Fourier Transform

Resmi N.G.Resmi N.G.Reference: Digital Image Processing

Rafael C. GonzalezRichard E. Woods

Mathematical Background:Complex Numbers

� A complex number x is of the form:

a: real part, b: imaginary part

, 1x a jb where j= + = −

a: real part, b: imaginary part

� Addition

� Multiplication

( ) ( ) ( ) ( )a jb c jd a c j b d+ + + = + + +

( ).( ) ( ) ( )a jb c jd ac bd j ad bc+ + = − + +

3/19/2012 2CS04 804B Image Processing - Module1

� Magnitude-Phase (Vector) representation

Magnitude:

Phase:

φ Phase – Magnitude notation:


� Multiplication using magnitude-phase representation

� Complex conjugate

� Properties


Sine and Cosine Functions

π

π


Shifting or translating the sine function by a constant b.

Note: Cosine is a shifted sine function.

cos( ) sin( )2

t tπ

= +


Image Transforms� Key steps:

(1) Transform the image(2) Perform the task(s) in the transformed domain.(3) Apply inverse transform to return to the spatial domain.(3) Apply inverse transform to return to the spatial domain.

3/19/2012 CS04 804B Image Processing - Module1 7

Fourier Series

� Any function that periodically repeats itself can beexpressed as the sum of sines and/or cosines of differentfrequencies each multiplied by a different coefficient.




Fourier Transform

� Functions that are not periodic but whose area under thecurve is finite can be expressed as the integral of sinesand/or cosines multiplied by a weighing function.and/or cosines multiplied by a weighing function.

� Transforms a function from spatial domain to frequencydomain.


Why is Fourier Transform Useful?

� Easier to remove undesirable frequencies.

� Faster to perform certain operations in the frequency� Faster to perform certain operations in the frequencydomain than in the spatial domain.


Example: Removing undesirable frequencies

frequenciesnoisy signal

remove highfrequencies

reconstructedsignal

To remove certain frequencies, set their corresponding F(u)coefficients to zero!


How do frequencies show up in an image?

� Low frequencies correspond to slowly varyinginformation (e.g., continuous surface).

� High frequencies correspond to quickly varyinginformation (e.g., edges)information (e.g., edges)

Original Image Low-passed


� A function expressed in either a Fourier Series or FourierTransform can be reconstructed completely via an inverseprocess.process.


Fourier Transforms of Some Simple Shapes




One Dimensional Fourier Transform and its Inverse

� The Fourier Transform F(u) of a single variablecontinuous function f(x) is defined by

where

2( ) ( ) (1)j uxF u f x e dxπ∞

−

−∞

= − − − −∫= −� where

� Given F(u), f(x) can be obtained by means of inverseFourier Transform

� (1) and (2) constitute the Fourier Transform pair.


−∞1j = −

2( ) ( ) (2)j uxf x F u e duπ∞

−∞

= − − − −∫

/22 2

/2

2 2/22 2 2

/2

2 2

( ) ( ) 1.

2 2

2sin

2

Tj ux j ux

T

T Tj u j uTj ux

T

T Tj u j u

F u f x e dx e dx

e e ej u j u

uTe e uT

π π

π ππ

π π

π π

ππ

∞− −

−∞ −

−− −−

−

−

= =

− = = − −

−

∫ ∫


2 22 2

sin22 sin

2 2

2sin

22

2

j u j ue e uT

u j

uT

TuT

π ππ

π

π

π

− − = = −

=

Q


/42 2

/4

2 2/42 4 4

/4

2 2

( ) ( ) 2.

22 2

2sin

2

Tj ux j ux

T

T Tj u j uTj ux

T

T Tj u j u

F u f x e dx e dx

e e ej u j u

uTe e uT

π π

π ππ

π π

π π

ππ

∞− −

−∞ −

−− −−

−

−

= =

− = = − −

−

∫ ∫


2 24 4

sin242 sin

2 4

2sin

422

4

j u j ue e uT

u j

uT

TuT

π ππ

π

π

π

− − = = −

=

Q


Discrete Fourier Transform (DFT)

� Fourier Transform of a discrete function of one variable,f(x), x = 0, 1, 2, …, M-1 is given by

� where u = 0, 1, 2, …, M-1.


