Download - Signal Processing Course : Wavelets
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WaveletProcessing
Gabriel Peyréwww.numerical-tours.com
![Page 2: Signal Processing Course : Wavelets](https://reader033.vdocument.in/reader033/viewer/2022052209/556aead7d8b42a86218b4d09/html5/thumbnails/2.jpg)
Overview
•Review : Fourier transforms
•1-D Multiresolutions
•1-D Wavelet Transform
•Filter Constraints
•2-D Multiresolutions
![Page 3: Signal Processing Course : Wavelets](https://reader033.vdocument.in/reader033/viewer/2022052209/556aead7d8b42a86218b4d09/html5/thumbnails/3.jpg)
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
![Page 4: Signal Processing Course : Wavelets](https://reader033.vdocument.in/reader033/viewer/2022052209/556aead7d8b42a86218b4d09/html5/thumbnails/4.jpg)
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)
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0
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0
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f0(t)
![Page 5: Signal Processing Course : Wavelets](https://reader033.vdocument.in/reader033/viewer/2022052209/556aead7d8b42a86218b4d09/html5/thumbnails/5.jpg)
Sampling and Periodization
(a)
(c)
(d)
(b)
1
0
![Page 6: Signal Processing Course : Wavelets](https://reader033.vdocument.in/reader033/viewer/2022052209/556aead7d8b42a86218b4d09/html5/thumbnails/6.jpg)
Sampling and Periodization: Aliasing
(b)
(c)
(d)
(a)
0
1
![Page 7: Signal Processing Course : Wavelets](https://reader033.vdocument.in/reader033/viewer/2022052209/556aead7d8b42a86218b4d09/html5/thumbnails/7.jpg)
Overview
•Review : Fourier transforms
•1-D Multiresolutions
•1-D Wavelet Transform
•Filter Constraints
•2-D Multiresolutions
![Page 8: Signal Processing Course : Wavelets](https://reader033.vdocument.in/reader033/viewer/2022052209/556aead7d8b42a86218b4d09/html5/thumbnails/8.jpg)
Multiresolutions: Approximation Spaces
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Multiresolutions: Approximation Spaces
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Multiresolutions: Approximation Spaces
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Haar Multiresolutions
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Multiresolutions: Detail Spaces
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Multiresolutions: Detail Spaces
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Multiresolutions: Detail Spaces
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Multiresolutions: Detail Spaces
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Multiresolutions: Detail Spaces
��j,n \ j � j0, 0 � n < 2�j
⇥�
�⇥j0,n \ 0 � n < 2�j0
⇥
![Page 17: Signal Processing Course : Wavelets](https://reader033.vdocument.in/reader033/viewer/2022052209/556aead7d8b42a86218b4d09/html5/thumbnails/17.jpg)
Haar Wavelets
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Haar Wavelets
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![Page 19: Signal Processing Course : Wavelets](https://reader033.vdocument.in/reader033/viewer/2022052209/556aead7d8b42a86218b4d09/html5/thumbnails/19.jpg)
Overview
•Review : Fourier transforms
•1-D Multiresolutions
•1-D Wavelet Transform
•Filter Constraints
•2-D Multiresolutions
![Page 20: Signal Processing Course : Wavelets](https://reader033.vdocument.in/reader033/viewer/2022052209/556aead7d8b42a86218b4d09/html5/thumbnails/20.jpg)
