Introductory Image Processing - KOCW

(1)

제 4 장 영상의 주파수 변환

[email protected]

Prof. Doo-Hyun Choi

Introductory Image Processing

Introductory image Processing IISL, School of EECS, KNU



영상의 주파수 변환

 Fourier Transform

 Fourier Transform

 Discrete Fourier Transform

 Properties of 2-D DFT

 Fast Fourier Transform

 Fourier Transform Application

 Inverse Fourier Transform

 Discrete Cosine Transform

 Discrete Cosine Transform

 DCT Application

 Discrete Sine Transform

 Walsh Transform

 Hadamard Transform

 Harr Transform

 Karhunen-Loeve Transform

 Wavelet Transform

Contents

(2)

Introductory image Processing Lecture Note 04

IISL, School of EECS, KNU (3/38)



Fourier Transform Pair

 Fourier transform of f (x) :

 Inverse Fourier transform:

(exist if f (x) : continuous and integrable fn. of a real variable x , F (u) : integrable)

 Euler’s formula:

∴ F (u) : an infinite sum of sine and cosine terms

Fourier Transform (1/8)

{ } ( ) = ( ) = ( )

²

where = − 1

ℑ f x F u 

₋^∞_∞

f x e

⁻^j ^π^ux

dx j { F u } ( ) f x −∞∞F u ej uxdu

−

= =

ℑ

¹

( ) ( )

²^π

) 2 sin(

) 2

2

cos( ux j ux

e

⁻^j ^π^ux

= π − π



Fourier Transform

 For f (x) , a real function,

where R(u) : real component, I(u) : imaginary component, u : frequency variable

 Magnitude(진폭, Fourier) spectrum of f (x) :

 Phase(위상) angle:

 Power(전력) spectrum (spectral density) of f (x) :

)

(

( ) ( ) ( )

( u R u jI u F u e

^j ^u

F = + =

^φ

) ( ) ( )

( u R

²

u I

²

u

F = +

 

 



=

⁻



) (

) tan (

)

(

¹

u R

u u I

φ

) ( ) ( )

( )

( u F u

²

R

²

u I

²

u

P = = +

(3)



Example: A simple function and its Fourier spectrum

 Fourier transform of f (x) :

 Fourier spectrum:

[ ] [ ]

[ ]

( )

^j ^uX

uX j uX j uX j

uX X j

ux X j ux j

ux j

e u uX

A

e e

u e j

A

u e j e A

u j dx A Ae

dx e

x f u

F

π

π π π

π π

π π π

π π

−

∞ −

∞

−

=

−

=

− −

− =

=

=  

sin 2

2 1 ) 2

( )

(

² ₀ ²

0

2 2

( ) ( )

( uX uX )

AX e

u uX u A

F

^j ^uX

π π π

π

^π

sin sin )

( =

⁻

=



Two-Dimensional Fourier Transform Pair

 Fourier transform of f (x,y):

 Inverse Fourier transform:

(exist if f (x,y) : continuous and integrable fn. of a real variable x,y , F (u,v) : integrable)

{ } ( ) = =  −∞∞ −∞∞ − ( + )

ℑ f ( x , y ) F u , v f ( x , y ) e

^j²^π ^ux ^vy

dxdy

{ } ( )  −∞∞ −∞∞ ( + )

−

= =

ℑ

¹

F ( u , v ) f x , y F ( u , v ) e

^j²^π ^ux ^vy

dudv

(4)



Fourier Transform

 For f (x,y) , a real function,

where R(u,v) : real component, I(u,v) : imaginary component, u,v : frequency variables

 Magnitude(진폭, Fourier) spectrum of f (x,y) :

 Phase(위상) angle:

 Power(전력) spectrum (spectral density) of f (x,y) :

) , ( ) , ( )

,

( u v R

²

u v I

²

u v

F = +

 

 



=

⁻



) , (

) , tan (

) ,

(

¹

v u R

v u v I

φ u

) , ( ) , ( )

, ( ) ,

( u v F u v

²

R

²

u v I

²

u v

P = = +

) ,

)