1 2

0

1( ) ( )

M j uxM

x

F u f x eM

π− −

=

= ∑

Inverse DFT1 2

0

1( ) ( )

M j uxM

u

f x F u eM

π−

=

= ∑

� where x = 0, 1, 2, …, M-1.

� The product of multipliers used in DFT and its inverseshould be equal to 1/M.


� Euler’s Formula :

� Therefore,

cos sinje jθ θ θ= +

( ) ( )11 2 2( ) ( ) cos sin

Mux uxF u f x jM M

π π−

= − ∑

� where u = 0, 1, 2, …, M-1.

� Each term of Fourier Transform F(u), for each value of u, is composed of sum of all values of function f(x).


( ) ( )0

2 2( ) ( ) cos sinx

ux uxF u f x jM MMπ π

=

= − ∑

� Values of f(x) are in turn multiplied by sines and cosinesof various frequencies.

� The domain (values of u) over which the values of F(u)range is called the frequency domain, because urange is called the frequency domain, because udetermines the frequency components of the transform.

� Each of the M terms of F(u) is called a frequencycomponent of the transform.


� Analogy –� Prism separates light into various color components each

depending on its wavelength or frequency content.

� Fourier Transform separates a function into variouscomponents based on frequency content.


� Fourier Transform in Polar Co-ordinates

� where

( )( ) ( ) j uF u F u e φ−=

12 2 2( ) ( ) ( )F u R u I u = +

� is called the magnitude or spectrum of Fourier Transformand

� is called the phase angle of Fourier Transform.


( ) ( ) ( )F u R u I u = +

1 ( )( ) tan

( )I u

uR u

φ − =

� Power Spectrum or Spectral Density� 2

2 2

( ) ( )

( ) ( )

P u F u

R u I u

=

= +


� Fourier Transform is centered at origin, but DFT is centered atM/2.


� DFT spectrum can be centered at u=0, by multiplying f(x) by(-1)x before taking the transform (Centering). F(0) will then beat u=M/2.



� Height of the spectrum doubles as area under the curve inx-domain doubles.

� Number of zeroes in the spectrum in the same intervaldoubles as the length of the function doubles.doubles as the length of the function doubles.


More on DFT� Samples are equally spaced.� Samples need not be at integer values of x in [0, M-1].� Can be spaced at x0, x0+∆x, …, x0+(M-1)∆x.� kth sample f(k) is at x0+k∆x.


0( ) (x x)f x f x= + ∆

( ) ( )F u F u u= ∆1

xu

M∆ =

∆

2D-Fourier Transform2 ( )( , ) ( , ) j ux vyF u v f x y e dxdyπ

∞ ∞− +

−∞ −∞

= ∫ ∫


2 ( )( , ) ( , ) j ux vyf x y F u v e dudvπ∞ ∞

+

−∞ −∞

= ∫ ∫

2D-Discrete Fourier Transform and its Inverse

� 2D-DFT of image f(x,y) of size MxN is given by

( )1 1 2

0 0

1( , ) ( , )

vyM N uxj M N

x y

F u v f x y eMN

π− − − +

= =

= ∑∑

� These equations constitute a 2D-DFT pair.� u, v – are transform or frequency variables.� x, y – are spatial or image variables.


( )1 1 2

0 0

( , ) ( , )vyM N uxj M N

u v

f x y F u v eπ− − +

= =

= ∑ ∑

� Fourier Spectrum:

� Phase Angle:

12 2 2( , ) ( , ) ( , )F u v R u v I u v = +

1 ( , )( , ) tan

( , )I u v

u vR u v

φ − =

� Power Spectrum:

�


( , )R u v

2

2 2

( , ) ( , )

( , ) ( , )

P u v F u v

R u v I u v

=

= +

� Centering is done by multiplying the input image functionby (-1)x+y prior to computing the transform. F(0,0) will thenbe at u=M/2, v=N/2.

M N


( , ).( 1) ( , )2 2

x y M Nf x y F u v+ ℑ − = − −

� gives the average of f(x,y).� That is, the value of Fourier Transform at the origin is

1 1

0 0

1(0,0) ( , )

M N

x y

F f x yMN

− −

= =

= ∑∑

� That is, the value of Fourier Transform at the origin isequal to the average gray level of the image.

� It corresponds to the dc component of the spectrum(frequencies are zero at the origin).