Computing the Wavelet Coefficients
![Page 21: Signal Processing Course : Wavelets](https://reader033.vdocument.in/reader033/viewer/2022052209/556aead7d8b42a86218b4d09/html5/thumbnails/21.jpg)
Computing the Wavelet Coefficients
![Page 22: Signal Processing Course : Wavelets](https://reader033.vdocument.in/reader033/viewer/2022052209/556aead7d8b42a86218b4d09/html5/thumbnails/22.jpg)
Computing the Wavelet Coefficients
![Page 23: Signal Processing Course : Wavelets](https://reader033.vdocument.in/reader033/viewer/2022052209/556aead7d8b42a86218b4d09/html5/thumbnails/23.jpg)
Computing the Wavelet Coefficients
![Page 24: Signal Processing Course : Wavelets](https://reader033.vdocument.in/reader033/viewer/2022052209/556aead7d8b42a86218b4d09/html5/thumbnails/24.jpg)
Computing the Wavelet Coefficients
![Page 25: Signal Processing Course : Wavelets](https://reader033.vdocument.in/reader033/viewer/2022052209/556aead7d8b42a86218b4d09/html5/thumbnails/25.jpg)
Discrete Wavelet Coefficients
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Discrete Wavelet Coefficients
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Fast Wavelet Transform
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Fast Wavelet Transform
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Fast Wavelet Transform
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Fast Wavelet Transform
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Fast Wavelet Transform
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Haar Refinement
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Haar Refinement
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Haar Transform
![Page 39: Signal Processing Course : Wavelets](https://reader033.vdocument.in/reader033/viewer/2022052209/556aead7d8b42a86218b4d09/html5/thumbnails/39.jpg)
Haar Transform
![Page 40: Signal Processing Course : Wavelets](https://reader033.vdocument.in/reader033/viewer/2022052209/556aead7d8b42a86218b4d09/html5/thumbnails/40.jpg)
Haar Transform
![Page 41: Signal Processing Course : Wavelets](https://reader033.vdocument.in/reader033/viewer/2022052209/556aead7d8b42a86218b4d09/html5/thumbnails/41.jpg)
Haar Transform
![Page 42: Signal Processing Course : Wavelets](https://reader033.vdocument.in/reader033/viewer/2022052209/556aead7d8b42a86218b4d09/html5/thumbnails/42.jpg)
Haar Transform
![Page 43: Signal Processing Course : Wavelets](https://reader033.vdocument.in/reader033/viewer/2022052209/556aead7d8b42a86218b4d09/html5/thumbnails/43.jpg)
Inverting the Transform
![Page 44: Signal Processing Course : Wavelets](https://reader033.vdocument.in/reader033/viewer/2022052209/556aead7d8b42a86218b4d09/html5/thumbnails/44.jpg)
Inverting the Transform
![Page 45: Signal Processing Course : Wavelets](https://reader033.vdocument.in/reader033/viewer/2022052209/556aead7d8b42a86218b4d09/html5/thumbnails/45.jpg)
Inverting the Transform
![Page 46: Signal Processing Course : Wavelets](https://reader033.vdocument.in/reader033/viewer/2022052209/556aead7d8b42a86218b4d09/html5/thumbnails/46.jpg)
Overview
•Review : Fourier transforms
•1-D Multiresolutions
•1-D Wavelet Transform
•Filter Constraints
•2-D Multiresolutions
![Page 47: Signal Processing Course : Wavelets](https://reader033.vdocument.in/reader033/viewer/2022052209/556aead7d8b42a86218b4d09/html5/thumbnails/47.jpg)
Approximation Filter Constraints
