Wavelets and Multiresolution Processing (Multiresolution ...
Signal Processing Course : Wavelets
88
Wavelet Processing Gabriel Peyré www.numerical-tours.com
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Transcript of Signal Processing Course : Wavelets
WaveletProcessing
Gabriel Peyréwww.numerical-tours.com
Overview
•Review : Fourier transforms
•1-D Multiresolutions
•1-D Wavelet Transform
•Filter Constraints
•2-D Multiresolutions
Infinite continuous domains:
Periodic continuous domains:
Infinite discrete domains:
Periodic discrete domains:
f0(t), t � R
f0(t), t ⇥ [0, 1] � R/Z
The Four Settings
f̂ [m] =N�1�
n=0
f [n]e�2i�N mn
f̂0(�) =� +⇥
�⇥f0(t)e�i�tdt
f̂0[m] =� 1
0f0(t)e�2i�mtdt
f̂(�) =�
n�Zf [n]ei�n
Note: for Fourier, bounded � periodic.
. . . . . .
. . .. . .f [n], n � Z
f [n], n ⇤ {0, . . . , N � 1} ⇥ Z/NZ
Sampling idealization:
Poisson formula:
f0 f
f̂0 f̂
sampling
periodization
cont. FT discr. FT
Commutative diagram:
f [n]
f̂0(�)
f̂(�)
Sampling and Periodization
f [n] = f0(n/N)
f̂(⇥) =�
k
f̂0(N(⇥ + 2k�))
(a)
(c)
(d)
(b)
1
0
(a)
(c)
(d)
(b)
1
0
(a)
(c)
(d)
(b)
1
0
(a)
(c)
(d)
(b)
1
0
f0(t)
Sampling and Periodization
(a)
(c)
(d)
(b)
1
0
Sampling and Periodization: Aliasing
(b)
(c)
(d)
(a)
0
1
Overview
•Review : Fourier transforms
•1-D Multiresolutions
•1-D Wavelet Transform
•Filter Constraints
•2-D Multiresolutions
Multiresolutions: Approximation Spaces
Multiresolutions: Approximation Spaces
Multiresolutions: Approximation Spaces
Haar Multiresolutions
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Multiresolutions: Detail Spaces
Multiresolutions: Detail Spaces
Multiresolutions: Detail Spaces
Multiresolutions: Detail Spaces
Multiresolutions: Detail Spaces
��j,n \ j � j0, 0 � n < 2�j
⇥�
�⇥j0,n \ 0 � n < 2�j0
⇥
Haar Wavelets
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Haar Wavelets
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Overview
•Review : Fourier transforms
•1-D Multiresolutions
•1-D Wavelet Transform
•Filter Constraints
•2-D Multiresolutions
Computing the Wavelet Coefficients
Computing the Wavelet Coefficients
Computing the Wavelet Coefficients
Computing the Wavelet Coefficients
Computing the Wavelet Coefficients
Discrete Wavelet Coefficients
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1.5
Discrete Wavelet Coefficients
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−1.5
−1
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0
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1
1.5
Discrete Wavelet Coefficients
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1
−0.2
−0.1
0
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−1.5
−1
−0.5
0
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1
1.5
Discrete Wavelet Coefficients
0
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0.6
0.8
1
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0
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−0.2
−0.1
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−1
−0.5
0
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1
1.5
Discrete Wavelet Coefficients
0
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0.6
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1
−0.2
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0
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−0.2
−0.1
0
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−0.5
0
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−1.5
