Haar Wavelet AnalysisHaar Wavelet Analysis

吳育德陽明大學放射醫學科學研究所台北榮總整合性腦功能實驗室

A First Course in Wavelets with Fourier AnalysisAlbert Boggess Francis J. Narcowich

Prentice-Hall, Inc., 2001

OutlinesOutlines

Why Wavelet Haar Wavelets

The Haar Scaling Function Basic Properties of the Haar Scaling Function The Haar Wavelet

Haar Decomposition and Reconstruction Algorithms

Decomposition Reconstruction Filters and Diagrams

Summary

4.1 Why Wavelet4.1 Why Wavelet

Wavelets were first applied in geophysics to analyze data from seismic surveys.

Seismic survey

geophonesgeophones

seismic traceseismic trace

Sesimic traceSesimic trace

Direct wave (along the surface)Direct wave (along the surface)Subsequent waves (rock layers below ground)Subsequent waves (rock layers below ground)

Fourier Transform (FT) is not a good tool – gives no direct information about when an oscillation occurred.

Short-time FT : equal time interval, high- frequency bursts occur are hard to detect.

Wavelets can keep track of time and frequency information. They can be used to “zoom in” on the short bursts, or to “zoom out” to detect long, slow oscillations

frequency

frequency + time (equal time intervals)

frequency + time

4.2 Haar Wavelets4.2 Haar Wavelets 4.2.1 The Haar Scaling Function 4.2.1 The Haar Scaling Function

Wavelet functionsWavelet functions Scaling function Scaling function ΦΦ (father wavelet) (father wavelet) Wavelet Wavelet ΨΨ (mother wavelet) (mother wavelet) These two functions generate a family of functions These two functions generate a family of functions

that can be used to break up or reconstruct a signal that can be used to break up or reconstruct a signal The Haar Scaling FunctionThe Haar Scaling Function

TranslationTranslation DilationDilation

Using Haar blocks to approximate a signalUsing Haar blocks to approximate a signal

High-frequency noise shows up as tall, thin blocks.High-frequency noise shows up as tall, thin blocks. Needs an algorithm that eliminates the noise and not Needs an algorithm that eliminates the noise and not

distribute the rest of the signal. distribute the rest of the signal. Disadvantages of Harr wavelet: discontinuous and Disadvantages of Harr wavelet: discontinuous and

does not approximate continuous signals very well.does not approximate continuous signals very well.

Figure 2Figure 2

Daubechies 8Daubechies 8

Dubieties 3Dubieties 3

Daubechies 4Daubechies 4

Chap 6Chap 6

4.2.2 Basic Properties of the Haar Scaling Function4.2.2 Basic Properties of the Haar Scaling Function

The Haar Scaling function is defined asThe Haar Scaling function is defined as

elsewhere ,0

1x0 if ,1)(x

ΦΦ(x-k)(x-k) : same graph but translated by to the right (if : same graph but translated by to the right (if k>0) by k unitsk>0) by k units

Let VLet V00 be the space of all functions of the form be the space of all functions of the form

Zk

kk Rakxa )(

VV00 consists of all piecewise constant functions whose consists of all piecewise constant functions whose

discontinuities are contained in the set of integers discontinuities are contained in the set of integers VV00 has compact support. has compact support.

Typical element in V0Typical element in V0

Figure 5Figure 5

Figure 6Figure 6

)3(2)2(3)1(3)(2)( xxxxxf

has discontinuitieshas discontinuities at x=0,1,3, and 4at x=0,1,3, and 4

Let VLet V11 be the space of piecewise constant functions of be the space of piecewise constant functions of

finite support with discontinuities at the half integers finite support with discontinuities at the half integers

Zk

kk Rakxa )2(

1)32()22(2)12(2)2(4)( Vxxxxxf

has discontinuities at x=0,1/2,3/2, and 2has discontinuities at x=0,1/2,3/2, and 2

)2( x

Suppose Suppose jj is any nonnegative integer. The space of step is any nonnegative integer. The space of step functions at level functions at level jj, denoted by , denoted by VVjj , , , is defined to be the , is defined to be the

space spanned by the setspace spanned by the set

,...}2/3,2/2,2/1,0,2/1{..., jjjj

}),22(),12(),2(),12(,{ xxxx jjjj

over the real numbers.over the real numbers.

