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ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 20111
Digital Image Processing
Image Transforms
1
Image Transforms
Digital Image ProcessingFundamentals of Digital Image
Processing, A. K. Jain
![Page 2: Image Transforms - Sharif University of Technology ...ee.sharif.edu/~dip/Files/DIPTransformForPrint.pdf · ee.sharif.edu/~dip E. Fatemizadeh, Sharif University of Technology, 2011](https://reader030.vdocument.in/reader030/viewer/2022021420/5acd92d67f8b9ab10a8dd5c4/html5/thumbnails/2.jpg)
ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 20112
Digital Image Processing
Image Transforms
2
• 2D Orthogonal and Unitary Transform:– Orthogonal Series Expansion:
– {ak,l(m,n)}: a set of complete orthonormal basis:– Orthonormality:– Completeness:
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ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 20113
Digital Image Processing
Image Transforms
3
• 2D Orthogonal and Unitary Transform:– v (m,n): Transformed coefficients– V={v (m,n)}: Transformed Image– Orthonormality requires:
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ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 20114
Digital Image Processing
Image Transforms
4
• Separable Unitary Transform:– Computational Complexity of former: O(N4)– With Separable Transform: O(N3)
– Orthonormality and Completeness:• A={a(k,m)} and B={b(l,n)} are unitary:
– Usually B is selected same as A (A=B):
– Unitary Transform!
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ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 20115
Digital Image Processing
Image Transforms
5
• Separable Unitary Transform:
– Basis Images:
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ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 20116
Digital Image Processing
Image Transforms
6
• Properties of Unitary Transform:
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ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 20117
Digital Image Processing
Image Transforms
7
• Two Dimensional Fourier Transform:
• Matrix Notation:
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ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 20118
Digital Image Processing
Image Transforms
8
• DFT Properties:– Symmetric Unitary– Periodic Extension– Sampled Fourier– Fast– Conjugate Symmetry– Circular Convolution
![Page 9: Image Transforms - Sharif University of Technology ...ee.sharif.edu/~dip/Files/DIPTransformForPrint.pdf · ee.sharif.edu/~dip E. Fatemizadeh, Sharif University of Technology, 2011](https://reader030.vdocument.in/reader030/viewer/2022021420/5acd92d67f8b9ab10a8dd5c4/html5/thumbnails/9.jpg)
ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 20119
Digital Image Processing
Image Transforms
9
• Basis of DFT (Real and Imaginary):
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ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 201110
Digital Image Processing
Image Transforms
10
• Basis of DFT (Real and Imaginary):
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ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 201111
Digital Image Processing
Image Transforms
11
• Discrete Cosine Transform (DCT):– 1D Cases:
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ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 201112
Digital Image Processing
Image Transforms
12
• Properties of DCT:– Real and Orthogonal: C=C* → C-1=CT
– Not! Real part of DFT– Fast Transform– Excellent Energy compaction (Highly Correlated Data)
• Two Dimensional Cases:– A=A*=C
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ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 201113
Digital Image Processing
Image Transforms
13
• DCT Basis:
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ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 201114
Digital Image Processing
Image Transforms
14
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ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 201115
Digital Image Processing
Image Transforms
15
• DCT Basis:
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ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 201116
Digital Image Processing
Image Transforms
16
• Discrete Sine Transform (DST):– 1D Cases:
– 2D Case: A=A*=AT=Ψ
![Page 17: Image Transforms - Sharif University of Technology ...ee.sharif.edu/~dip/Files/DIPTransformForPrint.pdf · ee.sharif.edu/~dip E. Fatemizadeh, Sharif University of Technology, 2011](https://reader030.vdocument.in/reader030/viewer/2022021420/5acd92d67f8b9ab10a8dd5c4/html5/thumbnails/17.jpg)
ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 201117
Digital Image Processing
Image Transforms
17
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ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 201118
Digital Image Processing
Image Transforms
18
• Properties of DST:– Real, Symmetric and Orthogonal: Ψ = Ψ*= ΨT=Ψ-1
– Forward and Inverse are identical– Not! Imaginary part of DFT– Fast Transform
