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Chapter 7Wavelets and
MultiresolutionProcessing
Background
Multiresolution Expansions
Wavelet Transforms in One Dimension
Wavelet Transforms in Two Dimensions
Image Pyramids
Subband Coding
The Haar Transform
The total number of elements in a P+1 level pyramid for P>0 is
( ) ( ) ( )2
212
34
41
41
411 NN P ≤⎟⎟
⎠
⎞⎜⎜⎝
⎛+⋅⋅⋅+++
Image Pyramids
Subband Coding
The Haar Transform
The Z-transform, a generalization of the discrete Fourier transform, is the ideal tool for studying discrete-time, sampled-data systems.The Z-transform of sequemce x(n) for n=0,1,2,…is
Where z is a complex variable.∑∞
∞−
−= nznxzX )()(
Downsampling by a factor of 2 in the time domain corresponds to the simple Z-domain operation
(7.1-2)Upsampling-again by a factor of 2---is defined by the transform pair
(7.1-3)
( )[ ]2/12/1 )(21)()2()( zXzXzXnxnx downdown −+=⇔=
)()(0
,...4,2,0)2/()( 2zxzX
otherwisennx
nx upup =⇔⎩⎨⎧ =
=
If sequence x(n) is downsampled and subsequently upsampled to yield , Eqs.(7.1-2) and (7.1-3) combine to yield
where is the downsampled-upsampled sequence. Its inverse Z-transform is
)(nx)
[ ])()(21)( zXzXzX −+=
)
[ ])()( 1 zXZnx)) −=
[ ] )()1()(1 nxzXZ n−=−−
We can express the system’s output as
The output of filter is defined by the transform pair
[ ]
[ ])()()()()(21
)()()()()(21)(
111
000
zXzHzXzHzG
zXzHzXzHzGzX
−−++
−−+=)
)(0 nh
∑ ⇔−=∗k
zXzHkxknhnxnh )()()()()()( 000
As with Fourier transform, convolution in the time (or spatial domain is equivalent to multiplication in the Z-domain.
[ ]
[ ] )()()()()(21
)()()()()(21)(
1100
1100
zXzGzHzGzH
zXzGzHzGzHzX
−−+−+
+=)
For error-free reconstruction of the input,and . Thus, we impose the
following conditions:
To get
Where denotes the determinant of .
)()( nxnx =))()( zXzX =
)
2)()()()(0)()()()(
1100
1100
=+=−+−
zGzHzGzHzGzHzGzH
( ) ⎥⎦
⎤⎢⎣
⎡−−
−=⎥
⎦
⎤⎢⎣
⎡)(
)()(det
2)()(
0
1
1
0
zHzH
zHzGzG
m
( ))(det zH m )(zH m
Letting , and taking the inverse Z-transform, we get
Letting , and taking the inverse Z-transform, we get
)12())(det( +−= km zzH α
2=α
)()1()(
)()1()(
01
1
10
nhng
nhngn
n
+−=
−=
2−=α
)()1()(
)()1()(
01
11
0
nhng
nhngn
n
−=
−= +
Three general solution:Quadrature mirror filters (OMFs)
Conjugate quadrature filters (CQFs)
Orthonormal
Image Pyramids
Subband Coding
The Haar Transform
The Haar transform can be expressed in matrix form
Where F is an N*N image matrix, H is an N*N transformation matrix, T is the resulting N*N transform.
THFHT =
For the Haar transform, transformation matrix H contains the Haar basis functions, .They are defined over the continuous, closed interval for k=0,1,2,…,N-1, where .To generate H, we define the integer k such that
where , or 1 for .