![Page 48: Signal Processing Course : Wavelets](https://reader033.vdocument.in/reader033/viewer/2022052209/556aead7d8b42a86218b4d09/html5/thumbnails/48.jpg)
Approximation Filter Constraints
![Page 49: Signal Processing Course : Wavelets](https://reader033.vdocument.in/reader033/viewer/2022052209/556aead7d8b42a86218b4d09/html5/thumbnails/49.jpg)
Approximation Filter Constraints
{⌅(·� n)}n orthogonal ⇥⇤ ⌅n, ⌅ ⇧ ⌅̄(n) = �[n] ⇥⇤�
k
|⌅̂(⇤ + 2k⇥)|2 = 1
![Page 50: Signal Processing Course : Wavelets](https://reader033.vdocument.in/reader033/viewer/2022052209/556aead7d8b42a86218b4d09/html5/thumbnails/50.jpg)
Approximation Filter Constraints
{⌅(·� n)}n orthogonal ⇥⇤ ⌅n, ⌅ ⇧ ⌅̄(n) = �[n] ⇥⇤�
k
|⌅̂(⇤ + 2k⇥)|2 = 1
![Page 51: Signal Processing Course : Wavelets](https://reader033.vdocument.in/reader033/viewer/2022052209/556aead7d8b42a86218b4d09/html5/thumbnails/51.jpg)
Approximation Filter Constraints
{⌅(·� n)}n orthogonal ⇥⇤ ⌅n, ⌅ ⇧ ⌅̄(n) = �[n] ⇥⇤�
k
|⌅̂(⇤ + 2k⇥)|2 = 1
![Page 52: Signal Processing Course : Wavelets](https://reader033.vdocument.in/reader033/viewer/2022052209/556aead7d8b42a86218b4d09/html5/thumbnails/52.jpg)
Approximation Filter Constraints
{⌅(·� n)}n orthogonal ⇥⇤ ⌅n, ⌅ ⇧ ⌅̄(n) = �[n] ⇥⇤�
k
|⌅̂(⇤ + 2k⇥)|2 = 1
![Page 53: Signal Processing Course : Wavelets](https://reader033.vdocument.in/reader033/viewer/2022052209/556aead7d8b42a86218b4d09/html5/thumbnails/53.jpg)
{�(·� n)}n orthogonal �⇥�
k
|⇥̂(⇤ + 2k�)|2 = 1�� �n, ⇥ ⇤ ⇥(n) = �[n]
Detail Filter Constraint
![Page 54: Signal Processing Course : Wavelets](https://reader033.vdocument.in/reader033/viewer/2022052209/556aead7d8b42a86218b4d09/html5/thumbnails/54.jpg)
{�(·� n)}n orthogonal �⇥�
k
|⇥̂(⇤ + 2k�)|2 = 1�� �n, ⇥ ⇤ ⇥(n) = �[n]
Detail Filter Constraint
![Page 55: Signal Processing Course : Wavelets](https://reader033.vdocument.in/reader033/viewer/2022052209/556aead7d8b42a86218b4d09/html5/thumbnails/55.jpg)
{�(·� n)}n orthogonal �⇥�
k
|⇥̂(⇤ + 2k�)|2 = 1�� �n, ⇥ ⇤ ⇥(n) = �[n]
Detail Filter Constraint
![Page 56: Signal Processing Course : Wavelets](https://reader033.vdocument.in/reader033/viewer/2022052209/556aead7d8b42a86218b4d09/html5/thumbnails/56.jpg)
{�(·� n)}n orthogonal �⇥�
k
|⇥̂(⇤ + 2k�)|2 = 1�� �n, ⇥ ⇤ ⇥(n) = �[n]
Detail Filter Constraint
![Page 57: Signal Processing Course : Wavelets](https://reader033.vdocument.in/reader033/viewer/2022052209/556aead7d8b42a86218b4d09/html5/thumbnails/57.jpg)
{�(·� n)}n orthogonal �⇥�
k
|⇥̂(⇤ + 2k�)|2 = 1�� �n, ⇥ ⇤ ⇥(n) = �[n]
Detail Filter Constraint
![Page 58: Signal Processing Course : Wavelets](https://reader033.vdocument.in/reader033/viewer/2022052209/556aead7d8b42a86218b4d09/html5/thumbnails/58.jpg)
Vanishing Moment Constraint
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Vanishing Moment Constraint
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Vanishing Moment Constraint
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Daubechies Family
![Page 62: Signal Processing Course : Wavelets](https://reader033.vdocument.in/reader033/viewer/2022052209/556aead7d8b42a86218b4d09/html5/thumbnails/62.jpg)
Daubechies Family
![Page 63: Signal Processing Course : Wavelets](https://reader033.vdocument.in/reader033/viewer/2022052209/556aead7d8b42a86218b4d09/html5/thumbnails/63.jpg)
Overview
•Review : Fourier transforms
•1-D Multiresolutions
•1-D Wavelet Transform
•Filter Constraints
•2-D Multiresolutions
