Review Questions

Jyun-Ming Chen

Spring 2001

Wavelet Transform

• What is wavelet?

• How is wavelet transform different from Fourier transform?

• Wavelets are building blocks that can quickly decorrelate data. – Wim Sweldens

• Wavelets are optimal bases for compressing, estimating, and recovering functions … - David Donoho

• Both try to represent a function in other basis (transform into other domain) and hope this transformation can reveal some insights. Yet, unlike Fourier transform, wavelet can choose many different basis.

On Details of Wavelet Transform

• Describe the concepts of filter banks – Analysis

– Synthesis

• MRA (multi-resolution analysis)

VN

VN-1 WN-1

VN-2 WN-2

VN-3 WN-3

1121 VWWWV NNN 1121 VWWWV NNN

Formal Definition of an MRAAn MRA consists of the nested linear vector space

such that• There exists a function (t) (called scaling function)

such that is a basis for V0

• If and vice versa• ;

• Remarks:– Does not require the set of (t) and its integer translates t

o be orthogonal (in general)

– No mention of wavelet

2101 VVVV

integer:)( kkt

1)2( then )( kk VtfVtf

)(lim 2 RLV jj

}0{jV

Details (cont)

• The roles of scaling functions and wavelets– Basis functions in V

and W

• Refinement (two-scale) relations

• Graphing by cascading • Computing wavelet

coefficients (orthogonal)

Important Properties of Fourier Transform

• Linearity:

• Time shifting:

• Time scaling:

• Parseval’s theorem

)()()(then

)()()( If

21

21

FFF

tftftf

aF

aF

atftf

a

a

1)(then

)()( If

)(),(2

1)(),(

GFtgtf

dFdttf22

)(2

1)(

FeF

ttftftj 0)(then

)()( If

0

00

dxxgxff

Note

)()( represents g,

space,function In :

On Wavelet Coefficients

24

23

22

21

12

11

01

01 02222622214

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

WWWWWWWV

f

1d 2c 2d0c 0d

)(d)(cf 22 xx 2 2 2 2 2 2

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

Orthogonal MRA

H~ H

G~ G

VN

VN-1 WN-1

VN-2 WN-2

VN-3 WN-3

Biorthogonal MRAVN

VN-1 WN-1

VN-2 WN-2

VN-3 WN-3

NV~

1-NV~

2-NV~

3-NV~

3-NW~

2-NW~

1-NW~

kkkk WVWV ~~kkkk WVWV ~~

Semiorthogonal MRA

• Common property:

• Differences: – if orthogonal: scaling functi

ons (and wavelets) of the same level are orthogonal to each other

– If semiorthogonal, wavelets of different levels are orthogonal (from nested space)

kkkkk WVVWV 1

VN

VN-1 WN-1

VN-2 WN-2

VN-3 WN-3

kkkk WWWV~

and ~

Dual and primal are the

same

Comparison of Different Types of Wavelet Transforms

• What’s the advantage of orthgonality?

• Why choose to design biorthogonal wavelets (instead of orthogonal wavelets)?

On Lifting

• What kind of wavelet transform does lifting produce?

• What are the advantages of lifting?

• In-place computation• Easy inversion• Extensible to 2nd

generation wavelets• More efficient

computations

Details of Lifting

• Types of predictors– Interpolating

– Average-interpolating

– B-spline

– …

• Types of Update– Number of vanishing

moments of the wavelets

• Characteristics of the transform– MRA order– Dual MRA order– Polynomial reproducibility

and vanishing moments

• Cascading algorithms• “Lifting” theory:

– why it ensures biorthogonality

– Exact reconstruction guaranteed

Applications of Wavelet Transform

• Denoising• Compression• Progressive Transmiss

ion• Geometric simplificati

on• MR Editing

• Feature recognition• Graphics related …

On Variations of Wavelet Transform

• What is continuous wavelet transform?

• What is fast wavelet transform?

• What is wavelet packet?

• What types of information does each one reveal?

• Derivatives of phi, psi?