Fourier Transform - Part II

Image Processing - Lesson 6

• Discrete Fourier Transform - 1D

• Discrete Fourier Transform - 2D

• Fourier Properties

• Convolution Theorem

• FFT

• Examples

Discrete Fourier Transform

Move from f(x) (x R) to f(x) (x Z) by sampling at equal intervals.

f(x0), f(x0+∆x), f(x0+2∆x), .... , f(x0+[n-1]∆x),

Given N samples at equal intervals, we redefine f as:

f(x) = f(x0+x∆x) x = 0, 1, 2, ... , N-1

4

3

2

1

4

3

2

1

0 0.25 0.5 0.75 1.0 1.25 0 1 2 3

f(x)

f(x0)

f(x0+∆x)

f(x0+2∆x) f(x0+3∆x)

f(xn) = f(x0 + x∆x)

∈ ∈

Discrete Fourier Transform

The Discrete Fourier Transform (DFT) is defined as:

u = 0, 1, 2, ..., N-1∑−

=

−

=1

0

2

)(1)(N

x

Niux

exfN

uFπ

∑−

=

=1

0

2

)()(N

u

Niux

euFxfπ

x = 0, 1, 2, ..., N-1

The Inverse Discrete Fourier Transform (IDFT) is defined as:

Matlab: F=fft(f);

Matlab: F=ifft(f);

4

3

2

1

Discrete Fourier Transform - Example

4

3

2

1

0 0.25 0.5 0.75 1.0 1.25 0 1 2 3

f(x)

f(x0)

f(x0+∆x)

f(x0+2∆x) f(x0+3∆x)

f(x) = f(x0 + x∆x)

F(0) = 1/4 Σ f(x) e = 1/4 Σ f(x) 13

x=0

-2πi0x4

= 1/4(f(0) + f(1) + f(2) + f(3)) = 1/4(2+3+4+4) = 3.25

3

x=0

F(1) = 1/4 Σ f(x) e = 1/4 [2e0+3e-iπ/2+4e−π i 4e-i3π/2] = [-2+i]3

x=0

-2πix4 1

4

F(3) = 1/4 Σ f(x) e = 1/4 [2e0+3e-i3π/2+4e−3πi 4e-i9π/2] = [-2-i]3

x=0

-6πix4 1

4

F(2) = 1/4 Σ f(x) e = 1/4 [2e0+3e-iπ+4e−2πi 4e-3πi] = [-1-0i]=3

x=0

-4πix4 -1

4-14

Fourier Spectrum:

|F(0)| = 3.25

|F(1)| = [(-1/2)2 + (1/4)2]0.5

|F(2)| = [(-1/4)2 + (0)2]0.5

|F(3)| = [(-1/2)2 + (-1/4)2]0.5

Discrete Fourier Transform - 2D

Image f(x,y) x = 0,1,...,N-1 y= 0,1,...,M-1

The Discrete Fourier Transform (DFT) is defined as:

The Inverse Discrete Fourier Transform (IDFT) is defined as:

x = 0, 1, 2, ..., N-1∑−

=∑

−

=

+=

1

0

1

0

)(2),(),(

N

u

Mvy

NuxM

v

ievuFyxf

π

y = 0, 1, 2, ..., M-1

u = 0, 1, 2, ..., N-1v = 0, 1, 2, ..., M-1

∑−

=

−

=∑

+−=

1

0

1

0

)(2),(1),(F

N

x

M

y

Mvy

Nuxi

eyxfMN

vuπ

Matlab: F=fft2(f);

Matlab: F=ifft2(f);

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

v

u

Fourier Transform - Image

Visualizing the Fourier Transform Image using Matlab Routines

• F(u,v) is a Fourier transform of f(x,y) and it has complex entries.

