Post on 25-Apr-2018
Wavelets and Multiresolution Processing
Thinh Nguyen
Multiresolution Multiresolution Analysis (MRA)Analysis (MRA)
A scaling function scaling function is used to create a series of approximations of a function or image, each differing by a factor of 2 from its neighboring approximations.
Additional functions called wavelets wavelets are then used to encode the difference in information between adjacent approximations.
Express a signal as
If the expansion is unique, the are called basis functionsbasis functions, and the expansion set
is called a basisbasis
Series ExpansionsSeries Expansions
( )f x( ) ( )k k
kf x xα ϕ=∑
expansion coefficientsexpansion functions
( )k xϕ
{ }( )k xϕ
All the functions expressible with this basis form a function spacefunction space which is referred to as the closed spanclosed span of the expansion set
If , then is in the closed span of and can be expressed as
Series ExpansionsSeries Expansions
{ }( )kk
V Span xϕ=
( )f x V∈ ( )f x{ }( )k xϕ
( ) ( )k kk
f x xα ϕ=∑
The expansion functions form an orthonormal basis for V
The basis and its dual are equivalent, i.e.,and
OrthonormalOrthonormal BasisBasis
0( ), ( )
1j k jkj k
x xj k
ϕ ϕ δ≠⎧
= = ⎨ =⎩
( ) ( )k kx xϕ ϕ= %
( ), ( ) ( ) ( )k k kx f x x f x dxα ϕ ϕ∗= = ∫
Consider the set of expansion functions composed
of integer translations and binary scalings of the
real square-integrable function defined by
for all and
By choosing the scaling function wisely,
can be made to span
Scaling FunctionsScaling Functions
( )xϕ
{ } { }/ 2, ( ) 2 (2 )j j
j k x x kϕ ϕ= −
,j k∈ 2( ) ( )x Lϕ ∈
{ }, ( )j k xϕ2( )L
( )xϕ
Index k determines the position of
along the x-axis, index j determines its width;
controls its height or amplitude.
By restricting j to a specific value the
resulting expansion set is a subset of
One can write
, ( )j k xϕ
/ 22 j
{ } { }/ 2, ( ) 2 (2 )j j
j k x x kϕ ϕ= −
oj j=
{ }, ( )j k xϕ{ }, ( )
oj k xϕ
{ }, ( )o oj j k
kV Span xϕ=
Example: The Example: The Haar Haar Scaling FunctionScaling Function
1,0 1,1 1,4( ) 0.5 ( ) ( ) 0.25 ( )f x x x xϕ ϕ ϕ= + −0, 1,2 1,2 1
1 1( ) ( ) ( )2 2k k kx x xϕ ϕ ϕ += +
0 1V V⊂
1 0 1( )
0x
xotherwise
ϕ≤ <⎧
= ⎨⎩
1. The scaling function is orthogonal to its integer translates
2. The subspaces spanned by the scaling function at low scales are nested within those spanned at higher scales:
MRA RequirementsMRA Requirements
1 0 1 2V V V V V V−∞ − ∞⊂ ⊂ ⊂ ⊂ ⊂ ⊂ ⊂L L
Given a scaling function which satisfies the MRA requirements, one can define a wavelet wavelet functionfunction which, together with its integer translates and binary scalings, spans the difference between any two adjacent scaling subspaces and
Wavelet FunctionsWavelet Functions
( )xψ
jV 1jV +
Define the wavelet set
for all that spans the spacesWe write
and, if
{ } { }/ 2, ( ) 2 (2 )j j
j k x x kψ ψ= −
Wavelet FunctionsWavelet Functions
k∈ jW
{ }, ( )j j kk
W Span xψ=
( ) jf x W∈
,( ) ( )k j kk
f x xα ψ=∑
This implies that
for all appropriateWe can write
and also
OrthogonalityOrthogonality:: 1j j jV V W+ = ⊕1j j jV V W+ = ⊕
, ,( ), ( ) 0j k j lx xϕ ψ =, ,j k l∈
20 0 1( )L V W W= ⊕ ⊕ ⊕L
22 1 0 1 2( )L W W W W W− −= ⊕ ⊕ ⊕ ⊕ ⊕ ⊕L L
(no need for scaling functions, only wavelets!)
