Reminder

entitb 2)(

n

b tats nn )()(

Fourier Basis:t[0,1]

nZ

Fourier Series:

Fourier Coefficient: )(),( tstba nn dttstbn )()(1

0

*

Example - Sinc

dtdttsak

k

ntintieen

12

1

0

2)(

otherwise0

11)(rect)(

ktktts

)2()2(2112

21 knikni

ni

kk

ntini

eee

n

nknki

ni

)2sin()2sin(2

21

)(sinc n

rect(t)

Sinc - Pictures

Discrete Fourier Transform

1

0

21 )()(

N

x

N

iux

N exfuF

Fourier Transform ( notations: f(x) = s(x/N), F(u) = au+Nk )

1

0

2

11 )()(

N

u

N

iux

euFxf

Inverse Fourier Transform

Complexity: O(N2) (106 1012)

FFT: O(N logN) (106 107)

fxfFN

xN e

1

0

01 )()0(

Fourier of Delta

ee N

iux

N

N

x

N

iux

N xxuF02

11

0

2

01 )()(

)()( 0xxxf

NuF 1)(

)()( xxf

2D Discrete Fourier

1

0

1

0

)(21 ),(),(

N

x

N

y

N

vyuxi

N eyxfvuF

Fourier Transform

Inverse Fourier Transform

1

0

1

0

)(21 ),(),(

N

u

N

v

N

vyuxi

N evuFyxf

fN

yxfyxfFN

x

N

yN

N

x

N

y

N

yxi

N e

1

0

1

0

11

0

1

0

)00(21 ),(),()0,0(

Display Fourier Spectrum as Picture

1)(log uF 1. Compute

2. Scale to full range

Original f 0 1 2 4 100Scaled to 10 0 0 0 0 10

Log (1+f) 0 0.69 1.01 1.61 4.62Scaled to 10 0 1 2 4 10

Example for range 0..10:

3. Move (0,0) to center of image (Shift by N/2)

Fourier Displays

)(uF

1)(log uF

Decomposition

1

0

1

0

)(21 ),(),(

N

x

N

y

N

vyuxi

N eyxfvuF

1

0

1

0

221 ),(),(

N

x

N

y

N

ivy

N

iux

N ee yxfvuF

1

0

21 ),(),(

N

x

N

iux

N vxFvuF e

Decomposition (II)

• 1-D Fourier is sufficient to do 2-D Fourier– Do 1-D Fourier on each column. On result:– Do 1-D Fourier on each row– (Multiply by N?)

• 1-D Fourier Transform is enough to do Fourier for ANY dimension

Decomposition Example

Translation

e N

yvxui

yxfvvuuF)(2

00

00

),(),(

e N

vyuxi

vuFyyxxf)(2

00

00

),(),(

Periodicity & Symmetry

),(

),(),(),(

NvNuF

NvuFvNuFvuF

),(),( * vuFvuF

),(),( vuFvuF

(Only for real images)

Rotation

),(),( 00 Frf

sincos

sincos

vu

ryrx

),(),(),(),( FvuFrfyxf

Linearity

),(),(),(),( 2121 yxfyxfyxfyxf

),(),( yxfayxfa

b

ya

uFab

byaxf ,1

),(

Derivatives I

u

e Niux

uFxf2

)()(

Inverse Fourier Transform

uN

i

uueee N

iuxNiux

Niux

uuFuFuFxf

222

)()()()(' 2'

'

Derivatives II

• To compute the x derivative of f (up to a constant):– Computer the Fourier Transform F– Multiply each Fourier coefficient F(u,v) by u– Compute the Inverse Fourier Transform

• To compute the y derivative of f (up to a constant): – Computer the Fourier Transform F– Multiply each Fourier coefficient F(u,v) by v– Compute the Inverse Fourier Transform

Convolution Theorem

GFgf

gfGFgf 11

GFgf

Convolution by Fourier:

Complexity of Convolution: O(N logN)

Filtering in the Frequency Domain

Low-Pass Filtering

Band-Pass Filtering

High-Pass Filtering

Picture Fourier FilterFilteredPicture

FilteredFourier

• (0 0 1 1 0 0) Sinc

• (0 0 1 1 0 0) * (0 0 1 1 0 0 ) = (0 1 2 1 0 0) Sinc2

• (0 1 4 6 4 1 0) = (0 0 1 1 0 0 ) 4 Sinc4

• Fourier (Gaussian) Gaussian

Low Pass: Frequency & Image

ex

xg 2

2

2

2

1)(

Continuous Sampling

· =

T

*=

1/T

· =Image:

*=

1/T

Fourier: