Chapter 4 Filtering in the Frequency Domain - BGUdip111/wiki.files/DIP3E_Chapter04_Art.pdf · Digital Image Processing, 3rd ed. Gonzalez & Woods Chapter 4 Filtering in the Frequency

Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.

www.ImageProcessingPlace.com

Gonzalez & Woods

Chapter 4Filtering in the Frequency Domain


+)sin(tA +++

=

© 1992–2008 R. C. Gonzalez & R. E. Woods



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Step FunctionStep FunctionThe unit step function, is a discontinuous function whose value is zero for negative argument and one for positive argument.

A-A

0000

)(tt

tH

Rectangle FunctionThe unit step function, is a discontinuous function whose value is zero for negative argument and one for positive argument.

AtA

Attct 1

0)(Re

A-A

At0

Impulse FunctionThe Dirac delta function, or δ function, is zero for all values of the parameter except when the parameter is zero, and its integral

th t f t i l tover the parameter from −∞ to ∞ is equal to one

000

)(tt

t

1)( dtt




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)()()(

)0()()(

tfdttttf

fdtttf

)()()( 00 tfdttttf

In discrete variable

0001

)(tx

x

)0()()( fxxfx


)()()( 00 tfxxxfx



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)()( Tntts T

n




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Complex NumberComplex Number

12

jIRCj

jIRC

jIRC

)t (

|| 22

IIRC

))sin()(cos(||

)tan(

jCCR


)sin()cos( je j



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Function Representationu c o ep ese o• Taylor Series

nn xxxfxf )()(1)( 00)(

...!

...!4!3!2!1

1432

xxxxxe

nx

n

xxxfn

xf )()(!

)( 000

• Fourier Series !!4!3!2!1 n

a

)sin()cos((2

)(1

0 ntbntaatf nn

n

ktktktktktf 0];)/2sin()/2sin()/[sin(2)( 11


ktktktkttf 0..];)/2sin()/2sin()/[sin()( 31

21

3



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xe




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Fourier TransformFor a give function g(x) of a real variable x, the F i t f ti f ( ) hi h i d t dFourier transformation of g(x) which is denoted as

2)(G(u){g(x)} dxexg uxj -

)(G(u){g(x)} dxexg j

-

21 )(G(u){g(x)} dxexg uxj




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SinceSince

F th t th F i t f F( ))2cos()2sin(2 tjte tj

For that reason the Fourier transform F(u)F(u) = R(u) + jI(u)

Which often appear as

|)(|)( )(euFuF uj

)())(t (

)()(|)(| 22

uIuIuRuF


)()())(tan(

uRu



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Spatial Domain Frequency Domain




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Spatial and FrequencySpatial and Frequency




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Spatial and Frequency RepresentationsSpatial and Frequency Representations

mai

n )()()()(' 321 xbxbxbxb

atia

lDom

)12(2sin(12

10

3

1xfn

nn

Spa

mai

n

)32sin(1)(

)2sin()( 01

Xfxb

Xfxb

uenc

y D

om

)52sin(1)(

)32sin(3

)(

03

02

Xfxb

Xfxb


Freq

u )52sin(5

)( 03 Xfxb



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2/2/ WW A

2/

2/22/

2/

22

2)()( W

WutjW

W

utjutj euj

AdtAedtetfuF




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2D Fourier Transform2D Fourier TransformFor a give function g(x) of a real variable x, the Fourier transformation of g(x) which is denoted as

dxdyeyxgvuGyxg vyuxj )(2),(),()},({

and

dudvevuGyxgvuG vyuxj )(21 ),(),()},({




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AAeeA

dyedxeAdxdyeyxfvuF

vYjuXjYvyjXuxj

Y uyjX uxjvyuxj

]1[]1[

),(),(

)(2)(2)(2)(2

0

)(2

0

)(2)_(2

vYuXAXY

evj

eujvjuj

A vYjuXj

)sin()sin(

]1[2

]1[222

)(2)(2

00


vYuX



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Discrete Fourier TransformDiscrete Fourier Transform• Let us discretize a continuous function f(x) into the N uniform

samples that generate the sequence f(x0) f(x0+Δx) f(x0+2Δx) f(x0+3Δx) f(x0+[N-1]Δx)f(x0), f(x0+Δx), f(x0+2Δx), f(x0+3Δx), …, f(x0+[N-1]Δx)

• Hence f(x) = f(x0+i Δx)• We could denote the samples as

f(0) f(1) f(2) f(N 1)f(0), f(1), f(2), …,f(N-1).• The Fourier Transform is

1

/2 1210;)(1)(N

Nuxj NuexfuF

1

0

1,...,2,1,0;)()(

N

x

j

and

NuexfN

uF


1

0

/2)()(N

u

NuxjeuFxf



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2D Discrete Fourier Transform

eyxfvuFM

NvyMuxjN

)(1)(1

)//(21

2D Discrete Fourier Transform

NvMu

eyxfNM

vuFx y

1,...,2,1,0;1,...,2,1,0

),(),(0 0

Ff

andM

NvyMuxjN

)()(1

)//(21

evuFyxfu

NvyMuxj

v

11

),(),(0

)//(2

0


yNv

xMu

;



