Fourier Transform

6th Lecture on Image Processing

Martina Mudrová2004

in Image ProcessingJean Baptiste Joseph Fourier

1768-1830

Motivation

Why should we use the Fourier transformation?• basic mathematical tool to process images

image compression(format .JPG)

edge detection and image segmentation

quality image processing

image reconstruction

objects detection

etc.

2

Basic problems solved with the FT:

M. Mudrová, 2004

Mathematical Minimum

= transformation between time (spatial) x(t) and frequency X(f) domain

)()( fXtx ⇔ t…time domain (spatial)f…frequency domainBasic Definitions:

inverse continuous FT:direct continuous FT:

∫+∞

∞−

−= dtetxfX tfi π2 )()( ∫+∞

∞−

+= dfefXtx tfi π2 )()(

Requests to f(t) : continuous, including the 1st derivation

integrateable : ∞

Example of Continuous FT

How does look FT of periodical functions?

) 2sin()( 0tftx π= ( ))()( 21)( 00 ffffi

fX +−−= δδ

Euler’s formulas: )2sin()2cos( 002 0 tfitfe tfi πππ ±=±

∫+∞

∞−

−= dtetxfX tfi π2 )()(Def. FT:

⇔d(f)… Dirac impulse

ff0

Time (spatial) domain x(t) Spectrum |X(f)| (amplitude fr. charact.)

T=1/f0 2T 3T t

x(t)

-f 0 f f

|X(f)

|

0 0

⇔

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From Continuous to discrete FT

How is defined the discrete FT ?

Discretization in frequency: f Y fk Y k

Discretization in time: tY tn Y n +

Discrete Fourier TransformationDirect discrete FT: Inverse discrete FT:

)()( kXnx ⇔∑−=

−=

1

0

2 )(1)(

N

n

Nnki

enxN

kXπ

∑−

=

+=

1

0

2 )()(

N

k

Nkni

ekXnxπ

0 20 40 60-1

0

1

x(n)

n

|X(k

)|

0 0 0

31 3.14 0.5

15 1.57 0.25

k...indexωk ...circular frequencyf …normalized freq.

63

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Function Restriction in Time (Spatial) Domain

= apodizationFunction with definite length...

0 10 20 30 40 50 60

-1

0

1

x(n)

n

Periodický signál a jeho spektrum

0 0.5 1 1.5 2 2.5 30

10

20

30

40

ω od 0 do π

|X( ω

)|

0 10 20 30 40 50 600

0.5

1

1.5Obdelnikové okno a jeho spektrum

x(n)

n0 0.5 1 1.5 2 2.5 3

0

10

20

30

40

ω od 0 do π

|X( ω

)|

-1

0

1

x(n)

n

Omezený signál a jeho spektrum

0 0.5 1 1.5 2 2.5 30

10

20

30

40

ω od 0 do π

|X( ω)

|

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Let’s Go to Two Dimensions…

0 10 20 30 40 50 60-101

x(n)

n

n -> [n, m] k -> [k, l]

|X(k

)|

0 31 15 k 63

2D DFT

),(),( lkXmnx ⇔∑∑−=

−

=

+−=

1

0

1

0

)(2 ),(1),(

N

n

M

m

Mlm

Nkni

emnxMN

lkXπ

∑∑−

=

−

=

++=

1

0

1

0

)(2 ),(),(

N

k

M

l

Mml

Nnki

elkXmnxπ

Direct 2D DFT: Inverse 2D DFT:

1

6 4

1

6 40

1

n

x (n ,m )

m-1

01

-1

0

1

f k

| X ( f k , f l) |

f ln

m

1 6 4

1 64

x (n ,m )

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Example of 2D DFT of Real Image

Symetry

0 1

1

0

1-1-1

1

n

m

1 366

1

270

x(n,m)

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Digital Filtering Principle

What happens in the case of spectrum processing?Spectrum |X(k)|Time space x(n)

1. Periodical function

2. Random noise

3. =1.+2.