VVj j is the space of piecewise constant functions of finite is the space of piecewise constant functions of finite

support whose discontinuities are contained in the setsupport whose discontinuities are contained in the set

...... 1110 jjj VVVVV means no information is lost as the resolution gets finer. VVj contains all relevant information up to a resolution scale order 2-j

A function f(x) belongs to V0 iff f(2jx) belongs to Vj

similar is converse theof proof The

.V ofmember a is thatmeanswhich

2 of nscombinatiolinear a is Therefore,

of nscombinatiolinear a is

then ,V tobelongs function a If

: Proof

j

o

x) f(2

},Zk),kx({x)f(2

}Zk),kx({f(x)

f

j

jj

A function f(x) belongs to Vj iff f(2-jx) belongs to V0

.V ofmember a is that meanswhich

},),22({ of nscombinatiolinear a is Therefore,

}),2({ of nscombinatiolinear a is

then ,V tobelongs function a If

: Proof

o

j

x)f(2

Zkkxx)f(2

Zkkxf(x)

f

j

jjj

j

How to decompose a signal into its VHow to decompose a signal into its V jj-components-components

When j is large, the graph of When j is large, the graph of Φ(2Φ(2jj x) x) is similar to one of is similar to one of the spikes of a signal that we may wish to filter out.the spikes of a signal that we may wish to filter out.

One way is to construct an orthonormal basis for VOne way is to construct an orthonormal basis for V j j using the Lusing the L2 2 inner productinner product

0

-L

1

-

22

L

Vfor basis lorthonormaan is }),({set theso

j ,0)()()( ),(

supportsdisjoint have )( and )( ,j if

1dx1dx)()(

2

2

Zkjx

kdxkxjxkxjx

kxjxk

kxkxk

k

Theorem:Theorem:

j2

-

j

2j

2/1))2(( : Note

Vfor basis lorthonormaan

is });2({2 functions ofset The

dxx

Zkjx

j

j

4.2.4 The Haar Wavelet4.2.4 The Haar Wavelet We want to isolate the spikes that belong to Vj, but that are not

members of Vj-1

The way is to decompose VVjj as an orthonormal sum of Vj-1 and its complement.

Start with V1, assume the orthonormal complement of Vo is generated by translates of some functions ΨΨ, , we needwe need::

k integers allfor 0k)-(xψ(x)

toequivalent is This .V toorthogonal is ψ

Ra of choices somefor )-(2xaψ(x)

as expressed becan ψ so and V of members a is ψ

0

1

l

ll l

Harr wavelet Harr wavelet

)Vψ : (2nd

overlapnot do (x) and ψ(x) of supports since ,0k)dx-(xψ(x),0

01/2-1/21dx-1dxk)dx-(xψ(x),0

)Vψ :(1st 1)-(2x -(2x)1/2))-(2(x -(2x)ψ(x)

0

-

1

2/1

2/1

0-

1

k

k

)12()2()( xxx

001

100

k

0

2

01

221

23

011

WV V

Vin V of complement orthogonal theis W

nonzero). are a ofnumber finite aonly (assume , ),(

form theof functions all of space thebe let W So

)( form theof isit

ifonly and if V toorthogonal is Vin function a s,other wordIn

)())122()22((

case, In this ....

ifonly and if )),( of nscombinatiolinear (

V toorthogonal is )2(function Any

Rakxa

kxa

kxakxkxaf

,-a, a-aa

Zllx

Vkxaf

kZk

k

k k

Zkk

Zkk

01

kk

Theorem 4.8 (extend to VTheorem 4.8 (extend to Vjj))

jj1j1jjj

j

WV V and Vin V of complement orthogonal theis W

R 2 form theof functions of space thebe Let W

kj

kk a)kx(a

0dxf(x)g(x)fg,

showmust We.V tobelongs that suppose and

W tobelongs )2( that Suppose

1.Part

. W tobelongmust V toorthogonal is that Vin function Any 2.

Vin function every toorthogonal is in Wfunction Each 1.

: facts twoshowmust we theorem,thisestablish To : Proof

-L

j

j

jj1j

jj

2

f

kxag j

Zk k

case. V as derived becan t requiremen second The

V any toorthogonal is Therefore

2

) letting(by 2

)V toorthogonal is (becasue 0

So V tobelongs function the,V tobelongs Since

1

j

j

j

0

0.j

.fg

.yd)yf()y(g

x2yyd)yf()ky(2a

dxx)f(2)kx(a

x)f(2 f(x)

-

-j

-

j

Zk k

-

j

Zk k

j

Decomposing VDecomposing Vjj

hs.other widt of spikes

of nscombinatiolinear as drepresente becannot that

1/2 width of of spikes"" therepresents y,Intuitivel

V tobelongs and 10 , W tobelongs each where

f

sum a asuniquely decomposed becan Vin each So

V

...