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Display F(u v)Display F(u,v)The Dynamic rang of Fourier spectra usually is much higher than the typical display device is able to reproduce fathfuly. Therefore, often use the logarithm function to perform the appropriate compression of the rang.

|)),(|1log(),( vuFcvuD

Where c is a scale factor




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Examples p


http://www.cs.unm.edu/~brayer/vision/fourier.html



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Properties of 2D Fourier TransformProperties of 2D Fourier Transform• Spatial and Frequency Domain

f(t, z) sampled from f(x, y) using the separation between samples as T and Z1

ZNv

TMu

1

1

• Translation and RotationMultiplying f(x,y) by the exponential shifts the original of DFT to (u0,v0).Multiplying F(u v) by the exponential shifts the original of f(x y) to (x0 y0);Multiplying F(u,v) by the exponential shifts the original of f(x, y) to (x0,y0);

)//(200

00),(),( NvyMuxjevuFyyxxf


)//(200

00),(),( NyvMxujeyxfvvuuF



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Properties of 2D Fourier Transform pPeriodicity

The Fourier transform and inverse are infinitely periodic on the u and vinfinitely periodic on the u and v directions. (k1 and k2 are integers).

),(),(),( 21 NkvuFvMkuFvuF

)(),(),(),(

),(

21

21

NkyMkxfNkyxfyMkxfyxf

NkvMkuF

),( 21 NkyMkxf To show the origin of F(u,v) at the center we shift the data by M/2 and N/2


)2/,2/()1)(,( NvMuFyxf yx



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Properties of 2D Fourier Transform pSymmetryAny real or complex function can be expressed as the sum of even and odd part

)()()( yxwyxwyxw

2),(),(),(

),(),(),(yxwyxwyxw

yxwyxwyxw

e

oe

2),(),(),( yxwyxwyxwe

Whi h h th t f ti t i d dd f tiWhich shows that even functions are symmetric and odd functions are antisymmetric

),(),( yxwyxw ee


),(),( yxwyxw oo



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Properties of 2D Fourier Transform pSymmetryThe Fourier transform of a real function f(x,y) is conjugate symmetric

)()(* vuFvuF ),(),( vuFvuF The Fourier transform of a imaginary function f(x,y) is conjugate anti-symmetric

),(),(* vuFvuF *

1

),(1),(

1 1

*1

0

)//(21

0

* eyxfNM

vuF

M N

M

x

NvyMuxjN

y

Proof

)(1

),(1

1)/][/]([2

1

0

)//(2

0

*

eyxf

eyxfNM

MNyvMxuj

N

x

NvyMuxj

y


),(

),(0 0

vuF

eyxfNM x y



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Sampling and Fourier TransformSampling and Fourier Transform1. Converting continuous function/signal into

a discrete one.2. The sampling is uniform at T intervalsTh l d f ti d th l f hThe sampled function and the value of each Sample are :

)()()()()(~ Tnttftstftf T

)()()(

)()()()()(

TkfdtTkttff

Tnttftstftf

k

nT




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The Fourier of Sampled functionThe Fourier of Sampled functionLet F() and be the Fourier transform of the continuous function f(t) and its equivalent Sampled function

)(~ F

)(~ tf

)()()}()({)}(~{)(~ SFtstftfF T

)(1)(Tn

TS

TT n

n

dSFSFF

1

)()()()()(~

dTnF

T

dTnF

T

n

n

)()(1

)()(1


TnF

T n

n

1



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Sampling TheoremSampling TheoremBand limited-A function f(t) whose Fourier transform is zero out of the interval [-max , max] is called band limited

We can recover a function f(t) from its sampled representation if we can isolate a copy of F() from the periodic sequence of copies.

Extracting from a single period that represents F() is possible in the separation between copies is sufficient , which is guaranteed if ½T > max

)(~ F

max21 T


Which is call the Nyquist Rate



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To extract a single copy we multiply )(~ Fg py p yby H().

)(F

0)( maxmax

otherwise

TH

)(~)()(

0

FHF

otherwise

Once we have F() we can recover f(t)

deFtftj2

)()(




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Aliasing gAppears when the sampling rate is less than the Nyquist rate – under-sampling. The inverse Fourier transform would then yield a corrupted f nction of t hich is kno n as freq encfunction of t, which is known as frequency aliasing or just aliasing.




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Gonzalez & Woods



Sampling in the 2D Domainp g




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Examples of Moire peffect




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Documents

Chapter 4 Filtering in the Frequency Domain - BGUdip111/wiki.files/DIP3E_Chapter04_Art.pdf · Digital Image Processing, 3rd ed. Gonzalez & Woods Chapter 4 Filtering in the Frequency