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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
ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 20112
Digital Image Processing
Image Transforms
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• 2D Orthogonal and Unitary Transform:– Orthogonal Series Expansion:
– {ak,l(m,n)}: a set of complete orthonormal basis:– Orthonormality:– Completeness:
ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 20113
Digital Image Processing
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• 2D Orthogonal and Unitary Transform:– v (m,n): Transformed coefficients– V={v (m,n)}: Transformed Image– Orthonormality requires:
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• 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!
ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 20115
Digital Image Processing
Image Transforms
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• Separable Unitary Transform:
– Basis Images:
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E. Fatemizadeh, Sharif University of Technology, 20116
Digital Image Processing
Image Transforms
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• Properties of Unitary Transform:
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E. Fatemizadeh, Sharif University of Technology, 20117
Digital Image Processing
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• Two Dimensional Fourier Transform:
• Matrix Notation:
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E. Fatemizadeh, Sharif University of Technology, 20118
Digital Image Processing
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• DFT Properties:– Symmetric Unitary– Periodic Extension– Sampled Fourier– Fast– Conjugate Symmetry– Circular Convolution
ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 20119
Digital Image Processing
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• Basis of DFT (Real and Imaginary):
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E. Fatemizadeh, Sharif University of Technology, 201110
Digital Image Processing
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• Basis of DFT (Real and Imaginary):
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E. Fatemizadeh, Sharif University of Technology, 201111
Digital Image Processing
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• Discrete Cosine Transform (DCT):– 1D Cases:
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E. Fatemizadeh, Sharif University of Technology, 201112
Digital Image Processing
Image Transforms
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• 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
ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 201113
Digital Image Processing
Image Transforms
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• DCT Basis:
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E. Fatemizadeh, Sharif University of Technology, 201114
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ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 201115
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• DCT Basis:
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E. Fatemizadeh, Sharif University of Technology, 201116
Digital Image Processing
Image Transforms
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• Discrete Sine Transform (DST):– 1D Cases:
– 2D Case: A=A*=AT=Ψ
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E. Fatemizadeh, Sharif University of Technology, 201117
Digital Image Processing
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ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 201118
Digital Image Processing
Image Transforms
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• 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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E. Fatemizadeh, Sharif University of Technology, 201119
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• Welsh‐Hadamard Transform (WHT): N=2n
– 1D Cases:
Walsh Function
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E. Fatemizadeh, Sharif University of Technology, 201120
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• Welsh‐Hadamard Transform (WHT): N=2n
– 2D Cases: A=A*=AT=H
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E. Fatemizadeh, Sharif University of Technology, 201121
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• Welsh‐Hadamard Transform Properties:– Real, Symmetric, and orthogonal: H=H*=HT= H-1
– Ultra Fast Transform (±1)– Good‐Very Good energy compactness
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• Welsh‐Hadamard Basis:
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• Welsh‐Hadamard Basis
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• Welsh‐Hadamard Basis
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• Haar Transform N=2n
– 1D Cases:
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E. Fatemizadeh, Sharif University of Technology, 201126
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• Haar Transform N=2n
– 1D Cases:
– 2D Cases: Hr*A*HrT
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• Haar Basis Function
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• Haar Transform Properties– Real and Orthogonal: Hr=Hr* , Hr-1=HrT
– Fast Transform– Poor energy compactness
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• Slant Transform (N=2n)
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• Slant Transform (N=2n)
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• Slant Basis Function
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• Slant Transform Properties:– Real and Orthogonal S=S* S-1=ST
– Fast– Very Good Compactness
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