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
VN
• Describe the concepts
of filter banks
– Analysis
– Synthesis
• MRA (multiresolution analysis)
VN-1
VN-2
WN-1
WN-2
VN-3 WN-3
VN  WN 1  WN 2    W1  V1
Formal Definition of an MRA
An MRA consists of the nested linear vector space
such that   V1  V0  V1  V2  
• There exists a function f(t) (called scaling function)
such that f (t  k ) : k integer  is a basis for V0
• If f (t ) Vk then f (2t ) Vk 1 and vice versa
• lim V j  L2 ( R) ;  V j  {0}
j 
• Remarks:
– Does not require the set of f(t) and its integer translates
to be orthogonal (in general)
– No mention of wavelet
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:
• Parseval’s theorem
If f (t )   f1 (t )   f 2 (t )
then F ( )   F1 ( )   F2 ( )
• Time shifting:
If f 0 (t )  f (t  t0 )
then F0 ( )  e  jt0 F  
• Time scaling:
If f a (t )  f (at )
then Fa ( ) 
1  
F 
a a
2
1 
 f (t ) dt  2  F ( ) d
1
f (t ), g (t ) 
F ( ), G ( )
2

2
Note : In function space,
f , g represents



f ( x)g ( x)dx
On Wavelet Coefficients
f  c ( x)  d ( x)
2
2
c 2 ( x)
d 2 ( x)
2
2
2
2
2
2
2
2
2
2
2
2
2
c 0 d 0 d1 cd 2
f  (4, 6,10,12, 8, 6, 5, 5)
 14 2 V10  2 2 W10  6 W11  2 W21  2 W12  2 W22  2 W32  0 W42
Orthogonal MRA
~
H
H


VN
VN-1
VN-2
VN-3 WN-3
WN-1
WN-2
~
G
G
Biorthogonal MRA
~
VN
VN
VN-1
VN-2
WN-2
VN-3 WN-3
~ ~
Vk  Wk Vk  Wk
~
WN-1
~
VN-1
WN-1
~
VN-2
~
~
VN-3 WN-3
~
WN-2
Semiorthogonal MRA
• Common property:
VN
Vk  Wk  Vk 1 Vk  Wk
VN-1
• Differences:
– if orthogonal: scaling
functions (and wavelets) of
the same level are
orthogonal to each other
– If semiorthogonal, wavelets
of different levels are
orthogonal (from nested
space)
WN-1
WN-2
VN-2
VN-3 WN-3
~
~
Vk  Wk and Wk  Wk
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
Transmission
• Geometric
simplification
• 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?