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ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 201119
Digital Image Processing
Image Transforms
19
• Welsh‐Hadamard Transform (WHT): N=2n
– 1D Cases:
Walsh Function
![Page 20: Image Transforms - Sharif University of Technology ...ee.sharif.edu/~dip/Files/DIPTransformForPrint.pdf · ee.sharif.edu/~dip E. Fatemizadeh, Sharif University of Technology, 2011](https://reader030.vdocument.in/reader030/viewer/2022021420/5acd92d67f8b9ab10a8dd5c4/html5/thumbnails/20.jpg)
ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 201120
Digital Image Processing
Image Transforms
20
• Welsh‐Hadamard Transform (WHT): N=2n
– 2D Cases: A=A*=AT=H
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ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 201121
Digital Image Processing
Image Transforms
21
• Welsh‐Hadamard Transform Properties:– Real, Symmetric, and orthogonal: H=H*=HT= H-1
– Ultra Fast Transform (±1)– Good‐Very Good energy compactness
![Page 22: Image Transforms - Sharif University of Technology ...ee.sharif.edu/~dip/Files/DIPTransformForPrint.pdf · ee.sharif.edu/~dip E. Fatemizadeh, Sharif University of Technology, 2011](https://reader030.vdocument.in/reader030/viewer/2022021420/5acd92d67f8b9ab10a8dd5c4/html5/thumbnails/22.jpg)
ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 201122
Digital Image Processing
Image Transforms
22
• Welsh‐Hadamard Basis:
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ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 201123
Digital Image Processing
Image Transforms
23
• Welsh‐Hadamard Basis
![Page 24: Image Transforms - Sharif University of Technology ...ee.sharif.edu/~dip/Files/DIPTransformForPrint.pdf · ee.sharif.edu/~dip E. Fatemizadeh, Sharif University of Technology, 2011](https://reader030.vdocument.in/reader030/viewer/2022021420/5acd92d67f8b9ab10a8dd5c4/html5/thumbnails/24.jpg)
ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 201124
Digital Image Processing
Image Transforms
24
• Welsh‐Hadamard Basis
![Page 25: Image Transforms - Sharif University of Technology ...ee.sharif.edu/~dip/Files/DIPTransformForPrint.pdf · ee.sharif.edu/~dip E. Fatemizadeh, Sharif University of Technology, 2011](https://reader030.vdocument.in/reader030/viewer/2022021420/5acd92d67f8b9ab10a8dd5c4/html5/thumbnails/25.jpg)
ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 201125
Digital Image Processing
Image Transforms
25
• Haar Transform N=2n
– 1D Cases:
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ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 201126
Digital Image Processing
Image Transforms
26
• Haar Transform N=2n
– 1D Cases:
– 2D Cases: Hr*A*HrT
![Page 27: Image Transforms - Sharif University of Technology ...ee.sharif.edu/~dip/Files/DIPTransformForPrint.pdf · ee.sharif.edu/~dip E. Fatemizadeh, Sharif University of Technology, 2011](https://reader030.vdocument.in/reader030/viewer/2022021420/5acd92d67f8b9ab10a8dd5c4/html5/thumbnails/27.jpg)
ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 201127
Digital Image Processing
Image Transforms
27
• Haar Basis Function
![Page 28: Image Transforms - Sharif University of Technology ...ee.sharif.edu/~dip/Files/DIPTransformForPrint.pdf · ee.sharif.edu/~dip E. Fatemizadeh, Sharif University of Technology, 2011](https://reader030.vdocument.in/reader030/viewer/2022021420/5acd92d67f8b9ab10a8dd5c4/html5/thumbnails/28.jpg)
ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 201128
Digital Image Processing
Image Transforms
28
• Haar Transform Properties– Real and Orthogonal: Hr=Hr* , Hr-1=HrT
– Fast Transform– Poor energy compactness
![Page 29: Image Transforms - Sharif University of Technology ...ee.sharif.edu/~dip/Files/DIPTransformForPrint.pdf · ee.sharif.edu/~dip E. Fatemizadeh, Sharif University of Technology, 2011](https://reader030.vdocument.in/reader030/viewer/2022021420/5acd92d67f8b9ab10a8dd5c4/html5/thumbnails/29.jpg)
ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 201129
Digital Image Processing
Image Transforms
29
• Slant Transform (N=2n)
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ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 201130
Digital Image Processing
Image Transforms
30
• Slant Transform (N=2n)
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ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 201131
Digital Image Processing
Image Transforms
31
• Slant Basis Function
![Page 32: Image Transforms - Sharif University of Technology ...ee.sharif.edu/~dip/Files/DIPTransformForPrint.pdf · ee.sharif.edu/~dip E. Fatemizadeh, Sharif University of Technology, 2011](https://reader030.vdocument.in/reader030/viewer/2022021420/5acd92d67f8b9ab10a8dd5c4/html5/thumbnails/32.jpg)
ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 201132
Digital Image Processing
Image Transforms
32
• Slant Transform Properties:– Real and Orthogonal S=S* S-1=ST
– Fast– Very Good Compactness