(

, ( ) , ( ) , ( ) ,

( u v R u v jI u v F u v e

^j ^u^v

F = + =

^φ



Example: A simple 2-D function and its Fourier spectrum (1/2)

Fourier Transform (6/8)

(5)



Example: A simple 2-D function and its Fourier spectrum (2/2)

 Fourier transform of f (x,y) :

 Fourier spectrum:

( ) ( )

[ ] [ ]

( )

( ) ( )

( )  

 



 



 



= 

− −

 =



 





 −



 





= −

=

−

− −

−

∞ −

∞

−

∞

−

+

−

 

vY e vY uX

e AXY uX

v e e j

u j

A v

j e u j A e

dy e

dx e

A dxdy e

y x f v

u F

vY j uX

j

vY j uX

j vy Y

X j ux j

X j ux Y j vy

vy ux j

π π π

π

π π

π π π

π

π π

π

sin sin

2 1 1 1 2

2 2

) , ( ,

2 2

0 2

0 0

2 2

2

( ) ( )

( ) vY vY e

uX e AXY uX

v u F

vY j uX

j

π π π

π

⁻ ^π ⁻ ^π

= sin sin

,



Example: Some 2-D functions and their Fourier spectra

(6)



Sampling of a continuous function

f

(

x

)

Discrete Fourier Transform (1/4)

{ }

{ ( 0 ), ( 1 ), ( 2 ), , ( 1 ) }

) ( is, that , ) (

) (

) ] 1 [ ( , ), 2 ( ), (

), (

0

0 0

−

∈ Δ

+

 =

Δ

− + Δ

+ Δ

+

N f f

f f x f x

x x f x f

x N

x f x x

f x x f x f



Discrete Fourier Transform (2/4)

 Discrete Fourier Transform Pair

 Discrete Fourier transform:

 Inverse Fourier transform

 Proof

( )

value average

: ) 1 (

) 0 (

1 ,) ( ) ( , 1 , , 1 , 0 for )

1 (

1

0

/ 1 2

0



−

=

− −

=

= Δ Δ

=

−

=

N x

N ux N j

x

x N f

F

x Δu N

u u F u F N

u e

x N f

u

F

^π



( )

¹

( )

² ^/

for 0 , 1 , , 1

0

−

=

= −

=

N x

e u F x

f

^j ^ux ^N

N

u

π



( )

{ }

[ ] =    =

=

 





 



= 

=



 



−

=

−

=

− −

=

− −

=

−

=

− −

=

otherwise.

0 if

since ) (

) 1 (

1

0

/ 2 / 2 1

0

/ 2 / 1 2

0

/ 1 2

0

/ 1 2

0 /

1 2 0

u r e N

e u

F

e e

r N F

e e

r N F

e x N f

u F

N x

N ux j N rx j N

x

N ux j N rx N j

r

N ux N j

x

N rx N j

r N

ux N j

x

π π

π

(7)

Discrete Fourier Transform (3/4)

 Two-Dimensional Discrete Fourier Transform Pair

 If M = N ,

( ) ( )

 

 





= Δ

= Δ Δ

+ Δ +

=

−

=

−

=

−

=

−

=

 

−

=

−

=

+

−

=

−

=

+

−

y Δv N

x Δu M

y y y x x x f y x f

N y

M x

e v u F y

x f

N v

M u

e y x MN f

v u F

M u

N v

N vy M ux j M

x N y

N vy M ux j

1 and 1

,) ,

( ) , (

1 , , 1 , 0 , 1 , , 1 , 0 for )

, ( ,

1 , , 1 , 0 , 1 , , 1 , 0 for )

, 1 (

,

0 0

1

0 1

0

/ / 2 1

0 1

0

/ / 2



π π

( ) ( )

( ) , 1 ( , ) ( ) for , 0 , 1 , , 1

1 , , 1 , 0 , for )

, 1 (

,

1

0 1

0

/ 2

1

0 1

0

/ 2

−

=

−

=

 

−

=

−

=

+

−

=

−

=

+

−

N y

x e

v u N F

y x f

N v

u e

y x N f

v u F

N u

N v

N vy ux j N

x N

y

N vy ux j



π π

Discrete Fourier Transform (4/4)