![Page 64: Signal Processing Course : Wavelets](https://reader033.vdocument.in/reader033/viewer/2022052209/556aead7d8b42a86218b4d09/html5/thumbnails/64.jpg)
Anisotropic Wavelet Transform
![Page 65: Signal Processing Course : Wavelets](https://reader033.vdocument.in/reader033/viewer/2022052209/556aead7d8b42a86218b4d09/html5/thumbnails/65.jpg)
Anisotropic Wavelet Transform
![Page 66: Signal Processing Course : Wavelets](https://reader033.vdocument.in/reader033/viewer/2022052209/556aead7d8b42a86218b4d09/html5/thumbnails/66.jpg)
Anisotropic Wavelet Transform
![Page 67: Signal Processing Course : Wavelets](https://reader033.vdocument.in/reader033/viewer/2022052209/556aead7d8b42a86218b4d09/html5/thumbnails/67.jpg)
Anisotropic Wavelet Transform
![Page 68: Signal Processing Course : Wavelets](https://reader033.vdocument.in/reader033/viewer/2022052209/556aead7d8b42a86218b4d09/html5/thumbnails/68.jpg)
Anisotropic Wavelet Transform
![Page 69: Signal Processing Course : Wavelets](https://reader033.vdocument.in/reader033/viewer/2022052209/556aead7d8b42a86218b4d09/html5/thumbnails/69.jpg)
2D Multi-resolutions
![Page 70: Signal Processing Course : Wavelets](https://reader033.vdocument.in/reader033/viewer/2022052209/556aead7d8b42a86218b4d09/html5/thumbnails/70.jpg)
2D Multi-resolutions
![Page 71: Signal Processing Course : Wavelets](https://reader033.vdocument.in/reader033/viewer/2022052209/556aead7d8b42a86218b4d09/html5/thumbnails/71.jpg)
2D Multi-resolutions
![Page 72: Signal Processing Course : Wavelets](https://reader033.vdocument.in/reader033/viewer/2022052209/556aead7d8b42a86218b4d09/html5/thumbnails/72.jpg)
2D Wavelet Basis
![Page 73: Signal Processing Course : Wavelets](https://reader033.vdocument.in/reader033/viewer/2022052209/556aead7d8b42a86218b4d09/html5/thumbnails/73.jpg)
Discrete 2D Wavelets Coefficients
![Page 74: Signal Processing Course : Wavelets](https://reader033.vdocument.in/reader033/viewer/2022052209/556aead7d8b42a86218b4d09/html5/thumbnails/74.jpg)
Discrete 2D Wavelets Coefficients
![Page 75: Signal Processing Course : Wavelets](https://reader033.vdocument.in/reader033/viewer/2022052209/556aead7d8b42a86218b4d09/html5/thumbnails/75.jpg)
Discrete 2D Wavelets Coefficients
![Page 76: Signal Processing Course : Wavelets](https://reader033.vdocument.in/reader033/viewer/2022052209/556aead7d8b42a86218b4d09/html5/thumbnails/76.jpg)
Discrete 2D Wavelets Coefficients
![Page 77: Signal Processing Course : Wavelets](https://reader033.vdocument.in/reader033/viewer/2022052209/556aead7d8b42a86218b4d09/html5/thumbnails/77.jpg)
Discrete 2D Wavelets Coefficients
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Examples of Decompositions
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Separable vs. Isotropic
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Fast 2D Wavelet Transform
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Fast 2D Wavelet Transform
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Fast 2D Wavelet Transform
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Fast 2D Wavelet Transform
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Inverse 2D Wavelet Transform
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Inverse 2D Wavelet Transform
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Conclusion
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Conclusion
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Conclusion