Fourier Series

1

Chapter 4Instructor: Hossein Pourghassem

Image Enhancement in theFrequency Domain

2

Fourier Series

2

Islamic Azad University of Najafabad, Department of Electrical Engineering, Dr. H. Pourghassem, 3

Fourier Series

Fourier series: a periodic function can be represented by the sum of sines/cosines of different frequencies, multiplied by a different coefficient (Fourier series)

Tu

uwnwtBnwtAxf

xfTxf

nn

12

)sin()cos()()()(

=

=

+=

=+

∑π


One dimensional Fourier TransformNon-periodic functions can also be represented as the

integral of sines/cosines multiplied by weighting function (Fourier transform)f(x): continuous function of a real variable x

Fourier transform of f(x):

{ } ∫∞

∞−

−==ℑ dxuxjxfuFxf ]2exp[)()()( π

where 1−=j

3


One dimensional Fourier Transform

Given F(u), f(x) can be obtained by the inverse Fourier transform:

)()}({1 xfuF =ℑ−

∫∞

∞−

= duuxjuF ]2exp[)( π

• The above two equations are the Fourier transform pair.


Discrete Fourier Transform

A continuous function f(x) is discretized into a sequence:

)}]1[(),...,2(),(),({ 0000 xNxfxxfxxfxf Δ−+Δ+Δ+

by taking N or M samples Δx units apart.

4



Where x assumes the discrete values (0,1,2,3,…,M-1) then

)()( 0 xxxfxf Δ+=

• The sequence {f(0),f(1),f(2),…f(M-1)} denotes any M uniformly spaced samples from a corresponding continuous function.



∫

∑

−

−

+∞

−∞=

−

=

=

π

ππdweeFxf

exfeF

jwxjw

x

jwxjw

)(21)(

)()(

Fourier Transform of discrete function is a continuous functionwhich is calculated as follows:

5



xwu

nFx

eFn

sjw

ΩΔ==Ω

Ω−ΩΔ

= ∑+∞

−∞=

π2

)(1)(

Relation between continuous and discrete Fourier Transform


Some Properties of Discrete Fourier Transform

Discrete Fourier Transform is periodic with period of 2π

continuous frequency of fs (sampling frequency) is mapped to discrete Fourier frequency of 2π

For real f(x)

)()( * jwjw eFeF −=

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The values u = 0, 1, 2, …, M-1 correspond to samples of the continuous transform at values 0, Δu, 2Δu, …, (M-1)Δu.

i.e. F(u) represents F(uΔu), where:

Δu =1

MΔx


Introduction to the Fourier Transform

The Fourier transform of a real function is generally complex and we use polar coordinates:

|F(u)| (magnitude function) is the Fourier spectrum of f(x) and φ(u) its phase angle.The square of the spectrum

is referred to as the power spectrum of f(x) (spectral density).

⎥⎦

⎤⎢⎣

⎡=

+=

=

+=

−

)()(tan)(

)]()([)(

)()()()()(

1

2/122

)(

uRuIu

uIuRuF

euFuFujIuRuF

uj

φ

φ

)()()()( 222 uIuRuFuP +==

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Two Dimensional Discrete Fourier Transform

In a 2-variable case, the discrete FT pair is:

∑∑−

=

−

=

+−=1

0

1

0

)]//(2exp[),(1),(M

x

N

y

NvyMuxjyxfMN

vuF π

∑∑−

=

−

=

+=1

0

1

0)]//(2exp[),(),(

M

u

N

vNvyMuxjvuFyxf π

For u=0,1,2,…,M-1 and v=0,1,2,…,N-1

For x=0,1,2,…,M-1 and y=0,1,2,…,N-1

AND:



Sampling of a continuous function is now in a 2-D grid (Δx, Δy divisions).