Properties of Fourier Transform� Linearity

� If F(u) and G(u) are Fourier transforms of f(x) and g(x)� If F(u) and G(u) are Fourier transforms of f(x) and g(x)respectively, then

� where a and b are constants.


[ ( ) ( )] ( ) ( )F af x bg x aF u bG u+ = +

� Proof:

2

2 2

[ ( ) ( )] [ ( ) ( )]

( ) ( )

j ux

j ux j ux

F af x bg x af x bg x e dx

af x e dx bg x e dx

π

π π

∞−

−∞

∞ ∞− −

+ = +

= +

∫

∫ ∫


2 2

( ) ( )

( ) ( )

( ) ( )

j ux j ux

af x e dx bg x e dx

a f x e dx b g x e dx

aF u bG u

π π

−∞ −∞

∞ ∞− −

−∞ −∞

= +

= +

= +

∫ ∫

∫ ∫

� Linearity� Additivity: The property that performing a linear process on

the sum of inputs is same as that of performing theoperations individually and then summing up the resuts.operations individually and then summing up the resuts.

� Homogeneity: The property that the response of a linearsystem to a constant times an input is same as the responseto the original input multiplied by a constant.


� Change of Scale Property� If F(u) is the Fourier transform of f(x), then

� Proof:

1[ ( )] , 0

uF f ax F a

a a = ≠

2[ ( )] ( ) j uxF f ax f ax e dxπ∞

−= ∫


2[ ( )] ( )

,

,

j uxF f ax f ax e dx

Let ax t

tOr x

adt

dxa

π−

−∞

=

=

=

∴ =

∫

2

2

[ ( )] ( )

1( )

1

j ux

tj u

a

u

F f ax f ax e dx

f t e dta

π

π

∞−

−∞

∞−

−∞

∞

=

=

∫

∫


21( )

1

uj t

af t e dta

uF

a a

π∞

−

−∞

=

=

∫

� Shifting Property� If F(u) is the Fourier transform of f(x), then

� Proof:

020[ ( )] ( )j uxF f x x e F uπ−− =

2[ ( )] ( ) j uxF f x x f x x e dxπ∞

−− = −∫


0 0

0

0

[ ( )] ( )

,

,

F f x x f x x e dx

Let x x t

Or x x t

dx dt

−∞

− = −

− =

= +

∴ =

∫

0

0

2 ( )0

22

[ ( )] ( )

( ) .

j u t x

j uxj ut

F f x x f t e dt

f t e e dt

π

ππ

∞− +

−∞

∞−−

−∞

− =

=

∫

∫


0

0

2 2

2

( )

( )

j ux j ut

j ux

e f t e dt

e F u

π π

π

−∞

∞− −

−∞

−

=

=

∫

Properties of 2D-DFT

� 1. Translation

0 02( , ) ( , ) (1)

u x v yj M Nf x y e F u u v vπ + ⇔ − − −− −


0 0( , ) ( , ) (1)M Nf x y e F u u v v ⇔ − − −− −

0 02

0 0( , ) ( , ) (2)u x v yj M Nf x x y y F u v e

π − + − − ⇔ −−−

� When

� Therefore, (1) and (2) can be written as:

0 02 2NMu and v= =

0 02( ) ( 1)

u x v yj M N j x y x ye eπ π + + + = = −

M N


( , )( 1) ,2 2

x y M Nf x y F u v+ − ⇔ − −

, ( , )( 1)2 2

u vM Nf x y F u v + − − ⇔ −

� 2. Distributivity

[ ] [ ] [ ]1 2 1 2( , ) ( , ) ( , ) ( , )f x y f x y f x y f x yℑ + = ℑ +ℑ

[ ] [ ] [ ]1 2 1 2( , ). ( , ) ( , ) . ( , )f x y f x y f x y f x yℑ ≠ ℑ ℑ

� 3. Scaling


( , ) ( , )af x y aF u v⇔

1( , ) ,

u vf ax by F

ab a b ⇔

� 4. Rotation� Representation in terms of polar coordinates

cos , sin

cos , sin

x r y r

u w v w

θ θφ φ

= =

= =

� Rotating f(x,y) by an angle θ0 rotates F(u,v) by the sameangle.

� Rotating F(u,v) rotates f(x,y) by the same angle.


0 0( , ) ( , )f r F wθ θ φ θ+ ⇔ +

� 5. Periodicity

� Inverse Transform is also periodic.

