Wavelets

Univ.Prof. Dr.-Ing. Markus RuppLVA 389.141 Fachvertiefung Telekommunikation

(LVA: 389.137 Image and Video Compression)

Last change: January 20, 2020

Outline

• Remember Dyadic Filter Banks

• An approach to Wavelets via Filter Banks

• Orthonormality Properties of Wavelets

2Univ.-Prof. Dr.-Ing. Markus Rupp

3Univ.-Prof. Dr.-Ing. Markus

Rupp

Remember Dyadic Filter Bank

• Thus:

• We find:

)()()()()(

)()()()()(

)()()()(

)()()(

42

3

42

2

2

1

0

zXzHzHzHzY

zXzHzHzHzY

zXzHzHzY

zXzHzY

HHH

LHH

LH

L

=

=

=

=

HH(z)

HL(z)

HH(z2)

HL(z2)

HH(z4)

HL(z4)

Y0(n) Y1(n) Y2(n)

Y3(n)

42 8

8x(n)

Dyadic Filter Bank

• In principle we work only with one filter HH(z)=1-HL(z)

• and then we repeat the filter by stretchedversions: HH(z2), HH(z4), HH(z8),…

4Univ.-Prof. Dr.-Ing. Markus

Rupp

Example 1: Haar

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Example II: Le Gall

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4321

01

12

111

8

1

4

1

4

3

4

1

8

1

)()()(

−−−−

−

++−+=

+=

zzzz

zGzzGzG

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Linear Phase

• Note that both Haar and LeGall Wavelets have symmetric (antisymmetric ) impulse responses h(t).

• This property is equivalent to having linear phase!

• Consider Fourier Transform H(ejW):

• If group delay is constant→linear phase.

Univ.-Prof. Dr.-Ing. Markus Rupp 8

g

jjj eeHeH

−=WW

= WWW

)(

)()( )(

Wavelets

• In all examples the various filter responses are all derived from a single one, that is the coefficients are identical (just stretching or compressing)

• If this filter has few multiplications (just adds/subs), all derived filters will have!

• →cool method to reduce complexity as search room is relatively moderate.

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10 Univ.-Prof. Dr.-Ing. Markus Rupp

Wavelets

• Consider the function:

• We thus have a two-dimensional transformation with modifications in position/location and scale

(Ger: Streckung/Granularität).

( )kttp jj

kj −= −− 22)( 2/

,

ShiftStretch

Wavelets

• For wavelets the function (t) is chosen so that

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( )

ljklkj

mkmjkj

jj

kj

dttptp

dttptp

kttp

−

−

−−

=

=

−=

)()(

)()(

22)(

,,

,,

2/

,

Orthonormality in two directions

12 Univ.-Prof. Dr.-Ing. Markus Rupp

Wavelets

• Note that, if (t) is normalized (||(t)||=1), then we also have ||pjk(t)||=1.

• We select the function (t) in such way that they build for all shifts n an orthonormal basis for a space:

The shifted functions thus build an orthonormal basis for V0.

ZnntV −= ),(span0

13 Univ.-Prof. Dr.-Ing. Markus Rupp

Wavelets

• Example: Consider the unit pulse

• With this basis function all functions f0(t), that are constant for an integer mesh can be described exactly. Continuous functions can be approximated with the precision of integer distance.

• We write:

• integer mesh (Ger: im Raster ganzzahliger Zahlen)

ZnntV

tutut

−=

−−=

),(

)1()()(

0

( ) )()()()()()( 0,0,00 tetftpdttptftfn

nn −==

14 Univ.-Prof. Dr.-Ing. Markus Rupp

Wavelets

• The so obtained coefficients

• Can also be interpreted as piecewise integrated areas over the function f(t).

dttptfc nn )()( ,0

)0(

=

15 Univ.-Prof. Dr.-Ing. Markus Rupp

Wavelets

• Stretching can also be used to define new bases for other spaces.

• If these spaces are nested (Ger:Verschachtelung):

course scale fine scale→

• then we call (t) a scaling function (Ger: Skalierungsfunktion) for a Wavelet.