 Example

( ) [ ] [ ]

( ) [ ]

( ) 3 4 1 ( ) 4 1 [ 2 3 4 4 ] 4 1 [ ] 2 F(3) 4 5

4 F(2) 1

4 4 1

4 3 4 2 ) 1

4 ( 2 1

4 F(1) 5

4 2 4 1

4 3

4 2 ) 1

4 ( 1 1

4 F(0) 13 25 . 3 4 4 3 4 2 ) 1 3 ( ) 2 ( ) 1 ( ) 0 4 ( ) 1 4 (

) 1 1 (

0

2 / 9 3

2 / 3 0 4

/ 3 6

0

3 2

0 4

/ 3 4

0

2 / 3 2

/ 0

4 / 3 2

0

3 0 0 0 , 4 / 1 2

0

 = +

−

= +

+ +

=

 =

−

= +

+ +

=

 = +

−

= +

+ +

=

 =

= + + +

=

−

=

−

=

−

=

= =

=

− −

=



j e

e e

x f F

e e

e e e

x f F

j e

e e

x f F

f f f f e x f e

x N f F

j j

j x

j j

j x

j j j

x j x

u x N N ux N j

x

π π

π

(8)

Properties of the 2-D DFT (1/5)

 Dynamic range adjustment

[ 1 ( , ) ] where : a scaling constant

log )

,

( u v c F u v c

D = +

] 255 , 0 [ ] 10 25 . 0 , 0

[ ×

⁶



Properties of the 2-D DFT (2/5)

 Separability

 2-D Fourier transform  2 successive application of 1-D Fourier transform

( ) ( )

( )   

 

−

=

− −

=

− −

=

−

=

−

=

−

=

−

=

+

−

=

−

=

−

− −

=

−

=

+

−



 





 



 





 



= 

 =

−

=

−

=

1 0

/ 1 2

0

/ 1 2

0

/ 2

1 0

/ 2 /

1 2 0

1 0

/ 2

1 0

/ 2 /

1 2 0

1 0

/ 2

) , 1 (

) 1 , 1 (

,

1 , , 1 , 0 , for )

, 1 (

) , 1 (

,

1 , , 1 , 0 , for )

, 1 (

) , 1 (

,

N

x

N ux N j

y

N vy N j

x

N ux j

N

u

N

v

N vy j N

ux N j

u N

v

N vy ux j

N

x

N

y

N vy j N

ux N j

x N

y

N vy ux j

e e

y x N f

N N e

v x N F

v u F

N y

x e

v u F N e

e v u N F

y x f

N v

u e

y x f N e

e y x N f

v u F

π π

π

π π

π

π π

π



(9)

Properties of the 2-D DFT (3/5)

 Translation

( Proof )

(

^u^x ^v ^y

)

^N

( ) ( )

^j

(

^ux ^vy

)

^N

j

F u u v v f x x y y F u v e

e y x

f ( , )

²^π ⁰ ⁺ ⁰ ^/

⇔ −

₀

, −

₀

, −

₀

, −

₀

⇔ ( , )

⁻ ²^π ⁰⁺ ⁰ ^/

( )

{ } ( ) ( ) { }

{ } ( )

( )

{ }

) ( 2

) ( 2 )

( 2 )

( ) ( 2

0 0

0 2 0 0

0

0 0

) ( ) ( 2 2

2 2

0 0

0 0 0

0

0 0 0

0 0

0

) , (

) , ( )

, (

,

and

, ,

) , ( )

, (

) , (

) , ( )

, ( )

, (

vy ux j

vy ux j vt

us j y

t v x s u j

vy ux j

y v v x u u j vy

ux j y v x u j y

v x u j

e v u F

e dsdt e

t s f dsdt

e t s f

y t y dt dy t y y x

s x ds dx s x x

dxdy e

y y x x f y

y x x f

v v u u F

dxdy e

y x f dxdy

e e

y x f e

y x f

+

−

+

− +

∞ −

∞

−

∞

− +

+ +

∞ −

∞

−

∞

−

+

∞ −

∞

−

∞

−

∞

−

∞

−

− +

−

− +

∞ −

∞

−

∞

−

+ +

=

⋅

=

+

=

− +

=

−

=

−

− ℑ

−

=

= ℑ

 