F = fft2(f);• In order to display the Fourier Spectrum |F(u,v)|

– Cyclically rotate the image so that F(0,0) is in the center:

F = fftshift(F);– Reduce dynamic range of |F(u,v)| by displaying

the log:

D = log(1+abs(F));

Example:

|F(u)| = 100 4 2 1 0 0 1 2 4

|F(u)| = 0 1 2 4 100 4 2 1 0Cyclic

|F(u)|/10 = 0 0 0 0 10 0 0 0 0

log(1+|F(u)|) = 0 0.69 1.01 1.61 4.62 1.61 1.01 0.69 0

0 1 2 4 10 4 2 1 0

Display in Range([0..10]):

log(1+|F(u)|)/0.462 =

Original

Visualizing the Fourier Image - Example

|F(u,v)|

|F(0,0)| at center log(1 + |F(u,v)|)

Original

Shifted Fourier Image Shifted Log Fourier Image = log(1+ |F(u,v)|)

Fourier Image = |F(u,v)|

Properties of The Fourier Transform

Distributive (addition)

LinearityF[a f(x,y)] = aF [f(x,y)]

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

CyclicF(u,v) = F(u+N,v) = F(u,v+N) = F(u+N,v+N)

F(x,y) = F(x+N,y+N)

SymmetricF(u,v) = F*(-u,-v)

thus:

|F(u,v)| = |F (-u,-v)| Fourier Spectrum is symmetric

DC (Average)1MF(0,0) = Σ Σ f(x) e0

N-1

x=0

1N

M-1

y=0

if f(x) is real:

F[ f1(x,y) + f2(x,y)] = [f1(x,y)] + F [f2(x,y)]F~ F

~F~

F~

F~

x

f(x)

ω

|F(ω)|

x

g(x)

ω

|G(ω)|

x

f+g

ω

|F(ω)+G(ω)|

Distributive: { } { } { }gFfFgfF ~~~ +=+

1 cycle

0 N/2 N-1N

-N/2

Cyclic and Symmetry of the Fourier Transform - 1D Example

|F(u)|

Translation

Image Transformations

f(x-x0,y-y0) F(u,v)e-2πi(ux0+vy0)

N

f(x,y)e F(u-u0,v-v0)2πi(u0x+v0y)

N

The Fourier Spectrum remains unchanged under translation:

|F(u,v)| = |F(u,v)e |-2πi(ux0+vy0)

N

Rotation

Rotation of f(x,y) by θ

Rotation of F(u,v)by θ

Scale

f(ax,by) F(u/a,v/b)1|ab|

x

f(x)

ω

|F(ω)|

x

f(x)

ω

|F(ω)|

Change of Scale- 1D:

x

f(x)

ω

|F(ω)|

( ){ } ( ) ( ){ }

==

aF

a1axfF~thenFxfF~if ωω

x

f(x)

ω

F(ω)

x

f(2x)

ω

0.5 F(ω/2)

Change of Scale

x

f(x/2)

ω

2 F(2ω)

2D Image 2D Image - Rotated

Fourier Spectrum Fourier Spectrum

Example - Rotation

Image Domain Frequency Domain

Fourier Transform Examples

Separabitity

F(u,v) = Σ Σ f(x,y) e-2πi(ux+vy)/n

N-1

x=0

1N

N-1

y=0

F(u,v) = Σ (Σ f(x,y) e-2π ivy/n) e-2π iux/nN-1

x=0

1N

N-1

y=0

F(u,v) = Σ F(x,v) e-2π iux/n

N-1

x=0

1N

Thus, to perform a 2D Fourier Transform is equivalent to performing 2 1D transforms:

1) Perform 1D transform on EACH column of image f(x,y).Obtain F(x,v).

2) Perform 1D transform on EACH row of F(x,v).

Higher Dimensions:Fourier in any dimension can be performed by applying 1D transform on each dimension.

Image Domain Frequency Domain

Fourier Transform Examples

Image Domain Frequency Domain

Fourier Transform Examples

Linear Systems and Responses

SpatialDomain

FrequencyDomain

Input f F

Output g G

ImpulseResponse h

Freq.Response H

Relationship g=f*h G=FH

The Convolution Theorem

g = f * h g = f h

implies implies

G = F H G = F * H

Convolution in one domain is multiplication in the other and vice

versa

The Convolution Theorem

Convolution in one domain is multiplication in the other and vice

versa

( ) ( ){ } ( ){ } ( ){ }xgFxfFxgxfF ~~~ =∗

and likewise

( ) ( ){ } ( ){ } ( ){ }xgFxfFxgxfF ~~~ ∗=

f(x,y) * g(x,y) F(u,v) G(u,v)

f(x,y) g(x,y) F(u,v) * G(u,v)

f(x)

-0.5 0.5

x

h(x)Example:

-1 1

f(x)

-0.5 0.5

*h(x) =

F(ω) F(ω)

H(ω) = .

