EE 4780 2D Discrete Fourier Transform (DFT). Bahadir K. Gunturk2 2D Discrete Fourier Transform 2D Fourier Transform 2D Discrete Fourier Transform (DFT)

EE 4780

2D Discrete Fourier Transform (DFT)

Bahadir K. Gunturk 2

2D Discrete Fourier Transform

2D Fourier Transform

2 ( )( , ) [ , ] j um vn

m n

F u v f m n e

1 1 2

0 0

1[ , ] [ , ]

k lM N j m nM N

m n

F k l f m n eMN

2D Discrete Fourier Transform (DFT)

2D DFT is a sampled version of 2D FT.

Bahadir K. Gunturk 3

2D Discrete Fourier Transform

Inverse DFT

1 1 2

0 0

1[ , ] [ , ]

k lM N j m nM N

m n

F k l f m n eMN

2D Discrete Fourier Transform (DFT)

1 1 2

0 0

[ , ] [ , ]k lM N j m nM N

k l

f m n F k l e

where and 0,1,..., 1k M 0,1,..., 1l N

Bahadir K. Gunturk 4

2D Discrete Fourier Transform

Inverse DFT

1 1 2

0 0

1[ , ] [ , ]

k lM N j m nM N

m n

F k l f m n eMN

It is also possible to define DFT as follows

1 1 2

0 0

1[ , ] [ , ]

k lM N j m nM N

k l

f m n F k l eMN

where and 0,1,..., 1k M 0,1,..., 1l N

Bahadir K. Gunturk 5

2D Discrete Fourier Transform

Inverse DFT

1 1 2

0 0

[ , ] [ , ]k lM N j m nM N

m n

F k l f m n e

Or, as follows

1 1 2

0 0

1[ , ] [ , ]

k lM N j m nM N

k l

f m n F k l eMN

where and 0,1,..., 1k M 0,1,..., 1l N

Bahadir K. Gunturk 6

2D Discrete Fourier Transform

Bahadir K. Gunturk 7

2D Discrete Fourier Transform

Bahadir K. Gunturk 8

2D Discrete Fourier Transform

Bahadir K. Gunturk 9

2D Discrete Fourier Transform

Bahadir K. Gunturk 10

Periodicity

1 1 2

0 0

1[ , ] [ , ]

k lM N j m nM N

m n

F k l f m n eMN

1 1 2

0 0

1[ , ] [ , ]

k M l NM N j m nM N

m n

F k M l N f m n eMN

1 1 2 2

0 0

1[ , ]

k l M NM N j m n j m nM N M N

m n

f m n e eMN

1

[M,N] point DFT is periodic with period [M,N]

[ , ]F k l

Bahadir K. Gunturk 11

Periodicity

1 1 2

0 0

[ , ] [ , ]k lM N j m nM N

k l

f m n F k l e

1

[M,N] point DFT is periodic with period [M,N]

1 1 2 ( ) ( )

0 0

[ , ] [ , ]k lM N j m M n NM N

k l

f m M n N F k l e

1 1 2 2

0 0

[ , ]k l k lM N j m n j M NM N M N

k l

F k l e e

[ , ]f m n

Bahadir K. Gunturk 12

Convolution

Be careful about the convolution property!

[ ]f m

m*

[ ]g m

m

[ ]* [ ]f m g m

m

Length=P Length=Q Length=P+Q-1

For the convolution property to hold, M must be greater than or equal to P+Q-1.

[ ]* [ ] [ ] [ ]f m g m F k G k

Bahadir K. Gunturk 13

Convolution

Zero padding

[ ]* [ ] [ ] [ ]f m g m F k G l

[ ]f m

m*

[ ]g m

m

[ ]* [ ]f m g m

m

[ ]F k

4-point DFT(M=4)

[ ]G k [ ] [ ]F k G k

Bahadir K. Gunturk 14

DFT in MATLAB

Let f be a 2D image with dimension [M,N], then its 2D DFT can be computed as follows:

Df = fft2(f,M,N);

fft2 puts the zero-frequency component at the top-left corner.

fftshift shifts the zero-frequency component to the center. (Useful for visualization.)