VVV

1

00

0021

j

0202j1j

2j2j1j1jjj

ll

ll

jj

fw

fjlw

fwww

f

WWW

WWW

Theorem:Theorem:

j000

0

2

21002

2

W tobelongs and V tobelongs where

asuniquely written becan (R)L each ,particularIn

.WWWV(R)L

sumdirect orthogonal infinitean as decomposed becan (R)L space The

jj

j wfwff

f

4.3 Haar Decomposition and Reconstruction 4.3 Haar Decomposition and Reconstruction AlgorithmsAlgorithms

noise. out thisfilter tozero toequal scommponent

set these noise, represents 6j with any ,20.012

noise represents 0.01 than less width of spikes that Suppose

: Example

7-6- jw

w

components its into Decompose

large)suitably (for function step aby eApproximat

problem filtering Noise

110 ll,jj

j

jj

Wwwff

f

jVff

ImplementationImplementation

.for )2/( to

leadswhich ,...,,0,1/22...,-1,-1/at x signal theSample jj

Zllfa jl

Zl

jlj )lx(a(x)f 2

Step 1 : Approximate the original signal f by a step function of the form

signal. theofdomain on the depends range:

signal. theof features essential thecapture enough to small:

l

j

(4.5) 1

(4.4) (

.for components Wits into )(2 Decompose

2 Stepj

ψ(x))/2(x)()(2x

(x))/2 x)(ψ(2x)

jllx l

(4.7) 1

(4.6) (

:Rx allfor holds relations following The

x

11

11

1

x))/2ψ(2x)(2()x(2

x))/2(2 x)2(ψx)(2

x2

jjj

jjj

j

Example 4.11Example 4.11

1 0 02, decompose f into W ,W ,V components.

f(x) 2 (4 ) 2 (4 1) (4 2) (4 3)

4 (2 2

4 1 2 (2

4 2 4( 1/ 2 ) (2( 1/ 2 ) 2( 1/ 2 )

4 3 4( 1/ 2 1) (2

j

x x x x

( x) (ψ x) ( x))/2

( x ) ( ( x) ψ x))/2

( x ) ( x ) (ψ x ) ( x )) /2

( x ) ( x ) (

1

1

1 0 0

( 1/ 2 2( 1/ 2 )

f(x) [ (2 2 ] [ 2 (2 ]

[ (2 1 2 1 ] / 2 [ 2 1 (2 1 ] / 2

(2 1 2 2

W -components: (2 1

V -components: 2 2

V V ,W

f(x) (2 1 (

x ) ψ( x )) /2

ψ x) ( x) ( x) ψ x)

ψ x ) ( x ) ( x ) ψ x )

ψ x ) ( x)

ψ x )

( x)

ψ x ) ψ x) (x)

General decomposition schemeGeneral decomposition scheme

11

11221122

1112

112

11

11

1

1111

122

22

22

222 222

(4.11) 22(122

(4.10) 22(22

2222

(4.7) 1 (4.6) (

(4.9) 12222

: termsodd andeven into 2 sum theDivide

: 1 Step

jj

Zk

jkkjkk

j

Zk

jk

j

Zk

jkj

jjj

jjj

jjj

jjjjj

Zk

jk

Zk

jkj

k

jkj

fw

)kx()aa

()kx(ψ)aa

(

/))kx(ψ)kx((a/))kx()kx(ψ(a(x)f

/))kx(ψk)x2()kx(

/))kx(k)x2ψ()kx(

)kx(kx

x))/2ψ(2x)(2()(2xx))/2(2 x)2(ψx)(2

)kx(a)kx(a(x)f

)kx(a(x)f

WWj-1j-1-component -component VVj-1j-1-component -component

Theorem 4.12 (Haar Decomposition)Theorem 4.12 (Haar Decomposition)

2

2

2

2

where

as decomposed becan Then

V 2

002122111

12211221

111

1

111

1

11

j

fwww...fwwfwf

aaa

aabwith

V)kx(af

W)kx(ψbw

,fwf

f

)kx(a(x)f

jjjjjjjj

jk

jkj

k

jk

jkj

k

Zkj

jjkj

Zkj

jjkj

jjj

j

Zk

jjkj

Example 4.13Example 4.13

12

0

888

888

001

8

22

1202

.components V,W, Winto f decompose

1,x0 813, Figure

k

k

k)()/kf()x(f

k),/kf(a

,j

VV88-component-component VV77-component-component VV66-component-component VV44-component-component