=

f(x,y) g(x,y) f * g (x,y)

F(u,v) G(u,v) F(u,v) G(u,v)

F[F(u,v) G(u,v)]

Convolution Theorem -2D Example

F-1ℑ

1−ℑ

Sampling the Image

x

f(x)

ω

|F(ω)|

F̂

x

g(x)

ω

F̂|G(ω)|

ω

T 1/T

x

*

F̂

-1/2T 1/2T

Undersampling the Image

x

g(x)

F̂|G(ω)|

ω

T 1/T

*

ω

-1/2T 1/2Tx

F̂

x

f(x)

ω

|F(ω)|

F̂

Critical Sampling• If the maximal frequency of f(x) is ωmax , it

is clear from the above replicas that ωmax should be smaller that 1/2T.

• Alternatively:

• Nyquist Theorem: If the maximal frequency of f(x) is ωmax the sampling rate should be larger than 2ωmax in order to fully reconstruct f(x) from its samples.

• If the sampling rate is smaller than 2ωmax overlapping replicas produce aliasing.

max21

ω>T

ω

|F(ω)|

-1/2T 1/2T

Optimal Interpolation

• It is possible to fully reconstruct f(x) from its samples:

ω

|F(ω)|

x

f(x)F-1^

-1/2T 1/2T

ω

|F(ω)|

x

f(x)F-1^

-1/2T 1/2T

ω

|F(ω)|

x

f(x)F-1^

-1/2T 1/2T

1

*

Note, that only N/2 different transform values are obtainedfor the N/2 point transforms.

Fast Fourier Transform - FFT

u = 0, 1, 2, ..., N-1∑−

=

−

=1

0

2

)(1)(N

x

Niux

exfN

uFπ

O(n2) operations

∑∑−

=

−

=

+−−

++=12/

0

12/

0

)12(222

)12(1)2(1)(N

x

N

x

Nxiu

Nxiu

exfN

exfN

uFππ

The Fourier transform of N inputs, can be performed as2 Fourier Transforms of N/2 inputs each + one complex multiplication and addition for each value i.e. O(N).

even x odd x

−−

∑ ++∑=−

=

−−

=

12/

0

212/

0

2/2

2/2

2/)12(

2/1)2(1

21 N

x

NiuN

x

Niux

Niux

Nexf

Neexf

πππ

Fourier Transform ofof N/2 even points

Fourier Transform ofof N/2 odd points

0 1 2 3 4 5 6 7 0 2 4 6 1 3 5 7

All sampling pointsOdd

sampling pointsEven

sampling points

For u = 0, 1, 2, ..., N/2-1

−−−

∑ ++∑=−

=

−

=

12/

0

12/

0

2/22

2/2

2/)12(

2/1)2(1

21)(

N

x

N

xN

Niux

Niu

Niux

Nexf

NeexfuF

πππ

−

+= )()(21)(

2/2/

2

uFeuFuF o

N

e

NNN

iuπ

For u' = u+N/2 :N

iuiN

iuN

NuiN

iu

eeeeeπ

ππππ 22)2/(2'2 −

−=−−

=

+−

=

−

obtain :

Thus: only one complex multiplication is needed for two terms.

−

+= )()(21)(

2/

2

2/uFeuFuF o

NN

iue

NN

π

−

−=+ )()(21)(

2/

2

2/2 uFeuFuF oN

Niu

eN

NN

π

)u(e2/NFCalculating and is done recursively by

calculating and .)u(o

2/NF)u(e

4/NF )u(o4/NF

F(0) F(1) F(2) F(3) F(4) F(5) F(6) F(7)

F(0) F(2) F(4) F(6) F(1) F(3) F(5) F(7)

F(0) F(4) F(2) F(6) F(1) F(5) F(3) F(7)

F(0) F(1) F(2) F(3) F(4) F(5) F(6) F(7)

2-pointtransform

4-pointtransform

FFT

FFT : O(nlog(n)) operations

FFT of NxN Image: O(n2log(n)) operations

FilteredImage

Image Transform

Filtered

FFT

FFT-1NewImage

Frequency Enhancement