)(zhk
[ ]1,0∈znN 2=
12 −+= qk P
10 −≤≤ np 0=q 0=ppq 21 ≤≤ 0≠pfor
Then the Haar basis functions are
and
[ ]1,0,1)()( 000 ∈== zN
zhzh
[ ]⎪⎩
⎪⎨
⎧
∈<≤−−
−<≤−
==1,0,0
2/2/)5.0(22/)5.0(2/)1(2
1)()( 2/
2/
zotherwiseqzq
qzq
Nzhzh ppp
ppp
pqk
The ith row of an N*N Haar transformation matrix contains the elements of
If N=4, for example k,q, and p assume the values./)1(,....,/2,/1,/0)( NNNNNzforzhi −=
213112101000qpk
The 4*4 transformation matrix, H4, is
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−−
−−=
220000221111
1111
41
4H
Background
Multiresolution Expansions
Wavelet Transforms in One Dimension
Wavelet Transforms in Two Dimensions
Series Expansion
Scaling Functions
Wavelet Functions
A signal of function f(x) can often be better analyzed as a linear combination of expansion functions
k is an interger index of the finite or infinite sum;
are real-valued expansion coefficients;are real-valued expansion functions.
∑=k
kk xxf )()( ϕα
kα
)(xkϕ
These coefficients are computed by taking the integral inner products of the dual ‘s and function f(x). That is
)(~ xkϕ
dxxfxxfx kkk )()(~)(),(~ *∫== ϕϕα
Series Expansion
Scaling Functions
Wavelet Functions
The set of expansion functions composed of integer translations and binary scaling of the real, square-integrable function ; that is, the set where
)(xϕ{ })(, xkjϕ
( )kxx jjkj −= 22)( 2/
, ϕϕ
If , it can be written
We will denote the subspace spanned over k for any j as
{ })(,00xSpanV kj
kj ϕ=
0)( jVxf ∈
∑=k
kjk xxf )()( ,0ϕα
{ })(, xSpanV kjk
j ϕ=
The simple scaling function in the preceding example obeys the four fundamental requirements of multiresolution analysis:
MRA Requirement 1: The scaling function is orthogonal to its integer translates;
MRA Requirement 2: The subspaces spanned by the scaling function at low scales are nested within those spanned at higher scales.
MRA Requirement 3: The only function that is common to all Vj is f(x)=0.
MRA Requirement 4: Any function can be represented with arbitrary precision.
Series Expansion
Scaling Functions
Wavelet Functions
Given a scaling function that meets the MRA requirements of the previous section, we can define a wavelet function that, together with its integer translates and binary scaling, spans the difference between any two adjacent scaling subspaces, and . We define the set of wavelets
)(xψ
jV 1+jV{ })(, xkjψ
{ })2(2)( 2/, kxx jjkj −= ψψ
As with scaling functions, we write
And note that if
The scaling and wavelet function subspaces are related by
{ })(, xSpanW kjk
j ψ=
jWxf ∈)(
∑=k
kjk xxf )()( ,ψα
jjj WVV ⊕=+1
We can now express the space of all measurable, square-integrable functions as
or...)( 100
2 ⊕⊕⊕= WWVRL
...)( 2112 ⊕⊕⊕= WWVRL
The Haar wavelet function is
⎪⎩
⎪⎨
⎧<≤−<≤
=elsewhere
xx
x0
15.015.001
)(ψ
Background
Multiresolution Expansions
Wavelet Transforms in One Dimension
Wavelet Transforms in Two Dimensions
The Wavelet Series Expansions
The Discrete Wavelet Transform
The Continuous Wavelet Transform
Defining the wavelet series expansion of function relative to wavelet and scaling function . f(x) can be written as
: the approximation or scaling coefficients;
: the detail or wavelet coefficients.
)()( 2 RLxf ∈ )(xψ
)(xϕ
∑∑∞
=
+=0
00)()()()()( ,,
jjkjj
kkjj xkdxkcxf ψϕ
skc j )'(0
skd j )'(
If the expansion functions form an orthonormal basis or tight frame, the expansion coefficients are calculated as
and
∫== dxxxfxxfkc kjkjj )()()(),()( ,, 000ϕϕ
∫== dxxxfxxfkd kjkjj )()()(),()( ,, ψψ
The Wavelet Series Expansions
The Discrete Wavelet Transform
The Continuous Wavelet Transform
If the function being expanded is a sequence of numbers, like samples of a continuous function f(x), the resulting coefficients are called the discrete wavelet transform(DWT) of f(x).
and
∑=x
kj xxfM
kjW )()(1),( ,0 0ϕϕ
∑=x
kj xxfM
kjW )()(1),( ,ψψ
∑∑∑∞
=
+=0
0)(),(1)(),(1)( ,,0
jj kkj
kkj xkjW
MxkjW
Mxf ψϕ ψϕ
Consider the discrete function of four points:f(0)=1, f(1)=4, f(2)=-3, and f(3)=0
Since M=4, J=2 and, with j0=0, the summations are performed over
x=0,1,2,3,j=0,1, and k=0 for j=0
or k=0,1 for j=1.