Example:f = imread(‘saturn.tif’); f = double(f);

Df = fft2(f,size(f,1), size(f,2));

figure; imshow(log(abs(Df)),[ ]);

Df2 = fftshift(Df);

figure; imshow(log(abs(Df2)),[ ]);

Bahadir K. Gunturk 15

DFT in MATLAB

f

Df = fft2(f)

After fftshift

Bahadir K. Gunturk 16

DFT in MATLAB

Let’s test convolution property

f = [1 1];

g = [2 2 2];

Conv_f_g = conv2(f,g); figure; plot(Conv_f_g);

Dfg = fft (Conv_f_g,4); figure; plot(abs(Dfg));

Df1 = fft (f,3);

Dg1 = fft (g,3);

Dfg1 = Df1.*Dg1; figure; plot(abs(Dfg1));

Df2 = fft (f,4);

Dg2 = fft (g,4);

Dfg2 = Df2.*Dg2; figure; plot(abs(Dfg2));

Inv_Dfg2 = ifft(Dfg2,4);

figure; plot(Inv_Dfg2);

Bahadir K. Gunturk 17

DFT in MATLAB

Increasing the DFT size

f = [1 1];

g = [2 2 2];

Df1 = fft (f,4);

Dg1 = fft (g,4);

Dfg1 = Df1.*Dg1; figure; plot(abs(Dfg1));

Df2 = fft (f,20);

Dg2 = fft (g,20);

Dfg2 = Df2.*Dg2; figure; plot(abs(Dfg2));

Df3 = fft (f,100);

Dg3 = fft (g,100);

Dfg3 = Df3.*Dg3; figure; plot(abs(Dfg3));

Bahadir K. Gunturk 18

DFT in MATLAB

Scale axis and use fftshift

f = [1 1];

g = [2 2 2];

Df1 = fft (f,100);

Dg1 = fft (g,100);

Dfg1 = Df1.*Dg1;

t = linspace(0,1,length(Dfg1));

figure; plot(t, abs(Dfg1));

Dfg1_shifted = fftshift(Dfg1);

t2 = linspace(-0.5, 0.5, length(Dfg1_shifted));

figure; plot(t, abs(Dfg1_shifted));

Bahadir K. Gunturk 19

Example

Bahadir K. Gunturk 20

Example

Bahadir K. Gunturk 21

DFT-Domain Filtering

a = imread(‘cameraman.tif');

Da = fft2(a);

Da = fftshift(Da);

figure; imshow(log(abs(Da)),[]);

H = zeros(256,256);

H(128-20:128+20,128-20:128+20) = 1;

figure; imshow(H,[]);

Db = Da.*H;

Db = fftshift(Db);

b = real(ifft2(Db));

figure; imshow(b,[]);

Frequency domain Spatial domain

H

Bahadir K. Gunturk 22

Low-Pass Filtering

81x8161x61 121x121

Bahadir K. Gunturk 23

Low-Pass Filtering

1 1 11

1 1 19

1 1 1

* =

DFT(h)

h

Bahadir K. Gunturk 24

High-Pass Filtering

* =1 1 1

1 8 1

1 1 1

DFT(h)

h

Bahadir K. Gunturk 25

High-Pass Filtering

High-pass filter

Bahadir K. Gunturk 26

Anti-Aliasing

a=imread(‘barbara.tif’);

Bahadir K. Gunturk 27

Anti-Aliasing

a=imread(‘barbara.tif’);b=imresize(a,0.25);c=imresize(b,4);

Bahadir K. Gunturk 28

Anti-Aliasing

a=imread(‘barbara.tif’);b=imresize(a,0.25);c=imresize(b,4);

H=zeros(512,512);H(256-64:256+64, 256-64:256+64)=1;

Da=fft2(a);Da=fftshift(Da);Dd=Da.*H;Dd=fftshift(Dd);d=real(ifft2(Dd));

Bahadir K. Gunturk 29

Noise Removal For natural images, the energy is concentrated mostly in

the low-frequency components.

