The Fourier Transform

Jean Baptiste Joseph Fourier

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Original histogram Equalized histogram

Image Operations in Different Domains

1) Gray value (histogram) domain

2) Spatial (image) domain

3) Frequency (Fourier) domain

- Histogram stretching, equalization, specification, etc...

- Average filter, median filter, gradient, laplacian, etc…

Original image Gradient magnitude22yx fff

f

Blurry Image Laplacian

+ =

Sharpened Image

Noisy image(Salt & Pepper noise)

3 X 3 Average 5 X 5 Average

7 X 7 Average Median

=

3 sin(x) A

+ 1 sin(3x) B A+B

+ 0.8 sin(5x) CA+B+C

+ 0.4 sin(7x) DA+B+C+D

A sum of sines and cosines

sin(x) A

Higher frequencies dueto sharp image variations

(e.g., edges, noise, etc.)

The Continuous Fourier Transform

u

iuxduxf 2F(u)e)(

)2sin()2cos2 uxiux(iux e

Complex Numbers

Real

ImaginaryZ=(a,b)

a

b|Z|

)(conjugate

(phase)

spectrum)(Fourier

vector)unit (a

i

i

eZibaZZ

abtg

baZ

ie

ii

*

1

22

2

)/(

sincos

)1(1

ieZ

ibaZiZZ

)Im()Re(

x

ux2cos

– The wavelength is 1/u .– The frequency is u .

1

The 1D Basis Functions iuxe 2

)2sin()Im(

)2cos()Re(2

2

uxe

uxeiux

iux

1/u

The Fourier Transform

))(( xf

x

iuxdxuF 2f(x)e)(

1D Continuous Fourier Transform:

))((1 uFThe InverseFourier Transform

The Continuous Fourier Transform

x y

uxi dxdyvuF vy)(2y)ef(x,),(

2D Continuous Fourier Transform:

u v

uxi dudvyxf vy)(2v)eF(u,),(

u

iuxduxf 2F(u)e)(

The Inverse Transform

The Transform

The wavelength is . The direction is u/v .22/1 vu

The 2D Basis Functions

u=0, v=0 u=1, v=0 u=2, v=0u=-2, v=0 u=-1, v=0

u=0, v=1 u=1, v=1 u=2, v=1u=-2, v=1 u=-1, v=1

u=0, v=2 u=1, v=2 u=2, v=2u=-2, v=2 u=-1, v=2

u=0, v=-1 u=1, v=-1 u=2, v=-1u=-2, v=-1 u=-1, v=-1

u=0, v=-2 u=1, v=-2 u=2, v=-2u=-2, v=-2 u=-1, v=-2

U

V

)(2 vyuxie

Discrete Functions

0 1 2 3 ... N-1

f(x)

f(x0)

f(x0+x)

f(x0+2x) f(x0+3x)

f(n) = f(x0 + nx)

x0 x0+x x0+2x x0+3x

The discrete function f:{ f(0), f(1), f(2), … , f(N-1) }

1

0

2

)()(N

u

N

iux

euFxf

1

0

2

)(1)(N

x

N

iux

exfN

uF

(u = 0,..., N-1)

(x = 0,..., N-1)

1D Discrete Fourier Transform:

The Discrete Fourier Transform

1

0

1

0

)(2),(

11),(N

x

M

y

Mvy

Nuxi

eyxfMN

vuF

2D Discrete Fourier Transform:

1

0

1

0

)(2),(),(

N

u

M

v

Mvy

Nuxi

evuFyxf

(x = 0,..., N-1; y = 0,…,M-1)

(u = 0,..., N-1; v = 0,…,M-1)

Fourier spectrum log(1 + |F(u,v)|) Image f

The Fourier Image

Fourier spectrum |F(u,v)|

Frequency Bands

Percentage of image power enclosed in circles (small to large) :

90%, 95%, 98%, 99%, 99.5%, 99.9%

Image Fourier Spectrum

Low pass Filtering

90% 95%

98% 99%

99.5% 99.9%

Noise Removal

Noisy image

Fourier Spectrum Noise-cleaned image

High Pass Filtering

Original High Pass Filtered

High Frequency Emphasis

+Original High Pass Filtered

High Frequency EmphasisOriginal High Frequency Emphasis

OriginalHigh Frequency Emphasis

Original High pass Filter

High Frequency Emphasis

High Frequency Emphasis +

Histogram Equalization

High Frequency Emphasis

Properties of the Fourier Transform –

Developed on the board…

(e.g., separability of the 2D transform, linearity, scaling/shrinking, derivative, rotation, shift phase-change, periodicity of the discrete transform, etc.)

We also developed the Fourier Transform of various commonly used functions, and discussed applications which are not contained in the slides (motion, etc.)

2D Image 2D Image - Rotated

Fourier Spectrum Fourier Spectrum

Image Domain

Frequency Domain

Fourier Transform -- Examples

Image Domain

Frequency Domain

Fourier Transform -- Examples

Image Domain

Frequency Domain

Fourier Transform -- Examples

Image Domain

Frequency Domain

Fourier Transform -- Examples

Image Fourier spectrum

Fourier Transform -- Examples

Image Fourier spectrum

Fourier Transform -- Examples

Image Fourier spectrum

Fourier Transform -- Examples

Image Fourier spectrum

Fourier Transform -- Examples