WW77-component-component

4.3.2 Reconstruction4.3.2 Reconstruction

jj

jl

Zl

jjl

'jk

'jk

'j

'j

/)l(x/l

af

)lx(af(x)

b

.b

W

W

'

212 interval the

over height offunction step a is done, is thisOnce

2 rebuild to

algorithmtion reconstruc a need modified,been have

larger keeponly out, thrown becan

small are that components- The :n compressio Data

out. thrown becan sfrequencie unwanted

the toingcorrespond components- The : filtering Noise

? then whatj,j'0

for components WandV into f decomposed Having j0

General reconstruction schemeGeneral reconstruction scheme

jl

l

jjl

kl

llkl

Zkk

llj

a

)lx(a)x(f

j-1l 0W)kx(ψbw

V)kx(a)x(f

whereWww)x(w)x(ff(x)

constants ofn computatio for the algorithm a find

and 2 rewrite To

: Goal

for 2

and

(x)

00

0

100

General reconstruction schemeGeneral reconstruction scheme

odd is 1 if

even is if ˆ

(4.16) )2(ˆ)( so

))122()22(( )()(

have we,by replaced with 4.12 Using

(4.15) 1222

(4.14) 1222

(4.13) 122

(4.12) 122

0

01

10

0000

1

1

2kla

2klaawhere

lxaxf

kxakxakxaxf

x-kx

)x(x)(x)ψ(

)x(x)(x)(

)x(x)(ψ(x)

)x(x)((x)

k

kl

Zkl

Zkkk

Zkk

jjj

jjj

odd is 1 if

even is if ˆˆ

))( ( )2()()(

odd is 1 if

even is if ˆ

(4.17) )2(ˆ)(

as written becan )( ,

00

00111

11

00

0

01

10

00

2klba

2klbabaawhere

xflxaxwxf

2klb

2klbbwhere

lxbxw

kxψawSimilarly

kk

kklll

Zll

k

kl

Zll

k k

manner. recursive ain on so and

ts,coefficien- thedetermine tscoefficien,

ts.coefficien- thedetermine tscoefficien,

odd is 1 if,

even is if,

) ( )2()()()(

sum the to)2( Add

211

100

11

112

222

10

11

lll

lll

kk

kkl

Zllo

k k

aba

aba

2klba

2klbaawhere

flxaxwxwxf

kxψbw

Theorem 4.14 (Haar Reconstruction)Theorem 4.14 (Haar Reconstruction)

odd is 1 if,

even is if,

algorithm by the j, untilon so and 2,then

1,for y recursivel determined are the

2

for 2

and

11

11

00

0

100

2klba

2klbaa

'j'j

'jawhere

V)lx(a)x(fThen

.j'j 0W)kx(ψb)x(w

V)kx(a)x(f

withwwffSuppose

'jk

'jk

'jk

'jk'j

l

'jl

Zlj

jjl

Zk'j

'j'jk'j

Zkk

j

Example 4.15Example 4.15

signal. t thereconstructhen

n),compressio (80% zero toequal 80%smallest set the

size,by theorderingAfter

.70for

)2()( and )(

with

120),2/(

1,x0 ,819, Figure

'

'''

'000

72100

888

jk

Zkj

jjkj0

k

b

j

WkxbxwV(x)axf

wwwwff

kkfa

j

80% compression80% compression 90% compression90% compressionsample signalsample signal

4.3.3 Filters and Diagrams4.3.3 Filters and Diagrams

1kkkk1kkkk x2

1x

2

1x)(L(x)x

2

1-x

2

1x)(hH(x)

xx and L(x)hH(x):

h

h

thusare sequences resulting The

. then ,}{x If

)02

1

2

10( ),0

2

1

2

10(

: and sequences thearewhich

responses, impulse their via

L and H operators)on (convoluti filters Discrete

2k

k=-1,0k=-1,0

Decomposition algorithm

1

0

12211221

][

2

2n

jjkjk

jk

jkj

k

jk

jkj

k

zyz*y

aaa

aab

D.by denoted ng,downsampli called is

sequence ain tscoefficien odd thediscarding ofoperation This

and

then ,subscriptseven only keep weIf

22222222 .x2

1x

2

1x)(L(x)x

2

1-x

2

1x)(hH(x) 1kkkk1kkkk

4.12) Theorem 2

2

(

.)( and )( jj'0for components

and into f sets.decompocoefficien wavelet and sacling

1-j level get the to tscoefficien scaling j-level from Go

12211221

11

'0

jk

jkj

k

jk

jkj

k

kjj

kkjj

k

j

jk

aaa

aab

aDLaaDHb

WV

a

downsampling downsampling operatoroperator

odd is 12 y

even is 2 y)