Profile along the red lineDFT of “Einstein”“Einstein”

Noise=40*rand(256,256);Signal vs Noise

Bahadir K. Gunturk 30

Noise Removal

At high-frequencies, noise power is comparable to the signal power.

Signal vs Noise

Low-pass filtering increases signal to noise ratio.

Bahadir K. Gunturk 31

Appendix

Bahadir K. Gunturk 32

Appendix: Impulse Train

■ The Fourier Transform of a comb function is

2, ,[ , ] [ , ] j um vn

M N M Nm n

F comb m n comb m n e

2[ , ] j um vn

k lm n

m kM n lN e

2[ , ] j um vn

m nk l

m kM n lN e

2j ukM vlN

k l

e

Bahadir K. Gunturk 33

Impulse Train (cont’d)

■ The Fourier Transform of a comb function is

2, [ , ] j ukM vlN

M Nk l

F comb m n e

2 ( ) ( )1 j uM k vN l

k l

e

(Fourier Trans. of 1)

( , )k l

uM k vN l

1

,k l

k lu v

MN M N

?

Bahadir K. Gunturk 34

Impulse Train (cont’d)

■ Proof

( ) ( ) ( ) ( )k k

uM k F u du uM k F u du

1( )

k

vv k F dv

M M

1

k

kF

M M

1( )

k

kv F v dv

M M

1( )

k

ku F u du

M M

Bahadir K. Gunturk 35

Appendix: Downsampling

m

n

u

v

[ , ]f m n( , )F u v

1 2

1 2

[ , ] [ , ]d m n f Mm Nn Question: What is the Fourier Transform of ?

Bahadir K. Gunturk 36

Downsampling

Let ,[ , ] [ , ] [ , ]M Ng m n f m n comb m n

Using the multiplication property:

1 1,

1( , ) ( , )* ( , )

M N

G u v F u v comb u vMN

1 1

2 2

1 1

2 2

1( , ) ,

k l

k lF x y u x v y dxdy

MN M N

, ( , ) ,M Nk l

comb x y x kM y lN

Bahadir K. Gunturk 37

Downsampling1 1

2 2

1 1

2 2

1( , ) ( , ) ,

k l

k lG u v F x y u x v y dxdy

MN M N

1,

k lk l

k lF u v

MN M N

1 1 such that

2 2k

kk u

M

where

1 1 such that

2 2l

ll v

N

Bahadir K. Gunturk 38

Example

umW

[ ]f m

11

( )F u

umW

2[ ] [ ] [ ]g m f m comb m

11

( )G u

0

0

Bahadir K. Gunturk 39

Example

umW

[ ]f m

11

( )F u

umW

2[ ] [ ] [ ]g m f m comb m

11

( )G u

0

0

m

[ ] [2 ]d m g m

0

?

Bahadir K. Gunturk 40

Downsampling

,[ , ] [ , ] [ , ]

[ , ] [ , , ]M Ng m n f m n comb m n

d m n g Mm N n

2 2( , ) [ , ] [ , ]j um vn j um vn

m n m n

D u v d m n e g Mm Nn e

1( , ) ,

k lk l

k lG u v F u v

MN M N

2 ' '

' ..., ,0, ,... ' ..., ,0, ,...

2 ' '

' '

[ ', ']

[ ', ']

u vj m n

M N

m M M n N N

u vj m n

M N

m n

g m n e

g m n e

'

'

Mm m

Nn n

1, ,

k lk l

u v u k v lG FM N MN M M N N

Bahadir K. Gunturk 41

Example

umW

[ ]f m

11

( )F u

umW

2[ ] [ ] [ ]g m f m comb m

11

( )G u

0

0

m

[ ] [2 ]d m g m

0u

2W 11

( )D u

Bahadir K. Gunturk 42

Example

umW

[ ]f m

11

( )F u

0

m

[ ] [ ]d m f Mm

0u

MW 11

( )D u

1

2MW No aliasing if