The discrete function f(x,y) represents samples of the function f(x0+xΔx,y0+yΔy) for x=0,1,2,…,M-1 and y=0,1,2,…,N-1.

yNv

xMu

Δ=Δ

Δ=Δ

1 ,1

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Introduction to the Fourier Transform

Fourier spectrum: [ ] 2/122 ),(),(),( vuIvuRvuF +=

• Phase: ⎥⎦

⎤⎢⎣

⎡= −

),(),(tan),( 1

vuRvuIvuφ

• Power spectrum: ),(),(),(),( 222 vuIvuRvuFvuP +==



When images are sampled in a square array, M=N and the FT pair becomes:

∑∑−

=

−

=

+−=1

0

1

0

]/)(2exp[),(1),(N

x

N

y

NvyuxjyxfN

vuF π

∑∑−

=

−

=

+=1

0

1

0]/)(2exp[),(1),(

N

u

N

vNvyuxjvuF

Nyxf π

For u,v=0,1,2,…,N-1

For x,y=0,1,2,…,N-1

AND:

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Basic Properties

F(0,0) is at u=M/2 and v=N/2Shifts the origin of F(u,v) to (M/2, N/2), i.e. the center of MxN of the 2-D DFT (frequency rectangle)Frequency rectangle: from u=0 to u=M-1, and v=0 to v=N-1 (u,v integers, M,N even numbers) In computers: summations are from u=1 to M and v=1 to N center of transform: u=(M/2) +1 and v=(N/2) +1

[ ] )2/,2/()1)(,( NvMuFyxf yx −−=−ℑ +

Common practice:


Basic Properties

Value of transform at (u,v)=(0,0):

which means that the value of FT at the origin = the average gray level of the imageFT is also conjugate symmetric:

F(u,v)=F*(-u,-v)so |F(u,v)|=|F(-u,-v)|

which means that the FT spectrum is symmetric.

F(0,0) =1

MNf (x, y)

y= 0

N −1

∑x= 0

M −1

∑

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Basic Properties


Image Enhancement in the Frequency Domain

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Basic steps for filtering in the frequency domain

1. Multiply input Image by (-1)x+y to center the transform,

2. Compute F(u,v) , the DFT of image

3. Multiply by a filter function H(u,v)

4. Compute inverse DFT of the result in (3)

5. Obtain the real part of result in (4)

6. Multiply the result in 5 by (-1)x+y

Summary: G(u,v) = H(u,v) F(u,v)

Filtered Image = ℑ−1 G(u,v)[ ]


Basic steps for filtering in the frequency domain

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Edges and sharp transitions (e.g., noise) in an image contributesignificantly to high-frequency content of FT.

Low frequency contents in the FT are responsible to the general appearance of the image over smooth areas.

Blurring (smoothing) is achieved by attenuating range of high frequency components of FT.

Frequency Domain Filtering


f(x,y) is the input imageg(x,y) is the filteredh(x,y): impulse response

Convolution Theorem

G(u,v)=F(u,v)●H(u,v)

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

• Filtering in Frequency Domain with H(u,v) is equivalent to filtering in Spatial Domain with h(x,y).

Multiplication in Frequency Domain

Convolution in Time Domain

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Example of Filters


Basic Filters

Notch filterTo force the average value of an image to 0:

F(0,0) gives the average value of an imagethen, since F(0,0)=0, take the inverse

⎩⎨⎧ =

=otherwise

, N/ M/if (u,v) vuH

1220

),(

14





Types of enhancement that can be done:

Lowpass filtering: reduce the high-frequency content--blurring or smoothing

Highpass filtering: increase the magnitude of high-frequency components relative to low-frequency components -- sharpening.

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Low Pass filtering or Smoothing in the Frequency Domain

G(u,v) = H(u,v) F(u,v)IdealButterworth (parameter: filter order); for high and low values of this parameter, the Butterworth approaches the form of the ideal filter and Gaussian filter, respectively. Gaussian

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Ideal low-pass filter (ILPF)

D0 is called the cutoff frequency.

(M/2,N/2): center in frequency domain

2 2 1/2( , ) [( / 2) ( / 2) ]D u v u M v N= − + −⎩⎨⎧

>≤

=0

0

),(0),(1

),(DvuDDvuD

vuH


Shape of ILPF

Spatial domain

Frequency domain

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Traditional Methodfor Calculation of Cutoff frequency

The summation is taken within the circle D0

⎥⎥⎦

⎤

⎢⎢⎣

⎡= ∑∑

u vTPvuP /),(100α

Calculate PT , the total Power of image

∑ ∑−

=

−

==

1

0

1

0),(

M

u

N

vT vuPP

A circle with radius D0, origin at the center of the frequency rectangle encloses a percentage of the power which is given by the expression



18

35

ringing and

blurring

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Ringing

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Butterworth Lowpass Filters (BLPF)

