No Slide Title - Sharifee.sharif.edu/~dip/Files/DIPTransformForView.pdfee.sharif.edu/~dip E....

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E. Fatemizadeh, Sharif University of Technology, 2011

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Digital Image Processing

Image Transforms

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Image Transforms


Fundamentals of Digital Image Processing, A. K. Jain

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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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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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• 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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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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• 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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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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Image Transforms

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• DFT Properties:

– Symmetric Unitary

– Periodic Extension

– Sampled Fourier

– Fast

– Conjugate Symmetry

– Circular Convolution

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Image Transforms

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• Basis of DFT (Real and Imaginary):

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Image Transforms

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• Basis of DFT (Real and Imaginary):

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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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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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Image Transforms

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• DCT Basis:

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Image Transforms

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Image Transforms

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• DCT Basis:

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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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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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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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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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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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Image Transforms

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• Welsh-Hadamard Basis:

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Image Transforms

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• Welsh-Hadamard Basis

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Image Transforms

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• Welsh-Hadamard Basis

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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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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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• 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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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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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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• 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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