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

( , ) ( , ) ( , ) ( , )f x y f x M y f x y N f x M y N= + = + = + +

� 6. Conjugate Symmetry


*( , ) ( , )F u v F u v= − −

� 7. Separability

1 12 2

0 0

12

1 1( , ) . ( , )

1( , )

M Nj ux M j vy N

x y

Mj ux M

F u v e f x y eM N

F x v eM

π π

π

− −− −

= =

−−

=

=

∑ ∑

∑

� F(x,v) à Fourier Transform along one row of f(x,y).� à 1Dimensional Fourier Transform is computed

for each value of x, with v varying from 0 to N-1.


0xM =∑

� By varying x from 0 to M-1, Fourier Transform iscomputed along all rows of f(x,y), with u remainingconstant.

� To complete the 2D transform, vary u from 0 to M-1.Compute 1D transform along each column of F(x,v).Compute 1D transform along each column of F(x,v).


( , ) ( , ) ( , )f x y F x v F u v⇒ ⇒

1D Row Transform 1D ColumnTransform

Fast Fourier Transform� 1D-transform of M points requires order of M2

multiplications / additions.

� FFT requires order of Mlog M operations.� FFT requires order of Mlog2M operations.

� where and M=2k; k is a positive integer.


1

0

1( ) ( )

MuxM

x

F u f x WM

−

=

= ∑2jM

MW eπ−

=

1

0

2 1

2

1( ) ( )

1( )

MuxM

x

kuxk

F u f x WM

f x W

−

=

−

=

=

∑

∑


20

1 1(2 ) (2 1)

2 20 0

( )2

1 1 1(2 ) (2 1)

2

kx

k ku x u xk k

x x

f x Wk

f x W f x Wk k

=

− −+

= =

=

= + +

∑

∑ ∑

2

2 (2 )

,

,

jM

M

j u x

We have

W e

Therefore

π

π

−

−

=


2 (2 )(2 ) 2

2

2

j u xu x kk

j uxk

uxk

W e

e

W

π

π

−

−

=

=

=

1 1(2 ) (2 )

2 2 20 0

1 1 1( ) (2 ) (2 1)

2

k ku x u x uk k k

x x

F u f x W f x W Wk k

− −

= =

= + +

∑ ∑

1 1

20 0

1 1 1(2 ) (2 1)

2

k kux ux uk k k

x x

f x W f x W Wk k

− −

= =

= + +

∑ ∑


0 02 x xk k= = ∑ ∑

Feven(u) Fodd(u)

2

1( ) ( )

2u

even odd kF u F u W = +

1( )

0

2 1

2 20

1( ) ( )

1( )

2

Mu k x

Mx

kux kxk k

x

F u k f x WM

f x W Wk

−+

=

−

=

+ =

=

∑

∑


1(2 ) (2 )

2 20

1(2 1) (2 1)

2 20

1 1(2 )

2

1 1(2 1)

2

ku x k xk k

x

ku x k xk k

x

f x W Wk

f x W Wk

−

=

−+ +

=

=

+ +

∑

∑

12 2

2 20

12 2

2 2 2 20

1 1(2 )

2

1 1(2 1)

2

kux kxk k

x

kux u kx kk k k k

x

f x W Wk

f x W W W Wk

−

=

−

=

=

+ +

∑

∑


0

2 22 22

2

22

2

2

, 1

1

x

j kxkx j xkk

j kk jkk

k

But W e e

and W e e

ππ

ππ

=

−−

−−

= = =

= = = −

1

0

1

20

1

1 1( ) (2 ) .1

2

1 1(2 1) .1.( 1)

2

1 1(2 )

kuxk

x

kux uk k

x

kux

F u k f x Wk

f x W Wk

f x W

−

=

−

=

−

+ =

+ + −

=

∑

∑

∑


0

1

20

2

1 1(2 )

2

1 1(2 1) ( )

2

1( ) ( )

2

uxk

x

kux uk k

x

ueven odd k

f x Wk

f x W Wk

F u F u W

=

−

=

=

+ + −

= −

∑

∑

2

1( ) ( ) ( )

2u

even odd kF u F u F u W = +

� Fast Fourier Transform Computation Steps


2

1( ) ( ) ( )

2u

even odd kF u k F u F u W + = −

� Requires only two M/2-point transforms.

Thank YouThank You


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Fourier Transform

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