−1
−0.5
0
0.5
1
1.5
Discrete Wavelet Coefficients
0
0.2
0.4
0.6
0.8
1
−0.2
−0.1
0
0.1
0.2
−0.2
−0.1
0
0.1
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−0.5
0
0.5
Fast Wavelet Transform
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Fast Wavelet Transform
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0
0.5
1
Fast Wavelet Transform
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1
0
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1
0
0.5
1
1.5
Fast Wavelet Transform
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0
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1
0
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2
Fast Wavelet Transform
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Haar Refinement
Haar Refinement
Haar Transform
Haar Transform
Haar Transform
Haar Transform
Haar Transform
Inverting the Transform
Inverting the Transform
Inverting the Transform
Overview
•Review : Fourier transforms
•1-D Multiresolutions
•1-D Wavelet Transform
•Filter Constraints
•2-D Multiresolutions
Approximation Filter Constraints
Approximation Filter Constraints
Approximation Filter Constraints
{⌅(·� n)}n orthogonal ⇥⇤ ⌅n, ⌅ ⇧ ⌅̄(n) = �[n] ⇥⇤�
k
|⌅̂(⇤ + 2k⇥)|2 = 1
Approximation Filter Constraints
{⌅(·� n)}n orthogonal ⇥⇤ ⌅n, ⌅ ⇧ ⌅̄(n) = �[n] ⇥⇤�
k
|⌅̂(⇤ + 2k⇥)|2 = 1
Approximation Filter Constraints
{⌅(·� n)}n orthogonal ⇥⇤ ⌅n, ⌅ ⇧ ⌅̄(n) = �[n] ⇥⇤�
k
|⌅̂(⇤ + 2k⇥)|2 = 1
Approximation Filter Constraints
{⌅(·� n)}n orthogonal ⇥⇤ ⌅n, ⌅ ⇧ ⌅̄(n) = �[n] ⇥⇤�
k
|⌅̂(⇤ + 2k⇥)|2 = 1
{�(·� n)}n orthogonal �⇥�
k
|⇥̂(⇤ + 2k�)|2 = 1�� �n, ⇥ ⇤ ⇥(n) = �[n]
Detail Filter Constraint
{�(·� n)}n orthogonal �⇥�
k
|⇥̂(⇤ + 2k�)|2 = 1�� �n, ⇥ ⇤ ⇥(n) = �[n]
Detail Filter Constraint
{�(·� n)}n orthogonal �⇥�
k
|⇥̂(⇤ + 2k�)|2 = 1�� �n, ⇥ ⇤ ⇥(n) = �[n]
Detail Filter Constraint
{�(·� n)}n orthogonal �⇥�
k
|⇥̂(⇤ + 2k�)|2 = 1�� �n, ⇥ ⇤ ⇥(n) = �[n]
Detail Filter Constraint
{�(·� n)}n orthogonal �⇥�
k
|⇥̂(⇤ + 2k�)|2 = 1�� �n, ⇥ ⇤ ⇥(n) = �[n]
Detail Filter Constraint
Vanishing Moment Constraint
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Vanishing Moment Constraint
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Vanishing Moment Constraint
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Daubechies Family
Daubechies Family
Overview
•Review : Fourier transforms
•1-D Multiresolutions
•1-D Wavelet Transform
•Filter Constraints
•2-D Multiresolutions
Anisotropic Wavelet Transform
Anisotropic Wavelet Transform
Anisotropic Wavelet Transform
Anisotropic Wavelet Transform
Anisotropic Wavelet Transform
2D Multi-resolutions
2D Multi-resolutions
2D Multi-resolutions
2D Wavelet Basis
Discrete 2D Wavelets Coefficients
Discrete 2D Wavelets Coefficients
Discrete 2D Wavelets Coefficients
Discrete 2D Wavelets Coefficients
Discrete 2D Wavelets Coefficients
Examples of Decompositions
Separable vs. Isotropic
Fast 2D Wavelet Transform
Fast 2D Wavelet Transform
Fast 2D Wavelet Transform
Fast 2D Wavelet Transform
Inverse 2D Wavelet Transform
Inverse 2D Wavelet Transform
Conclusion
Conclusion
Conclusion
![Linear algebra, signal processing, and wavelets. A unified ...linear algebra. One example is [33], which goes further in using matrix notation than many signal processing textbooks.](https://static.fdocuments.net/doc/165x107/5f085f1d7e708231d421ae49/linear-algebra-signal-processing-and-wavelets-a-uniied-linear-algebra.jpg)