4. After spectrum processing

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2D Filtering

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Discrete Convolution Principle

What is happening in the image space during the spectrum processing?Frequency space:

Multiplication

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Time (image) space:Convolution

∑=

−==n

jjnhjxnhnxny

0)()()(*)()(

kkHkXkY pro )( . )()( ∀=

y(n)…requested functionx(n)...given (damaged) functionh(n) …digital filter

Where:

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Discrete Convolution Example

What is a meaning of convolution?

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∑=

−=

=n

jjnhjx

nhnxny

0)()(

)(*)()(

x(n) = 10; 20; 30; 40; 50h(n) = 0,5; 0,5

n=0 j=0 f(0).h(0) 10 . 0,5 5 y(0)= 5n=1 j=0 f(0).h(1-0) 10 . 0,5 5

j=1 f(1).h(1-1) 20 . 0,5 10 y(1)=5+10 = 15n=2 j=0 f(0).h(2-0) nedef.

j=1 f(1).h(2-1) 20 . 0,5 10j=2 f(2).h(2-2) 30 . 0,5 15 y(2)=10+15= 25

n=3 j=0 f(0).h(3-0) nedef.j=1 f(1).h(3-1) nedef.j=2 f(2).h(3-2) 30 . 0,5 15j=3 f(3).h(3-3) 40 . 0,5 20 y(3)=15+20= 35

n=4 j=0,1,2 nedef.j=3 f(4).h(4-3) 40 . 0,5 20j=4 f(4).h(4-4) 50 . 0,5 25 y(4)=25+20= 45

Example:(Sliding average)

y(n) = 5; 15; 25; 35; 45

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2D Discrete Convolution

How does it seem the convolution in 2D?

∑∑= =

−−==n

j

m

jjmjnhjjxmnhmnxmny

0 02121

1 2

),(),(),(*),(),(

0 1 0

1 -4 1

0 1 0

filter matrixh(n,m)

image matrixx(n,m)

Element evaluatedin (n,m) position

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Application I – Edge Detection

Goal: Edge detection

(=>image segmentation)

Orig

inal

Imag

eIm

age

afte

r im

age

dete

ctio

n

Edge = abrupt image function change

Principle:Application of edge detectors= highpass filters

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Example of filtr spectrum |H(k,l)|Filter matrix example:h(n,m) (Prewitt filtr)

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−− 111000111

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Application II – Noise Removing

Goal: Image Enhancement

Orig

inal

Imag

e

Principle:Application of low pass filter

Filter Spetrum |H(k,l)|

Pre

cess

edIm

age

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Application III – Sharpening

Goal: Corrupted image reconstruction

Cor

rupt

ed im

age

Rec

onst

ruct

ed im

age

Principle:

• Inverse filtering• Wiener filtering

Types of corruption:

• Defocussing due to object movement • Bad objective focussing• Atmosphere turbulentions• etc.

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Application IV – Object Detection

Goal: Finding the coordinates of a given object in the image

Principle:

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• Evaluation of correlation function = convolution in the spatial domain

• Using an algorithm of FFT the computation gets faster-> multiplication in the

frequency domain

Imag

ex(

n,m

)

Filtr

h(n,

m)

Kor

elač

ní fu

nkce

h(n

,m)

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Advanced Methods of Image Analysis

Short-time Fourier Transform- compromise between time (image)-frequency resolution

Wavelet transform-use time (image) window with various length- used in image analysis, denoising, compression

Radon transform-used for conversion from cylindric coordinate system-used mainly for biomedical image processing

…

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Fourier TransformMotivationMathematical MinimumExample of Continuous FTFrom Continuous to discrete FTFunction Restriction in Time (Spatial) DomainLet’s Go to Two Dimensions…Example of 2D DFT of Real ImageDigital Filtering Principle2D FilteringDiscrete Convolution PrincipleDiscrete Convolution Example2D Discrete ConvolutionApplication I – Edge DetectionApplication II – Noise RemovingApplication III – SharpeningApplication IV – Object DetectionAdvanced Methods of Image Analysis