ee 7730 2d fourier transform. bahadir k. gunturk ee 7730 - image analysis i 2 summary of lecture 2...
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EE 7730
2D Fourier Transform
Bahadir K. Gunturk EE 7730 - Image Analysis I 2
Summary of Lecture 2
We talked about the digital image properties, including spatial resolution and grayscale resolution.
We reviewed linear systems and related concepts, including shift invariance, causality, convolution, etc.
Bahadir K. Gunturk EE 7730 - Image Analysis I 3
Fourier Transform
What is ahead? 1D Fourier Transform of continuous signals 2D Fourier Transform of continuous signals 2D Fourier Transform of discrete signals 2D Discrete Fourier Transform (DFT)
Bahadir K. Gunturk EE 7730 - Image Analysis I 4
Fourier Transform: Concept
■ A signal can be represented as a weighted sum of sinusoids.
■ Fourier Transform is a change of basis, where the basis functions consist of sines and cosines.
Bahadir K. Gunturk EE 7730 - Image Analysis I 5
Fourier Transform
Cosine/sine signals are easy to define and interpret. However, it turns out that the analysis and manipulation of
sinusoidal signals is greatly simplified by dealing with related signals called complex exponential signals.
A complex number: z = x + j*y
A complex exponential signal: r*exp(j*a) =r*cos(a) + j*r*sin(a)
Bahadir K. Gunturk EE 7730 - Image Analysis I 6
Fourier Transform: 1D Cont. Signals■ Fourier Transform of a 1D continuous signal
2( ) ( ) j uxF u f x e dx
■ Inverse Fourier Transform
2( ) ( ) j uxf x F u e du
2 cos 2 sin 2j uxe ux j ux “Euler’s formula”
Bahadir K. Gunturk EE 7730 - Image Analysis I 7
Fourier Transform: 2D Cont. Signals■ Fourier Transform of a 2D continuous signal
■ Inverse Fourier Transform
2 ( )( , ) ( , ) j ux vyf x y F u v e dudv
2 ( )( , ) ( , ) j ux vyF u v f x y e dxdy
f F
■ F and f are two different representations of the same signal.
Bahadir K. Gunturk EE 7730 - Image Analysis I 8
Examples
Magnitude: “how much” of each componentPhase: “where” the frequency component in the image
Bahadir K. Gunturk EE 7730 - Image Analysis I 9
Examples
Bahadir K. Gunturk EE 7730 - Image Analysis I 10
Fourier Transform: Properties■ Linearity
■ Shifting
■ Modulation
■ Convolution
■ Multiplication
■ Separable functions
( , ) ( , ) ( , ) ( , )af x y bg x y aF u v bG u v
( , )* ( , ) ( , ) ( , )f x y g x y F u v G u v
( , ) ( , ) ( , )* ( , )f x y g x y F u v G u v
( , ) ( ) ( ) ( , ) ( ) ( )f x y f x f y F u v F u F v
0 02 ( )0 0( , ) ( , )j ux vyf x x y x e F u v
0 02 ( )0 0( , ) ( , )j u x v ye f x y F u u v v
Bahadir K. Gunturk EE 7730 - Image Analysis I 11
Fourier Transform: Properties■ Separability
2 ( )( , ) ( , ) j ux vyF u v f x y e dxdy
2 2( , ) j ux j vyf x y e dx e dy
2( , ) j vyF u y e dy
2D Fourier Transform can be implemented as a sequence of 1D Fourier Transform operations.
Bahadir K. Gunturk EE 7730 - Image Analysis I 12
Fourier Transform: Properties■ Energy conservation
2 2( , ) ( , )f x y dxdy F u v dudv
Bahadir K. Gunturk EE 7730 - Image Analysis I 13
Fourier Transform: Properties■ Remember the impulse function (Dirac delta function) definition
0 0( ) ( ) ( )x x f x dx f x
■ Fourier Transform of the impulse function
2 ( )( , ) ( , ) 1j ux vyF x y x y e dxdy
Bahadir K. Gunturk EE 7730 - Image Analysis I 14
Fourier Transform: Properties■ Fourier Transform of 1
2 ( )1 ( , )j ux vyF e dxdy u v
1 2 ( ) 2 (0 0)( , ) ( , ) 1j ux vy j x vF u v u v e dudv e
Take the inverse Fourier Transform of the impulse function
Bahadir K. Gunturk EE 7730 - Image Analysis I 15
Fourier Transform: 2D Discrete Signals■ Fourier Transform of a 2D discrete signal is defined as
where
2 ( )( , ) [ , ] j um vn
m n
F u v f m n e
1 1
,2 2
u v
1/ 2 1/ 22 ( )
1/ 2 1/ 2
[ , ] ( , ) j um vnf m n F u v e dudv
■ Inverse Fourier Transform
Bahadir K. Gunturk EE 7730 - Image Analysis I 16
Fourier Transform: Properties■ Periodicity: Fourier Transform of a discrete signal is periodic with period 1.