π

π π

π

π π



( ) [ ( ) ] [ ( ) ] ( )

( ) 



 



 − −

⇔

−

=

−

= + +

+

=

+ +

+ + +

, 2 2

)

1 )(

, ( )

, (

1 sin

cos

2 ,

If

) ( /

2

) ( )

( /

0 2 0

0 0

v N u N

F y

x f e

y x f

y x j

y x e

N e v u

y x N

y v x u j

y x y

x j N y v x u j

π

π π

( ) ( , ) : A shift does not affect the magnitude )

,

( u v e

² ⁰ ⁰ ^/

F u v

F

⁻^j ^π ^u^x⁺^v ^y ^N

=

 Periodicity

) , ( ) , ( ) , ( ) ,

( u v F u N v F u v N F u N v N

F = + = + = + +

) , ( ) , ( and ) , ( ) , ( , real is

If f(x,y) F u v = F

^*

− u − v F u v = F − u − v

(10)

 Shuffling

n

m (0,0)

(M-1,N-1)

j

k A

D C

B

Fast Fourier Transform (1/8)

 Computational Complexity

 Direct Fourier transform: O ( N

²

)

 FFT: O ( Nlog

₂

N )

( )



 



−

=

−

− −



=

results the

of additions

1

by of tions multiplica complex

, of values

the of each For

) 1 (

/ 2 /

1 2 0

N

e u u N

N e x N f

u F

N ux j N

ux N j

x

π π

N N²(Direct FT) Nlog2N (FFT) N/log2N(Computational Advantage)

2 4 2 2.00

4 16 8 2.00

8 64 24 2.67

16 256 64 4.00

32 1,024 160 6.40

64 4,096 384 10.67

128 16,384 896 18.29

256 65,536 2,048 32.00

512 262,144 4,608 56.89

1024 1,048,576 10,240 102.40

2048 4,194,304 22,528 186.18

4096 16,777,216 49,152 341.33

(11)

Fast Fourier Transform (2/8)

 Successive doubling method

 An N -point Fourier transform  Two ( N/2 )-point Fourier transforms

( ) [ ]

( )

[ ]

( ) ( )

( ) 2 1 [ ( ) ( ) ] for 0 , 1 , 2 , , 1 0 , 1 , 2 , , 2 1

,

and

Since

1 , , 2 , 1 , 0 for ) 1 2 1 (

and ) 2 1 (

Defining

) ( ) 2 (

) 1 1 2 1 (

) 2 1 (

2 1

)

1 2 1 (

) 2 1 (

2 1

) 2 (

) 1 1 (

, 2 2 that Assume

/ 2 exp

where ) 1 (

2 2 2

1 0 1

0

2 2

1 0 1

0

22 ) 2 2( )

1 2 2( 1

0 )

2 2( 1

0

2 1

2 0 1

0 1 0

−

=

−

=

−

= +

−

=

−

= +

=

+

 =

 



 



 + +

=

 =

 





 



 + +

=

−

=

+ +

−

=

−

=

−

=

−

=

− +

=

−

=

−

=

−

=

−

=



N M

u W

u F u F M u F

W W

M u

W x M f

u F W

x M f

u F

W u F u F W

W x M f

W W

x M f

W x M f

W x N f

u F

M N

N j W

W x N f

u F

uM odd

even

uM M

uM Mu

M Mu

Mux M

x ux odd

M M

x even

uM odd

u even ux M M M

x Mux

M

x

Mux Mux

x uM x

uM M

x x

uM M

x

uxM M

x Nux

N

x

n ux N N N

x



 π

Fast Fourier Transform (3/8)

 Let m ( n ) and a ( n ): # of complex multiplications and additions (1/4)

 for n = 1, N = 2, M = 1

( ) [ ]

( ) ( F ) [ F F W ] Nn Nn

F

W F

F F F W

F F

F

M odd

even

M odd

odd even M

odd even

=

 ∴

−

= +

=

 