ZnntV

ZnntV

jj

j −=

−=

−−

−

),2(2span

),2(2span

2/

1

...... 1012 − VVVV

16Univ.-Prof. Dr.-Ing. Markus

Rupp

Wavelets

• Example: Consider the unit impulse

)2(2)(

),2(2

)12()2()2(

,1

1

nttp

ZnntV

tutut

n −=

−=

−−=

−

−

)12()2()( −ttt

1 ½ 1

17 Univ.-Prof. Dr.-Ing. Markus Rupp

Wavelets

• Example: Consider the unit impulse

• With this function we can resemble all functions f(t), that are constant in a half-integer (n/2) mesh. All continuous functions can be approximated by a half-integer mesh.

)2(2;),2(2

)12()2()2(

,11 ntpZnntV

tutut

n −=−=

−−=

−−

( ) )()()()()()( 1,1,11 tetftpdttptftfn

nn −−−− −==

18 Univ.-Prof. Dr.-Ing. Markus Rupp

Wavelets

• The function f-1(t) thus is an even finer approximation of f0(t) in V0. Since V0 is a subset of V-1we have:

• With a suitable basis y0,n(t) from W0 withW0 U V0 =V-1

+=

+=

=

−

−

−

−

n

nn

n

nn

n

nn

n

nn

tdtpc

tetpc

tpctf

)()(

)()(

)()(

,0

)0(

,0

)0(

1,0,0

)0(

,1

)1(

1

y

19Univ.-Prof. Dr.-Ing. Markus

Rupp

Wavelets

• In other words, the set Wj complements the set Vj in such a way that:

Wj U Vj =Vj-1

With Vj-1 the next finer approximation can be built.

• Hereby, Wj is the orthogonal complement of Vj:

1in −

⊥= jjj VVW

20 Univ.-Prof. Dr.-Ing. Markus Rupp

Wavelets

• These functions yj,n(t) are called Wavelets.

• Thus, we can decompose any function at an arbitrary scaling step into two components yj,n and {pj,n}.

• Very roughly, one can be considered a high pass, the other a low pass.

• By finer scaling the function can be approximated better and better.

• The required number of coefficients is strongly dependent on the Wavelet- or the corresponding scaling function.

• Note that for coding as well as for transmission only the coefficients are required. The fewer the better…

21Univ.-Prof. Dr.-Ing. Markus

Rupp

Wavelets

• Wavelets have also the scaling property:

• As well as orthonormal properties:

• causing that they are energy preserving!

nkljnlkj

jj

nj

dttt

ntgt

−−

−−

=

−=

yy

y

)(),(

)2(2)(

,,

2/

,

22Univ.-Prof. Dr.-Ing. Markus

Rupp

Wavelets

• Example: Haar Wavelets

)()(2

1)(

)()(2

1)(

12,2,,1

12,2,,1

tptpt

tptptp

nmnmnm

nmnmnm

++

++

−=

+=

y

)12()2()()2(2)()()( 0,00,10,0 −−=== − tttttpttp y

23 Univ.-Prof. Dr.-Ing. Markus Rupp

Example

f-1(t)

f0(t)

e0,-1(t) )(

)(

0,0

0,0

t

tp

y

Modern Wavelets

• In the 80ies wavelets were just viewed as an alternative description of subband filtering.

• Ingrid Daubechies introduced new families of wavelets, some of them not having the orthogonality property but a so-called bi-orthogonal property.

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25 Univ.-Prof. Dr.-Ing. Markus Rupp

Vector Spaces

• Definition : If there are two bases,

that span the same space with the additional property:

then these bases are said to be dual or biorthogonal(biorthonormal for ki,j=1).

},...,,{};,...,,{2121 mm

qqqUpppT ==

jijijikqp −= ,,

26 Univ.-Prof. Dr.-Ing. Markus Rupp

Vector Spaces

• Example: let:

These pairs build a dual basis in R2. Consider

then:

;1,0;1,1;1,1;0,12121

==−==TTTTqqpp

],[ bafT

=

212211

212211

)(,,

)(,,

qbaqaqpfqpff

pbpbapqfpqff

+−+=+=

++=+=

Modern Wavelets

• Daubechies and LeGall wavelets share this biorthogonal property which makes them of linear phase.

• Unfortunately, they lose the orthogonality and thus the energy preserving property (not unitary).

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Specific Basis Functions

• Find the right basis for your problem:

• Wavelet or DCT

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Curvelet Basis

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Ridgelet Basis

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Summary

• Orthogonal wavelets can classically be interpreted as specific solutions of dyadic filter trees

• They offer further advantages in complexity when selecting the proper wavelet families.

• Biorthogonal Wavelets are the method of choice for the JPEG 2000 standard.

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