• Assume discrete signal (analog function occurring at discrete instants)

• Assume equally spaced samples (number of samples 2n)

• Decompose the signal into two sub-signals of half its length– Running average (trend)

– Running difference (fluctuation)

2c 2d

• Running difference

• Denoted by:

– Meaning of superscript explained later

Haar transform, 1-level

• Running average

• Multiplication by is needed to ensure energy conservation (see later)

2

mc

Example

2c

2d

Inverse Transform

2c 2d

c c c c

Small Fluctuation Feature

• Magnitudes of the fluctuation subsignal (d) are often significantly smaller than those of the original signal

• Logical: samples are from continuous analog signal with very short time increment

• Has application to signal compression

2c 2d

Energy Concerns

• Energy of signals

• The 1-level Haar transform conserves energy2c 2d

2c

2d

Proof of Energy Conservation

c

c

c c

Haar Transform, multi-level2c

1c

2c

1d

1c 2d1d

1d 2d0d0c

1d2d0d

0c2c 1cf

• Compare with 1-level

• Can be seen more clearly by cumulative energy profile

Compaction of Energy

1c

0c

Cumulative Energy Profile

• Definition

Algebraic Operations

• Addition & subtraction• Constant multiple• Scalar product

Haar Wavelets

• 1-level Haar wavelets– “wavelet”: plus/minus

wavy nature

– Translated copy of mother wavelet

– support of wavelet =2 • The interval where func

tion is nonzero

2

2

2

2

2

Property 1. If a signal f is (approximately) constant over the support of a Haar wavelet, then the fluctuation value is (approximately) zero.

2

2

2

2c

Haar Scaling Functions

• 1-level scaling functions

• Graph: translated copy of father scaling function

• Support = 2

Haar Wavelets (cont)

• 2-level Haar scaling functions

• support = 4

• 2-level Haar wavelets• support = 4

1

1

1

1 1 11

1

1

1

1 1 11c

Multiresolution Analysis (MRA)

Natural basis:

Therefore:

3

3

3

3 3 3

MRAc c c c c c

)(d)(cf 22 xx cc c c c c

c c c

)(c2 x

2 2 2)(c2 x

2 2 2

)(d2 x

)(d2 x

2 2 2 2 2 2

2 2 2 2 2 2)(d2 x)(c2 x

),,,(c 2/212

Nccc ),,,(d 2/21

2Nddd

Note: the coefficient vectors

MRA

01

01

0 )( VVfc If do it all the way through,representing the average of all data

)(d)(c)(c

)(d)(cf112

22

xxx

xx

nn- Nxxx 2 where)(d)(d)(cf 100

)(c1 x 1 1 1 1 1 1

1 1 1 1 1 1)(d1 x

Example

1d 2c 2d1c

)(c1 x

)(d1 x

)(c2 x

)(d2 x

2c2c 2d

2 2 2 2

2 2 2 2

)(c2 x

)(d2 x

Example (cont)

24

23

22

21

12

11

01

01 02222622214

)5,5,6,8,12,10,6,4(

WWWWWWWV

f

)2

1,

2

1,

2

1,

2

1,0,0,0,0(

)0,0,0,0,2

1,

2

1,

2

1,

2

1(

)22

1,

22

1,

22

1,

22

1,

22

1,

22

1,

22

1,

22

1(

)22

1,

22

1,

22

1,

22

1,

22

1,

22

1,

22

1,

22

1(

12

11

01

01

W

W

W

V

)2

1,

2

1,0,0,0,0,0,0(

)0,0,2

1,

2

1,0,0,0,0(

)0,0,0,0,2

1,

2

1,0,0(

)0,0,0,0,0,0,2

1,

2

1(

24

23

22

21

W

W

W

W

1d 2c 2d0c 0d

Decomposition coefficients obtained by inner product with basis function

Haar MRA

0c1c2c3c4c

5c6c7c8c

)(xf

More on Scaling Functions (Haar)

• They are in fact related

• Pj is called the synthesis filter (more later)

2

2

2

1

1

1

case 8for )22

1,

22

1,

22

1,

22

1,

22

1,

22

1,

22

1,

22

1(V0

8/ NN

jjj PVV 1

Ex: Haar Scaling Functions

1

1

1

1

1

1

1

1

2

1

1

1

1

1

1

1

1

1

21

21

21

21

21

21

21

21

SynthesisFilter P3

Ex: Haar Scaling Functions

1

1

1

1

2

1

21

21

21

21

21

21

21

21

21

21

21

21

21

21

21

21

SynthesisFilter P2

1

1

2

1

21

21

21

21

21

21

21

21

221

221

221

221

221

221

221

221

SynthesisFilter P1

More on Wavelets (Haar)• They are in fact relate

d

• Qj is called the synthesis filter (more later)

1

1

1

2

2

2

case 8for )22

1,

22

1,

22

1,

22

1,

22

1,

22

1,

22

1,

22

1(W0

8/

NN

jjj QVW 1

Ex: Haar Wavelets

1

1

1

1

1

1

1

1

2

1

1

1

1

1

1

1

1

1

21

21

21

21

21

21

21

21

SynthesisFilter Q3

1

1

2

1

21

21

21

21

21

21

21

21

221

221

221

221

221

221

221

221

Ex: Haar Wavelets

SynthesisFilter Q2

1

1

1

1

2

1

21

21

21

21

21

21

21

21

21

21

21

21

21

21

21

21

SynthesisFilter Q1

Analysis Filters

• There is another set of matrices that are related to the computation of analysis/decomposition coefficient

• In the Haar case, they are the transpose of each other

• Later we’ll show that this is a property unique to orthogonal wavelets

Analysis/Decomposition (Haar)

5

5

6

8

12

10

6

4

11

11

11

11

2

1

25

27

211

25

25

27

211

25

11

11

2

1

12

16

12

1611

2

1214

5

5

6

8

12

10

6

4

11

11

11

11

2

1

0

2

2

2

25

27

211

25

11

11

2

1

2

6

12

1611

2

122

AnalysisFilter Aj

AnalysisFilter Bj

A3

A2

A1B3

B2

B1

Synthesis Filters

• On the other hand, synthesis filters have to do with reconstructing the signal from MRA results

Synthesis/Reconstruction (Haar)

SynthesisFilter Pj

SynthesisFilter Qj

0

2

2

2

1

1

1

1

1

1

1

1

2

1

55

27

211

25

1

1

1

1

1

1

1

1

2

1

5

5

6

8

12

10

6

4 Q3P3

2

6

1

1

1

1

2

1

12

16

1

1

1

1

2

1

55

27

211

25

Q2P2

221

1

2

1214

1

1

2

1

12

16

Q1P1

Conclusion/Exercise

Haar (N=8) j=3 j=2 j=1 j=0In general

N=2n

support 1 2 4 8 2n-j

translation 1 2 4 8 2n-j