 

 + 

=

2 a(1) 2 , 1 1 m(1) addition one

) 0 ( ) 0 2 (

1 1 0 1

addition one

and ) ) 0 ( ( tion multiplica one

: ) 0 (

addition and

tion multiplica no

: ) 0 (

addition and

tion multiplica no

: ) 0 ( )

0 ( ) 0 2 (

0 1

20

20 20

(12)

Fast Fourier Transform (4/8)

 for n = 2, N = 4, M = 2

( ) ( ) ( ) ( )

Nn )

a(

) a(

Nn )

m(

) m(

F F F

F

a m

=

= +

=

= +

=

∴ , 2 2 1 4 8

2 4 1 2 1 2 2

additions

more 2 : 3 , 2 additions,

and tions multiplica

further

2 : 1 , 0

) 1 ( 2 ), 1 ( 2 : transform

point - 2 Two

 for n = 3, N = 8, M = 4

( ) ( ) ( ) ( )

Nn )

a(

) Nn, a(

) m(

F F

a m

=

= +

=

= +

=

∴ 3 2 2 8 24

2 12 1 4 2 2 3

additions

more 4 : 7 , , 4 additions,

and tions multiplica further

4 : 3 , , 0

) 2 ( 2 ), 2 ( 2 : nsform point tra -

4 Two



(13)

 For N -point FFT, n = log

₂

N, M = N/2

 Number of Operations

 (Proof by induction)

0 ) 0 ( and 0 ) 0 ( where 2 1 2 2

1

2 − +

¹

= − + = =

=

∴ m(n) m(n )

ⁿ⁻

, a(n) a(n )

ⁿ

m a

1 for log

2 log 2 ) ( 2 ,

log 1 2 2 1 log 2 2

) 1

( n =

₂

= N

₂

N = Nn a n =

₂

= N

₂

N = Nn n ≥

m

ⁿ ⁿ ⁿ ⁿ

( ) ( ) ( )

true.

is ) ( and 2

) 1 (

1 2

2 2 2 2 2

2 ) ( 2 ) 1 (

1 2 2

1 1 2 2 2 2

2 1 2 2

2 1 2 ) ( 2 ) 1 (

, 1 For

true.

is ) ( and 2

) 1 ( that Assume

. 2 ) 1 )(

2 ( ) 1 ( and 1 ) 1 )(

2 2 ( ) 1 1 ( , 1 For

1 1 1

1

Nn n a Nn n

m

n n

Nn n

a n

a

n n

n Nn

n m n

m n

Nn n a Nn n

m

a m

n

n n n n

n

n n

n n n

n

=

∴

+

= +

+

= +

 

 



=  +

 

 



=  +

= + +

=

+ + +

+

Fast Fourier Transform (7/8)

 Implementation

 Bit reversal

(14)

 2-D FFT Example

원영상 진폭 스펙트럼 위상

Fourier Transform Application

 Frequency in an image: variation of intensity levels

 Edges have relatively high frequency components

(15)

Inverse Fourier Transform

 Inverse FFT

 Inverse FFT = Take the complex conjugate and multiply by N

 2-D Inverse FFT

( ) ( )

( )

^N ^j ^ux ^N

x

N ux N j

x N

ux N j

x

e u N F

x N f

e u F x

f e

x N f

u F

/ 1 2

0

*

/ 1 2

0 /

1 2 0

) 1 (

1

) (

and )

1 (

π

π π

− −

=

−

=

− −

=



 =

=

( ) ( )

( ) , 1 * ( , ) ( ) for , 0 , 1 , , 1

1 , , 1 , 0 , for )

, 1 (

,

1 , , 1 , 0 , for )

, 1 (

,

1

0 1

0

/ 2

*

1

0 1

0

/ 2

1

0 1

0

/ 2

−

=

 =

−

=

−

=

 

−

=

−

=

+

−

=

−

=

+

−

=

−

=

+

−

N y

x e

v u N F

y x f

N y

x e

v u N F

y x f

N v

u e

y x N f

v u F

N u

N v

N vy ux j N

u N

v

N vy ux j N

x N y

N vy ux j



π π

π

Discrete Cosine Transform (1/4)