~(

odd is 12 x-

even is 2 x)

~(

then0, all are entries odd hein which t sequences arey and x If

.)~

( and )~

( have we},{x sequence aFor

)01 10(~

),01- 10(

: ~ and

~ sequences thearewhich

responses, impulse their via

L~

and H~

operators)on (convoluti filters Discrete

2k

2k

2k

2k

k

kl

kly

kl

klxh

xxx-xxxh

h

h

ll

k-1kkk-1kk

Reconstruction

k=0,1 k=0,1

odd is 1 if,

even is if, 11

11

2klba

2klbaa

'jk

'jk

'jk

'jk'j

l

11

1-j2

1-j1

1-j0

1-j1-

11

11

1'1'

1'1''

and sequences

theof upsampling called are and sequences The

y.for similarly and )0 0 0 0 0(x

is,that ; and set weso

choose, toours are ' and ' the

0, are ' and ' that assumed have eAlthough w

odd is 1 if,

even is if, 4.14. Theorem

in given formulatin reconstruc for thepattern almost the is This

)~

()~

(

us gives then ~ and

~ sequences two theAdding

jj

jk2k

jk2k

2k2k

2k2k

jk

jk

jk

jkj

l

2k2k

2k 2k ll

ab

yx

bbb b

aybx

sysx

sysx

2klba

2klbaa

1 is odd2k l-xy

2k is even lyxyxh

yxh

1j1jj

1j1j

UbHUaLa

UayUbx

~~

and so operator, upsampling thedenote to Uuse We

upsampling upsampling operatoroperator

SummarySummary

waveletandfunction scalingHaar thebe and Let

, signalGiven

f(t)y

Zk

JJkJ

JJ

JJk

J

k)xφ(2a(x)f

f

)2k-1 (or 12k1, 0t0

kkfa

2J j

ion toapproximat level-highest

step. thisskip : signal discrete

ex.

signal. theof

duration by the determined interval finite a : )2/(Let

rate.Nyquist the

n larger tha is and level topchoose : signal continuous

Sample Step1.

reached. is level 0the

until continue willstated, otherwised Unlesscontinue. totscoefficien

few twoare or there purpose somefor ry satisfactoeither is reaached

level theuntil on, so and , becomesThen . from determined

are and and becomesThen 1. Step from valuessignal

sampled theare which by determined are and ,When

filters. pass low and -high theare L and H

(4.19) (4.18)

algorithm by they recursivel the

from determined are tscoefficien The

2 2

where Decompose

ionDecomposit 2 Step

1

22

11

11

11

111

111

0011

j

J-2j a

baJ-1j

,abaJ j

where

)a(DHb)a(DLa

a

a,b

)lx(af)lx(ψbw

fwwwwf

Jk

Jk

Jk

Jk

Jk

Jk

kjj

lkjj

l

j

jl

jl

Zl

jjlj

Zl

jjlj

jjJJ

zero. set to be would

resholdcertain th a below are that the:n compressio data c.

filtered. be tois of values

particular toingcorrespond signal a ofsegment certain aonly b.

lue.certain va a above for zero

set to be would theall : sfrequenciehigh allout Filter a.

. tscoefficien

wavelet themodifyingby filtered be nowcan signal The

(4.21) )())2((

(4.20) )(

form in the signal theion,decompositAfter .Processing 3 Step

1

0

011

1

00

jk

jk

jk

J

j Zkk

Zk

jjk

J

jjJ

b

k

j

b

b

kxalxψb

fwxf

. at time signal processed theof

valueeapproximat therepresents level), top(the When

signal. theofduration timeby the determined is of range The

forth. so and and thefrom computed are the2,When

. and thefrom computed are the,When

.for (4.22) ~~

algorithmtion reconstruc by the edaccomplish is This

).2(

asit t reconstruc and

(4.21) )())2(()(

signal, modified theTake

tionReconstruc 4. Step

112

001

1

0

011

J

Jk

kkk

kkk

1j1jj

Zk

JJkJ

J

j Zkk

Zk

jjkJ

J

k/2x

aJj

k

baaj

baa1j

1,...,JjUbHUaLa

kxaf

kxalxψbxf

f