Example: Example: Haar Haar Wavelet Functions in Wavelet Functions in WW00 and and WW11
( ) ( ) ( )a df x f x f x= +
low frequencies
high frequencies
A function can be expressed as
Wavelet Series ExpansionsWavelet Series Expansions
0 0
0
,,( ) ( ( )) ( ) ( )j jj j kk k
kj j
f x c kx xk d ψϕ∞
== +∑ ∑ ∑
2( ) ( )f x L∈
approximation or scaling coefficientsapproximation or scaling coefficientsdetail or wavelet coefficientsdetail or wavelet coefficients
00 ,( ) ( ), ( )jj k xc k f x ϕ=,( ) ( ), ( )j kjd x xk f ψ=
0 00
21( ) j jj WV WL += ⊕ ⊕ ⊕L
ConsiderIf , the expansion coefficients are
Example: The Example: The Haar Haar Wavelet Series Wavelet Series Expansion of Expansion of yy==xx22
2 0 10x xy
otherwise
⎧ ≤ <= ⎨⎩
1 12 2
0 00 01 1
2 21 1
0,0
1,0
,
0
0 0
01,1
1 1(0) (0)3 4
2 3 2(0) (1)32
( )
( (
)
) )
(
32
c x xdx d x dx
xd x dx d x xx d
x ψ
ψ ψ
ϕ= = = = −
= = − = = −
∫ ∫
∫ ∫
0 0 1
1 0 0
2 1 1 0 0 1
0,0 1,0 10 0 ,1, ( )1 1 2 3 23 4
( ) ( ) (2 32
)3
V W WV V W
V V W V W W
x xx xy ψ ψ ψϕ
= ⊕
= ⊕ = ⊕ ⊕
⎡ ⎤⎡ ⎤= + − + − − +⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦L
14243 1442443 144444244444314444244443
14444444444244444444443
0 0j =
Example: The Example: The Haar Haar Wavelet Series Wavelet Series Expansion of Expansion of yy==xx22
Let denote a discrete function Its DWT is defined as
Let and so that
The Discrete Wavelet Transform (DWT)The Discrete Wavelet Transform (DWT)( ), 0,1, , 1f x x M= −K
00 ,
,
1( , ) ( )
1( , ) (
(
) ( )
)x
j
k
xk
jW j k f xM
W j k f x xM
xϕ
ψ ψ
ϕ=
=
∑
∑ 0j j≥
0
00 ,, ( )1 1( )( ) ( , ) ( , ) j kk j j k
j kf x W j k W j kM M
xxϕ ψ ψϕ∞
== +∑ ∑ ∑
approximation approximation coefficientscoefficients
detail detail coefficientscoefficients
2JM =0oj =0,1, , 1,0,1, , 1,0,1, ,2 1j
x Mj Jk
= −⎧⎪ = −⎨⎪ = −⎩
K
K
K
Consider the discrete function
It isThe summations are performed over
and for andfor
Use the Haar scaling and wavelet functions
Example: Computing the DWTExample: Computing the DWT
(0) 1, (1) 4, (2) 3, (3) 0f f f f= = = − =24 2 2M J= = ⎯⎯→ =
0,1,2,3x = 0k = 0j =0,1k = 1j =
Example: Computing the DWTExample: Computing the DWT
[ ]
[ ]
3
03
03
0
0,0
1
0,
3
0
,
1,
0
0
1
1(0,0) ( ) 1 4 3 0 12
1(0,0) ( ) 1 4 3 ( ) 0 ( ) 42
1(1,0) ( ) 1
1( ) 1 1 1 12
1( ) 2 2 0 0
1( ) 1 1 1
4 ( ) 3 0 1.5 22
1(1,1) ( ) 1
2
1( ) 0 0 2
12
32
4 0 (2
x
x
x
x
x
x
W f x
W f x
W f x
W f x x
xϕ
ψ
ψ
ψ
ψ