 1-D Discrete Cosine Transform Pair ( N = 2

ⁿ

)

 2-D DCT Pair ( N = 2

ⁿ

)

 Separable DCT kernels  2D DCT = successive applications of 1D DCT

( ) [ ]

( ) [ ]  



 



−

=

−

 ∈

 

 +

=

−

 ∈

 

 +

=



−

=

−

=

. 1 , , 2 , 1 2 for

0 1 for )

( where 1 , , 1 , 0 for 2

) 1 2 cos ( ) ( ) (

1 , , 1 , 0 for 2

) 1 2 cos ( )

( ) (

1 0

1

0

N N u

N u u

N N x

u u x

C u x

f

N N u

u x x

f u u C

N

u N

x

 

 π α

α α π

( ) [ ]

( ) , ( ) ( ) ( , ) cos ( 2 2 1 ) cos ( 2 2 1 ) for , [ 0 , 1 , , 1 ]

1 , , 1 , 0 , for 2

) 1 2 cos ( 2

) 1 2 cos ( , )

( ) ( ) , (

1 0

−

 ∈

 

 +

 

  +

=

−

 ∈

 

 +

 

  +

=

 

−

=

−

=

−

=

−

=

N y

N x v y N

u v x

u C v u y

x f

N v

N u v y N

u y x

x f v

u v u C

N

u N

v N

x N

y



 π

α π α

π α π

α

(16)

Discrete Cosine Transform (2/4)

 Implementation

( )

FFT point 2

2 :

exp 2 ) ( that Note 2

exp 2 ) 2 (

exp Re ) (

1 2 , , 1 , when 0

) (

and 1 , , 1 , 0 where

2 ) 1 2 exp ( ) ( Re

) (

2 ) 1 2 exp ( ) ( Re

) (

2 ) 1 2 cos ( )

( ) (

1 2

0 1

2

0 1

2

0 1

0

N N- ux x j

N f ux x j

N f u u j

N N

N x x

f

N u

N u x x j

f u

N u x x j

f u

N u x x

f u u

C

N

x N

x



−

=

−

=

−

=

−

=

−

=

 

 



 −

 



 

 



 



 −

 ⋅



 



 



 



 −

⋅

=

− +

=

−

=

 



 

 



 



 − +

⋅

=

 



 

 



 



 − +

⋅

=

 

  +

=

π π

α π α π α π α π



Discrete Cosine Transform (3/4)

 Example

( ) ( )

( ) { }

( ) ( )

( )

0.112

8 cos 21 ) 4 8 ( cos 15 ) 2 8 ( cos 9 ) 1 8 ( cos 3 ) 0 4 ( 2 4

2 3 ) 1 2 cos ( )

1 ( ) 1 (

5 . 8 0 cos 14 ) 4 8 ( cos 10 ) 2 8 ( cos 6 ) 1 8 ( cos 2 ) 0 4 ( 2 4

2 2 ) 1 2 cos ( )

2 ( ) 2 (

577 . 8 1 cos 8 ) 4 8 ( cos 5 ) 2 8 ( cos 3 ) 1 8 ( cos ) 0 4 ( 2 4

2 1 ) 1 2 cos ( )

1 ( ) 1 (

5 . 6 ) 4 ( ) 2 ( ) 1 ( ) 0 4 ( 1 4

2 0 ) 1 2 cos ( )

0 ( ) 0 (

4 2

) 1 2 cos ( )

2 ( ) 1 2 cos ( )

( ) (

1 4

0 1 4

0

1 4

0 1

0

=





 



+

  + 





+

 

⋅ 

=

 



⋅

= +

−

=





 



+

  + 





+

 

⋅ 

=

 



⋅

= +

−

=





 



+

  + 





+

 

⋅ 

=

 



⋅

= +

= + + +

⋅

=

 



⋅

= +





⋅

= +



 +

=



−

=

−

=

−

=

−

=

−

=

−

=

π π

π π π

α

π π

α π

π π

α π α π

f f

f x f

x f C

f f

f x f

x f C

f f

f x f

x f C

f f f x f

x f C

u x x

f N u

u x x

f u u C

x x x x

x N

x

(17)

Discrete Cosine Transform (4/4)

 Example

DCT Application (1/4)

 JEPG and MPEG

 Steps of the lossy sequential DCT-based coding mode

 Not DFT, but DCT! Why?

