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E. Fatemizadeh, Sharif University of Technology, 2011
1
Digital Image Processing
Image Transforms
1
Image Transforms
Digital Image Processing
Fundamentals of Digital Image Processing, A. K. Jain
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E. Fatemizadeh, Sharif University of Technology, 2011
2
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:
1 1
,
0 0
1 1*
,
0 0
, , , 0 , 1
, , , 0 , 1
N N
k l
m n
N N
k l
k l
v k l u m n a m n k l N
u m n v k l a m n m n N
1 1*
, ,
0 0
1 1*
, ,
0 0
, , ,
, , ,
N N
k l k l
m n
N N
k l k l
k l
a m n a m n k k l l
a m n a m n m m n n
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E. Fatemizadeh, Sharif University of Technology, 2011
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Digital Image Processing
Image Transforms
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• 2D Orthogonal and Unitary Transform:
– v (m,n): Transformed coefficients
– V={v (m,n)}: Transformed Image
– Orthonormality requires:
11*
, ,
0 0
21 1
,ˆ
0 0
, , , ,
ˆ, arg min , ,
QP
P Q k l
k l
N N
P Qu
m n
u m n v k l a m n P N Q N
u m n u m n u m n
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E. Fatemizadeh, Sharif University of Technology, 2011
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Digital Image Processing
Image Transforms
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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!
,
One Dimensional Complete Orthonorma
, ,
l B s: i
,
, as
k l k l
k l
a m n a m b n a k m b l n
a m b n
* *T
TA A A A I
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E. Fatemizadeh, Sharif University of Technology, 2011
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Digital Image Processing
Image Transforms
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• Separable Unitary Transform:
– Basis Images:
1 1
0 0
1 1* *
0 0
, , , ,
, , , ,
N N
m n
N N
k l
v k l a k m u m n a l n
u m n a k m v m n a l n
TTT
T* *
V = AUA A AU
U = A VA
*
* *
1 12
0 0
, : Linear Combination of N mat ices
:
r
k
T
k l
N
k l
h
N
t
v
k Colum
l
n
k
T*
*
k,l
*
k,l
A a
A a a
U A
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Digital Image Processing
Image Transforms
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• Properties of Unitary Transform:
1 1 1 1
2 2
0 0 0 0
, ,N N N N
m n k l
u m n v k l
2 2v = Au v = u
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E. Fatemizadeh, Sharif University of Technology, 2011
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Digital Image Processing
Image Transforms
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• Two Dimensional Fourier Transform:
• Matrix Notation:
1 1
0 0
1 1
20 0
1 1
0 0
1 1
0 0
2, , , exp
1, ,
1, ,
1,
Unitary DFT Pai
,
r
N Nkm ln
N N N
m n
N Nkm ln
N N
k l
N Nkm ln
N N
m n
N Nkm ln
N N
k l
jv k l u m n W W W
N
u m n v k l W WN
v k l u m n W WN
u m n v k l W WN
1
, 0
1N
kn
N
k n
WN
* *V = FUF U = F VF
F =
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E. Fatemizadeh, Sharif University of Technology, 2011
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Digital Image Processing
Image Transforms
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• DFT Properties:
– Symmetric Unitary
– Periodic Extension
– Sampled Fourier
– Fast
– Conjugate Symmetry
– Circular Convolution
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E. Fatemizadeh, Sharif University of Technology, 2011
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Digital Image Processing
Image Transforms
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• Basis of DFT (Real and Imaginary):
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Digital Image Processing
Image Transforms
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• Basis of DFT (Real and Imaginary):
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Digital Image Processing
Image Transforms
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• Discrete Cosine Transform (DCT):
– 1D Cases:
1
0
1
0
,
10, 0 1
,2 12
cos 1 1, 0 12
2 1cos , 0 1
2
1 20 , , 1 1
2 1cos , 0 1
2
N
n
N
k
c k n
k n NN
c k nn k
k N n NNN
n kv k k u n k N
N
k k NN N
n ku n k v k n N
N
C
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E. Fatemizadeh, Sharif University of Technology, 2011
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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
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E. Fatemizadeh, Sharif University of Technology, 2011
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Digital Image Processing
Image Transforms
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• DCT Basis:
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Digital Image Processing
Image Transforms
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E. Fatemizadeh, Sharif University of Technology, 2011
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Digital Image Processing
Image Transforms
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• DCT Basis:
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E. Fatemizadeh, Sharif University of Technology, 2011
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Digital Image Processing
Image Transforms
