Chapter 7 Wavelets and Multiresolution Processingcs.tju.edu.cn/faculty/hyh/image analysis/chapter...
Transcript of Chapter 7 Wavelets and Multiresolution Processingcs.tju.edu.cn/faculty/hyh/image analysis/chapter...
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Chapter 7Wavelets and
MultiresolutionProcessing
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Background
Multiresolution Expansions
Wavelet Transforms in One Dimension
Wavelet Transforms in Two Dimensions
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Image Pyramids
Subband Coding
The Haar Transform
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The total number of elements in a P+1 level pyramid for P>0 is
( ) ( ) ( )2
212
34
41
41
411 NN P ≤⎟⎟
⎠
⎞⎜⎜⎝
⎛+⋅⋅⋅+++
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Image Pyramids
Subband Coding
The Haar Transform
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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 )()(
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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 =⇔⎩⎨⎧ =
=
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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−=−−
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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
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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
−−+−+
+=)
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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
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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
−=
−= +
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Three general solution:Quadrature mirror filters (OMFs)
Conjugate quadrature filters (CQFs)
Orthonormal
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Image Pyramids
Subband Coding
The Haar Transform
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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 =
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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
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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
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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
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The 4*4 transformation matrix, H4, is
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−−
−−=
220000221111
1111
41
4H
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Background
Multiresolution Expansions
Wavelet Transforms in One Dimension
Wavelet Transforms in Two Dimensions
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Series Expansion
Scaling Functions
Wavelet Functions
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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
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These coefficients are computed by taking the integral inner products of the dual ‘s and function f(x). That is
)(~ xkϕ
dxxfxxfx kkk )()(~)(),(~ *∫== ϕϕα
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Series Expansion
Scaling Functions
Wavelet Functions
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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
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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=
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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.
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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.
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Series Expansion
Scaling Functions
Wavelet Functions
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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
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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
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We can now express the space of all measurable, square-integrable functions as
or...)( 100
2 ⊕⊕⊕= WWVRL
...)( 2112 ⊕⊕⊕= WWVRL
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The Haar wavelet function is
⎪⎩
⎪⎨
⎧<≤−<≤
=elsewhere
xx
x0
15.015.001
)(ψ
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Background
Multiresolution Expansions
Wavelet Transforms in One Dimension
Wavelet Transforms in Two Dimensions
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The Wavelet Series Expansions
The Discrete Wavelet Transform
The Continuous Wavelet Transform
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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 )'(
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If the expansion functions form an orthonormal basis or tight frame, the expansion coefficients are calculated as
and
∫== dxxxfxxfkc kjkjj )()()(),()( ,, 000ϕϕ
∫== dxxxfxxfkd kjkjj )()()(),()( ,, ψψ
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The Wavelet Series Expansions
The Discrete Wavelet Transform
The Continuous Wavelet Transform
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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
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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.
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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
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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
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The Wavelet Series Expansions
The Discrete Wavelet Transform
The Continuous Wavelet Transform
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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
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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
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The Mexican hat wavelet
2/24/1 2
)1(3
2)( xexx −− −⎟⎟⎠
⎞⎜⎜⎝
⎛= πψ
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Background
Multiresolution Expansions
Wavelet Transforms in One Dimension
Wavelet Transforms in Two Dimensions
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In two dimensions, a two-dimensional scaling function, , and three two-dimensional wavelet, , and , are required.
),( yxϕ
),( yxHψ ),( yxVψ ),( yxVψ
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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 =
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The scaled and translated basis functions:
)2,2(2),( 2/,, nymxyx jjjnmj −−= ϕϕ
{ }DVHinymxyx jjjnmj
i ,,),2,2(2),( 2/,, =−−= ψψ
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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
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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
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Fig. 7.24 (Con’t)
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