Smooth transfer functionno sharp discontinuity no clear cutoff frequency

n

DvuD

vuH 2

0

),(1

1),(

⎥⎦

⎤⎢⎣

⎡+

=

21


Butterworth Lowpass Filters (BLPF)

20

39

No serious ringing artifacts


• Smooth transfer function• Smooth impulse response• No ringing• Its inverse is also Gaussian

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2

2),(

),( DvuD

evuH−

=

Gaussian Lowpass Filters (GLPF)

220

220

2

20

2 2)()( xDDu

AeDxhAeuH ππ −−

==

21


Gaussian Lowpass Filters (GLPF)

42

No serious ringing artifacts

22



Sharpening High-pass Filters

Hhp(u,v)=1-Hlp(u,v)

Ideal:

Butterworth:

Gaussian:

n

vuDD

vuH 20

2

),(1

1|),(|

⎥⎦

⎤⎢⎣

⎡+

=

⎩⎨⎧

≤>

=0

0

),(0),(1

),(DvuDDvuD

vuH

20

2 2/),(1),( DvuDevuH −−=

23

45


High-pass Filters

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Ideal High-pass Filtering

ringing artifacts


Butterworth High-pass Filtering

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Gaussian High-pass Filtering


Laplacian

∇2 f =∂ 2 f∂x 2 +

∂ 2 f∂y 2

∂ 2 f∂ 2x 2 = f (x +1, y) + f (x −1, y) − 2 f (x, y)

∂ 2 f∂ 2y 2 = f (x,y +1) + f (x,y −1) − 2 f (x, y)

∇2 f = [ f (x +1,y) + f (x −1,y) + f (x,y +1) + f (x,y −1)]− 4 f (x,y)

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Laplacian in the Frequency Domain

),()(),(

),()(),(

vuFjvy

yxf

vuFjux

yxf

nn

n

nn

n

=⎥⎦

⎤⎢⎣

⎡∂

∂ℑ

=⎥⎦

⎤⎢⎣

⎡∂

∂ℑ

It can be shown

[ ] ),()(),( 222 vuFvuyxf +−=∇ℑ


The Laplacian can be implemented in the FD by using the filter

FT pair:


)(),( 22 vuvuH +−=

),(])2/()2/[(),( 222 vuFNvMuyxf −+−−⇔∇

27

53



Subtract Laplacian from the Original Image to Enhance It

),()(),(),( 22 vuFvuvuFvuG ++=

new operator

Spatial domain

Original image

enhanced image

Laplacian output

Frequency domain

Laplacian

),(1)(1),( 122

2 vuHvuvuH −=++=

),(),(),( 2 yxfyxfyxg ∇−=

28

55



Unsharp Masking, High-boost Filtering

Unsharp masking: fhp(x,y)=f(x,y)-flp(x,y)Hhp(u,v)=1-Hlp(u,v)

High-boost filtering: fhb(x,y)=Af(x,y)-flp(x,y)fhb(x,y)=(A-1)f(x,y)+fhp(x,y)Hhb(u,v)=(A-1)+Hhp(u,v)Hhfe(u,v)=a+bHhp(u,v)

One more parameter to

adjust the enhancement

High frequency emphasis

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Image Enhancement in Frequency Domain


An image formation modelWe can view an image f(x,y) as a product of two components:

i(x,y): illumination. It is determined by the illumination source.r(x,y): reflectance. It is determined by the characteristics of imaged objects.

( ) ( ) ( )

1),(0),(0

,,,

<<∞<<

⋅=

yxryxi

yxryxiyxf

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Homomorphic Filtering

In some images, the quality of the image has reduced because of non-uniform illumination.Homomorphic filtering can be used to perform illumination correction.

The above equation cannot be used directly in order to operate separately on the frequency components of illumination and reflectance.

( ) ( ) ( )yxryxiyxf ,,, ⋅=


( ) ( ) ( )vuFvuFvuZ ri ,,, +=

( ) ( ) ( ) ( )yxryxiyxfyxz ,ln,ln,ln, +==

),(),(),(),(),(),(

00),(

''

yxryxieyxgyxryxiyxs

yxs ==

+=


( )vuZvuHvuS ,),(),( =

ln :

DFT :

H(u,v) :

(DFT)-1 :

exp :

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By separating the illumination and reflectance components, homomorphic filter can then operate on them separately.Illumination component of an image generally has slow variations, while the reflectance component vary abruptly. By removing the low frequencies (highpass filtering) the effects of illumination can be removed .