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• Discrete Sine Transform (DST):
– 1D Cases:
– 2D Case: A=A*=AT=Ψ
1
0
1
0
,
1 12, sin , 0 , 1
1 1
1 12sin , 0 1
1 1
1 12sin , 0 1
1 1
N
n
N
k
k n
n kk n k n N
N N
n kv k u n k N
N N
n ku n v k n N
N N
ψ
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E. Fatemizadeh, Sharif University of Technology, 2011
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Digital Image Processing
Image Transforms
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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, 2011
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Digital Image Processing
Image Transforms
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• Welsh-Hadamard Transform (WHT): N=2n
– 1D Cases:
1
1 1
1 1 1 1
1 1
3
1 11
1 12
1
2
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 11
1 1 1 1 1 1 1 18
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1
0
7
3
4
1 1 1
6
2
51
1
n n
n n n
n n
H
H HH H H H H
H H
H
Walsh Function
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E. Fatemizadeh, Sharif University of Technology, 2011
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Digital Image Processing
Image Transforms
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• Welsh-Hadamard Transform (WHT): N=2n
– 2D Cases: A=A*=AT=H
1,
0
1,
0
1
0
1 1
0 0
, , 2
11 0 1
11 0 1
, , 0,1
2 , 2
n
Nb k m
m
Nb k m
k
n
i i i i
i
n ni i
i i
i i
N
v k u m k NN
u m v k m NN
b k m k m k m
k k m m
nv = Hu u = Hv H = H
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E. Fatemizadeh, Sharif University of Technology, 2011
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Digital Image Processing
Image Transforms
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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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E. Fatemizadeh, Sharif University of Technology, 2011
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Digital Image Processing
Image Transforms
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• Welsh-Hadamard Basis:
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E. Fatemizadeh, Sharif University of Technology, 2011
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Digital Image Processing
Image Transforms
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• Welsh-Hadamard Basis
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E. Fatemizadeh, Sharif University of Technology, 2011
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Digital Image Processing
Image Transforms
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• Welsh-Hadamard Basis
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Digital Image Processing
Image Transforms
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• Haar Transform N=2n
– 1D Cases:
0 0,0
2
2
,
, 0,1 , 0, , 1
2 1
0 1; 0,1 0
1 2 0
1, 0,1
1 1 22
2 2
1 1 22
2 2
0 O.W.
, 0,1, , 1
k
p
p
p
p p
p
k p q p p
h x x k N
k q
p n q p
q p
h x h x xN
q qx
q qh x h x x
N
mx m N
N
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E. Fatemizadeh, Sharif University of Technology, 2011
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Digital Image Processing
Image Transforms
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• Haar Transform N=2n
– 1D Cases:
– 2D Cases: Hr*A*HrT
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
2 2 2 2 0 0 0 0
1 0 0 0 0 2 2 2 2
8 2 2 0 0 0 0 0 0
0 0 2 2 0 0 0 0
0 0 0 0 2 2 0 0
0 0 0 0 0 0 2 2
0
1
2
2
2
2
2
2
rH
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E. Fatemizadeh, Sharif University of Technology, 2011
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Digital Image Processing
Image Transforms
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• Haar Basis Function
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Digital Image Processing
Image Transforms
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• Haar Transform Properties
– Real and Orthogonal: Hr=Hr* , Hr-1=HrT
– Fast Transform
– Poor energy compactness
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Digital Image Processing
Image Transforms
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• Slant Transform (N=2n)
1
2 2 2 2 1 2
2
2 1
2 2 2 2
1 11S
1 12
1 0 1 0
01
0 1 0 1 02
a b a b SS
S
b a b a
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E. Fatemizadeh, Sharif University of Technology, 2011
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Digital Image Processing
Image Transforms
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12
( / 2) 2 ( / 2) 2 1
1
( / 2) 2 ( / 2) 2
1 12 22 2
1 12 2
1 0 1 00 0
0 0 01
20 1 0 1 0
0 0
0 0
3 12 , ,
4 1 4 1
n n n n
N N n
n
n
n n n n
N N
n
n n
a b a b
I I S
S
S
b a b a
I I
N NN a b
N N
• Slant Transform (N=2n)
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E. Fatemizadeh, Sharif University of Technology, 2011
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Digital Image Processing
Image Transforms
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• Slant Basis Function
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Digital Image Processing
Image Transforms
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• Slant Transform Properties:
– Real and Orthogonal S=S* S-1=ST
– Fast
– Very Good Compactness