 DCT uses smaller coefficients

 DCT have more efficient energy compaction capability

(18)

DCT Application (2/4)

 1-D Example

 2-D Example (1/2)

















21 21 22 171 229 229 222 218

21 20 17 104 221 224 228 225

20 21 13 43 196 210 217 215

20 19 14 26 143 213 212 208

15 18 19 56 109 148 195 174

18 17 38 98 119 122 132 110

17 25 82 118 118 116 113 109

19 61 108 122 118 114 113 107

















⎯

⎯ →

⎯

















⎯

⎯ →

⎯

















0 0 0 0 0 0 0 0

0 0 0 0 0 90 335 253

0 0 0 0 0 44 567 855

0 0 0 0 0 626 2818 6831

567 44 226 114 307 277 1168 855

335 90 110 122 146 228 189 253

186 128 55 90 10 58 287 212

184 108 50 35 50 108 184 169

287 58 10 90 55 128 186 212

189 228 146 122 110 90 335 253

1168 277 307 114 226 44 567 855

2818 626 732 533 732 626 2818 6831

21 21 22 171 229 229 222 218

21 20 17 104 221 224 228 225

20 21 13 43 196 210 217 215

20 19 14 26 143 213 212 208

15 18 19 56 109 148 195 174

18 17 38 98 119 122 132 110

17 25 82 118 118 116 113 109

19 61 108 122 118 114 113 107

n CompressioTruncationfor DFT

















−

⎯

⎯ →

⎯

















−

⎯

⎯ →

⎯

















0 0 0 0 0 0 0 0

0 0 0 0 0 96 60 61

0 0 0 0 0 17 195 149

0 0 0 0 0 81 513 853

6 6 3 3 11 5 2 2

0 7 4 4 4 10 14 6

5 15 1 1 6 9 0 1

14 0 7 16 9 17 10 17

25 18 38 31 12 32 18 18

19 7 12 36 42 96 60 61

1 20 35 35 95 17 195 149

11 23 18 23 98 81 513 853

21 21 22 171 229 229 222 218

21 20 17 104 221 224 228 225

20 21 13 43 196 210 217 215

20 19 14 26 143 213 212 208

15 18 19 56 109 148 195 174

18 17 38 98 119 122 132 110

17 25 82 118 118 116 113 109

19 61 108 122 118 114 113 107

n CompressioTruncationfor DCT

(19)

 2-D Example (2/2)

















110 72 87 117 137 154 166 152

111 66 87 121 141 159 171 158

101 59 81 112 130 149 162 148

95 53 71 103 125 146 156 140

87 42 69 107 130 145 151 132

70 38 78 112 126 133 135 116

67 53 86 108 114 120 123 106

90 69 88 108 119 132 141 126

















9 26 85 148 196 224 234 235

8 20 70 126 175 210 229 237

6 13 49 95 143 186 219 236

2 10 35 71 116 161 200 221

4 15 36 66 103 141 173 191

12 27 52 81 108 129 142 148

20 41 74 105 122 124 115 107

24 51 90 122 134 122 99 81

















21 21 22 171 229 229 222 218

21 20 17 104 221 224 228 225

20 21 13 43 196 210 217 215

20 19 14 26 143 213 212 208

15 18 19 56 109 148 195 174

18 17 38 98 119 122 132 110

17 25 82 118 118 116 113 109

19 61 108 122 118 114 113 107

Practice

 Fourier Transform을 구현하시오.

 Discrete Fourier transform을 구현하시오.

 Fast Fourier transform을 구현하시오 (강의노트 p. 27 참조)

 Discrete Cosine Transform을 구현하시오.

 Discrete Cosine Transform을 구현하시오.

 FFT를 이용하여 DCT를 구현하시오.

 DCT Application의 Example들 (강의노트 pp. 35-37)을 구현해서 확인하시오.