ψ
ψ
ϕ=
=
=
=
= = ⋅ + ⋅ − ⋅ + ⋅ =
= = ⋅ + ⋅ − ⋅ + ⋅ =
⎡ ⎤= = ⋅ + ⋅ − ⋅ + ⋅ = −⎣ ⎦
= = ⋅ + ⋅ − ⋅ + ⋅
− −
−
−
∑
∑
∑
∑ ) 12 .5 2⎡ ⎤ = −⎣ ⎦
The DWT of the 4-sample function relative to the Haar wavelet and scaling functions thus is
The original function can be reconstructed as
for
Example: Computing the DWTExample: Computing the DWT
{ }1,4, 1.5 2, 1.5 2− −
0,0
1,0 1
0,
,1
0 ( )1( ) (0,0) (0,0)2
(1 (,0) (1,1 (
)
) )
(
)
f x W W
W W
x
x x
xϕ ψ
ψ ψ
ψϕ
ψ ψ
⎡= + +⎣
⎤+ ⎦0,1,2,3x =
In 2-D, one needs one scaling function
and three wavelets
is a 1-D scaling function and is its corresponding wavelet
Wavelet Transform in 2Wavelet Transform in 2--DD
( , ) ( ) ( )x y x yϕ ϕ ϕ=
( , ) ( ) ( )( , ) ( ) ( )( , ) ( ) ( )
H
V
D
x y x yx y x yx y x y
ψ ψ ϕψ ϕ ψψ ψ ψ
⎧ =⎪ =⎨⎪ =⎩
•detects horizontal details
•detects vertical details
•detects diagonal details
(.)ϕ(.)ψ
Define the scaled and translated basis functions
Then
22--D DWT: DefinitionD DWT: Definition
{ }
/ 2
/
,
2
,
, ,
2 (2 ,2 )
2 (2 ,2 ),( ,
( )
,
,
) ,ij m n
j j j
j i j
j m n
j
x m y n
x m yx n i H V D
x y
y
ϕϕ
ψψ
= − −
= − − =
{ }
0
1 1
00 0
1 1
0 0,
,
,
,
( ,
( ,
)
1( , , ) ( , )
1( , , ) ( , , ,
)
, )
M N
x yj m
M Ni
x
ij n
ym
nW j m n f x yMN
W j x ym n f
x y
x y i H V DMN
ϕ
ψ ψ
ϕ− −
= =
− −
= =
=
= =
∑ ∑
∑ ∑
0
0
0
,,
,
, ,
,
1( , ) ( ,
(
( , )
, )
, )
1 ( , , )
m n
i
i
j m
H V D j j m nm
n
ij n
f x y W j m nMN
W j m nM
x y
xN
y
ϕ
ψ ψ
ϕ
∞
= =
=
+
∑∑
∑ ∑ ∑∑
Filter bank implementation of 2Filter bank implementation of 2--D D waveletwavelet
analysis FBanalysis FB
synthesis FBsynthesis FB
resulting resulting decompositiondecomposition
( , )f x y
LL
LHHL
HH
HHHL
LHLL
Example: A ThreeExample: A Three--Scale FWTScale FWT
Analysis and Synthesis FiltersAnalysis and Synthesis Filters
scaling scaling functionfunction
wavelet wavelet functionfunction
analysis analysis filtersfilters
synthesis synthesis filtersfilters
“An Introduction to Wavelets,” by Amara Graps
Amara’s Wavelet Page (with many links to other resources) http://www.amara.com/current/wavelet.html
“Wavelets for Kids,” (A Tutorial Introduction), by B. Vidakovic and P. Mueller
Gilbert Strang’s tutorial papers from his MIT webpage
http://www-math.mit.edu/~gs/
Wavelets and Subband Coding, by Jelena Kovacevic and Martin Vetterli, Prentice Hall, 2000.
Want to Learn More About Wavelets?Want to Learn More About Wavelets?