We find that
[ ] 11013141121)()(
21)0,0(
3
00,0 =⋅+⋅−⋅+⋅== ∑
=xxxfW ϕϕ
[ ] 4)1(0)1(3141121)0,0( =−⋅+−⋅−⋅+⋅=ψW
[ ] 25.10003)2(42121)0,1( −=⋅+⋅−−⋅+⋅=ψW
[ ] 25.1)2(023040121)1,1( −=−⋅+⋅−⋅+⋅=ψW
For x=0,1,2,3. If x=0, for instance,
[ ])()1,1()()0,1()()0,0()()0,0(21)( 1,10,10,00,0 xWxWxWxWxf ψψψϕ ψψψϕ +++=
[ ] 1025.1)2(25.1141121)0( =⋅−⋅−⋅+⋅=f
The Wavelet Series Expansions
The Discrete Wavelet Transform
The Continuous Wavelet Transform
The continuous wavelet transform of a continuous, square-integrable function, f(x), relative to a real-valued wavelet, ,is
Where
And s and are called scale and translation parameters.
)(xψ
∫∞
∞−= dxxxfsW s )()(),( ,τψ ψτ
⎟⎠⎞
⎜⎝⎛ −
=s
xs
xsτψψ τ
1)(,
τ
Given , f(x) can be obtained using the inverse continuous wavelet transform
Where
And is the Fourier transform of .
),( τψ sW
dsds
xsW
Cxf s τ
ψτ τ
ψψ∫ ∫∞ ∞
∞−=
0 2, )(
),(1)(
∫∞
∞−
Ψ= du
uu
C2)(
ψ
)(uΨ )(xψ
The Mexican hat wavelet
2/24/1 2
)1(3
2)( xexx −− −⎟⎟⎠
⎞⎜⎜⎝
⎛= πψ
Background
Multiresolution Expansions
Wavelet Transforms in One Dimension
Wavelet Transforms in Two Dimensions
In two dimensions, a two-dimensional scaling function, , and three two-dimensional wavelet, , and , are required.
),( yxϕ
),( yxHψ ),( yxVψ ),( yxVψ
Excluding products that produce one-dimensional results, like , the four remaining products produce the separable scaling function
And separable, “directionally sensitive”wavelets
)()( xx ψϕ
)()(),( yxyx ϕϕϕ =
)()(),( yxyxH ϕψψ =
)()(),( yxyxV ψϕψ =
)()(),( yxyxD ψψψ =
The scaled and translated basis functions:
)2,2(2),( 2/,, nymxyx jjjnmj −−= ϕϕ
{ }DVHinymxyx jjjnmj
i ,,),2,2(2),( 2/,, =−−= ψψ
The discrete wavelet transform of function f(x,y) of size M*N is then
( ) ∑∑−
=
−
=
=1
0
1
0,,0 ),(),(1,,
0
M
x
N
ynmj yxyxf
MNnmjW ϕϕ
( ) { }∑∑−
=
−
=
==1
0
1
0,, ,,),(),(1,,
M
x
N
ynmj
ii DVHiyxyxfMN
nmjW ψψ
Given the and , f(x,y) is obtained via the inverse discrete wavelet transform
ϕW iWψ
∑ ∑∑∑
∑∑
=
∞
=
+
=
DVHi jj m n
inmj
i
m nnmj
yxnmjWMN
yxnmjWMN
yxf
,,,,
,,0
0
0
),(),,(1
),(),,(1),(
ψ
ϕ
ψ
ϕ
Fig. 7.24 (Con’t)
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