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Digital Image Processing. 7 Wavelets and Multiresolution Processing. Preview. 7.1 Background. Multiresolution Objects, which are of small size or of low contrast, require high resolution; Objects, which are of large size or of high contrast, often only require low resolution. - PowerPoint PPT Presentation
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Digital Image Processing
7 Wavelets and Multiresolution Processing
2
Preview
Fourier transform
Wavelet transform
Basis functions Sinusoids Small waves
Time duration Infinite Finite
Frequency information
Known Known
Temporal information
Unknown Known
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7.1 Background Multiresolution
Objects, which are of small size or of low contrast, require high resolution;
Objects, which are of large size or of high contrast, often only require low resolution.
Statistics features
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7.1 Background
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7.1.1 Image pyramids
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7.1.1 Image pyramids
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7.1.2 Subband coding
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7.1.2 Subband coding
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7.1.2 Subband coding
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7.1.3 The Haar transform Principle
Basis functions of the Haar transform are the oldest and simplest known orthonormal wavelets.
Expression of the Haar transform T = HFH where F is an image, H is the Haar transform.
An instance of the Haar transform
2200
0022
1112
1111
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4H
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7.1.3 The Haar transform
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7.2 Multiresolution expansion Series expansion
Scaling functionsInteger translationBinary scaling
kkk
kkk
kkk
xxfxxf
dxxfxxfxa
xaxf
)()(),()(
)()()(),(
)()(
)2(2)( 2, kxx j
j
kj
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7.2 Multiresolution expansion
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7.2 Multiresolution expansion Wavelet functions
Definition
An example: the Haar wavelet function)2(2)( 2
, kxx jj
kj
elsewhere0
15.01
1.001
)( x
x
x
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7.2 Multiresolution expansion
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7.3 Wavelet transform in one dimension The wavelet series expansions
Expression
Approximation coefficients
Wavelet coeffients
0
00)()()()()( ,,
jj kkjj
kkjj xkdxkcxf
dxxxfxxfkc kjkjj )()()(),()( ,, 000
dxxxfxxfkd kjkjj )()()(),()( ,,
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7.3 Wavelet transform in one dimension An example of the Haar wavelet series expansion
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7.3 Wavelet transform in one dimension The discrete wavelet transform
Definition
0
0
0
)(),(1)(),(1)(
)()(1),(
)()(1),(
,,0
,
,0
jj kkj
kkj
xkj
xkj
xkjWM
xkjWM
xf
xxfM
kjW
xxfM
kjW
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7.3 Wavelet transform in one dimension The continuous wavelet transform
Definition
The inverse continuous wavelet transform
s
xs
x
dxxxfsW
s
s
1)(
)()(),(
,
,
duu
uC
dsds
xsW
Cxf s
2
0 2,
)(
)(),(1)(
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7.3 Wavelet transform in one dimension
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7.4 The fast wavelet transform (skipped)
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7.5 Wavelet transform in two dimension Two dimensional scaling function
(x, y) = (x) (y)
Two dimensional wavelet functions H(x, y) = (x) (y) V(x, y) = (x) (y) D(x, y) = (x) (y)
The scaled and translated basis functions
},,{),2,2(2),(
)2,2(2),(
2,,
2,,
DVHinymxyx
nymxyx
jjij
inmj
jjj
nmj
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7.5 Wavelet transform in two dimension Definition
The inverse discrete wavelet transform
},,{,),(),(1),,(
),(),(1),,(
1
0
1
0,,0
1
0
1
0,,0 0
DVHiyxyxfMN
nmjW
yxyxfMN
nmjW
M
x
N
y
inmj
i
M
x
N
ynmj
DVHi jj m n
inmj
i
m nnmj
yxnmjWMN
yxnmjWMN
yxf
,,,,0
,,0
0
0
),(),,(1
),(),,(1),(
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