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•Brought out the details of objects inside the shelter and balanced the gray levels of the outside wall.


Some Properties of the 2-D Fourier Transform

TranslationDistributivity and ScalingRotationPeriodicity and Conjugate SymmetrySeparabilityConvolution and Correlation

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Translation

),(),( 00)//(2 00 vvuuFeyxf NyvMxuj −−⇔+π

)//(200

00),(),( NvyMuxjevuFyyxxf +−⇔−− π

and

)2/,2/()1)(,( NvMuFyxf yx −−⇔− +


Translation

The previous equations mean:Multiplying f(x,y) by the indicated exponential term and taking the transform of the product results in a shift of the origin of the frequency plane to the point (u0,v0).

Multiplying F(u,v) by the exponential term shown and taking the inverse transform moves the origin of the spatial plane to (x0,y0).

A shift in f(x,y) doesn’t affect the magnitude of its Fourier transform.

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Distributivity and ScalingDistributive over addition but not over multiplication.

),(),( vuaFyxaf ⇔

)},({)},({)},(),({ 2121 yxfyxfyxfyxf ℑ+ℑ=+ℑ

)},({)},({)},(),({ 2121 yxfyxfyxfyxf ℑ⋅ℑ≠⋅ℑ

For two scalars a and b,

)/,/(1),( bvauFab

byaxf ⇔


Rotation

Polar coordinates:

Which means that:

),(),,( become ),(),,( ϕωθ FrfvuFyxf

ϕϕθθ sincossincos wvwuryrx ====

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Rotation

Which means that rotating f(x,y) by an angle θ, rotates F(u,v) by the same angle (and vice versa).

),(),( 00 θϕωθθ +⇔+ Frf


Periodicity & Conjugate Symmetry

The discrete FT and its inverse are periodic with period N:

),(),(),(),( NvMuFNvuFvMuFvuF ++=+=+=

For real f(x,y), FT also exhibits conjugate symmetry:

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

vuFvuFvuFvuF

−−=

−−=

36


Periodicity & Conjugate Symmetry


SeparabilityThe discrete FT pair can be expressed in separable forms which (after some manipulations) can be expressed as:

F(u,v) =1M

F(x,v)exp[− j2πux / M]x= 0

M −1

∑

Where: F(x,v) =1N

f (x, y)exp[− j2πvy /N]y= 0

N−1

∑⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

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Separability

The discrete FT pair can be expressed in separable forms which can be expressed as:

∑

∑ ∑−

=

−

=

−

=

−=

−−=

1

0

1

0

1

0

]/2exp[),(1),(

]/2exp[),(1]/2exp[1),(

M

x

M

x

M

y

MuxjvxFM

vuF

MvyjyxfN

MuxjM

vuF

π

ππ

Where: F(x,v) =1N

f (x, y)exp[− j2πvy /N]y= 0

N−1

∑⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥


Convolution

Convolution theorem with FT pair:

),(),(),(*),( vuGvuFyxgyxf ⇔

),(*),(),(),( vuGvuFyxgyxf ⇔

H(u,v)vuFMN

y)f(x,y)h(x,

H(u,v)vuF,y)f(x,y)*h(x

*),(1),(

⇔

⇔

Convolution theorem with FT pair: (Matlab)

38

75

Periodicity: the Need for Padding

76


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• For convolution or correlation of two images f(x,y) and h(x,y) of sizes A*B and C*D, Wraparound error is avoided if we create images with the size of P*Q using following equation:

⎩⎨⎧

≤≤≤≤−≤≤−≤≤

=

⎩⎨⎧

≤≤≤≤−≤≤−≤≤

=

QyDorPxCDyandCxyxh

yxh

QyBorPxAByandAxyxf

yxf

e

e

01010),(

),(

01010),(

),(

11

−+≥−+≥

DBQCAP


Correlation (Cross correlation)

Correlation of two functions f(x,y) and g(x,y):

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

vuGvuFyxgyxfvuGvuFyxgyxf

οο

⇔⇔

Fourier Transform and Correlation:

∑∑−

=

−

=

++

=1

0

1

0

* ),(),(1),(),(

M

m

N

nnymxgnmf

MN

yxgyxf ο

40


Correlation (Auto correlation)

Autocorrelation:

Applicable for Template Matching

80

Correlation (Application)

Fourier Series

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