2 ( ) ( )( , ) [ , ] j u k m v l n
m n
F u k v l f m n e
2 2 2[ , ] j um vn j km j ln
m n
f m n e e e
2 ( )[ , ] j um vn
m n
f m n e
1 1
( , )F u v
Arbitrary integers
Bahadir K. Gunturk EE 7730 - Image Analysis I 17
Fourier Transform: Properties■ Linearity, shifting, modulation, convolution, multiplication, separability, energy conservation properties also exist for the 2D Fourier Transform of discrete signals.
Bahadir K. Gunturk EE 7730 - Image Analysis I 18
Fourier Transform: Properties■ Linearity
■ Shifting
■ Modulation
■ Convolution
■ Multiplication
■ Separable functions
■ Energy conservation
[ , ] [ , ] ( , ) ( , )af m n bg m n aF u v bG u v
0 02 ( )0 0[ , ] ( , )j um vnf m m n n e F u v
[ , ] [ , ] ( , )* ( , )f m n g m n F u v G u v
[ , ]* [ , ] ( , ) ( , )f m n g m n F u v G u v
0 02 ( )0 0[ , ] ( , )j u m v ne f m n F u u v v
[ , ] [ ] [ ] ( , ) ( ) ( )f m n f m f n F u v F u F v 2 2
[ , ] ( , )m n
f m n F u v dudv
Bahadir K. Gunturk EE 7730 - Image Analysis I 19
Fourier Transform: Properties■ Define Kronecker delta function
■ Fourier Transform of the Kronecker delta function
1, for 0 and 0[ , ]
0, otherwise
m nm n
2 2 0 0( , ) [ , ] 1j um vn j u v
m n
F u v m n e e
Bahadir K. Gunturk EE 7730 - Image Analysis I 20
Fourier Transform: Properties■ Fourier Transform of 1
To prove: Take the inverse Fourier Transform of the Dirac delta function and use the fact that the Fourier Transform has to be periodic with period 1.
2( , ) 1 ( , ) 1 ( , )j um vn
m n k l
f m n F u v e u k v l
Bahadir K. Gunturk EE 7730 - Image Analysis I 21
Impulse Train
■ Define a comb function (impulse train) as follows
, [ , ] [ , ]M Nk l
comb m n m kM n lN
where M and N are integers
2[ ]comb n
n
1
Bahadir K. Gunturk EE 7730 - Image Analysis I 22
Impulse Train
, [ , ] [ , ]M Nk l
comb m n m kM n lN
1, ,
k l k l
k lm kM n lN u v
MN M N
1 1,
( , )M N
comb u v, [ , ]M Ncomb m n
, ( , ) ,M Nk l
comb x y x kM y lN
Fourier Transform of an impulse train is also an impulse train:
Bahadir K. Gunturk EE 7730 - Image Analysis I 23
Impulse Train
2[ ]comb n
n u
1 12
1
2
1( )
2comb u
1
2
Bahadir K. Gunturk EE 7730 - Image Analysis I 24
Impulse Train
1, ,
k l k l
k lx kM y lN u v
MN M N
1 1,
( , )M N
comb u v, ( , )M Ncomb x y
, ( , ) ,M Nk l
comb x y x kM y lN
In the case of continuous signals:
Bahadir K. Gunturk EE 7730 - Image Analysis I 25
Impulse Train
2 ( )comb x
x u
1 12
1
2
1( )
2comb u
1
22
Bahadir K. Gunturk EE 7730 - Image Analysis I 26
Sampling
x
x
M
( )f x
( )Mcomb x
u
( )F u
u
1( )* ( )M
F u comb u
u1
M
1 ( )M
comb u
x
( ) ( )Mf x comb x
Bahadir K. Gunturk EE 7730 - Image Analysis I 27
Sampling
x
( )f x
u
( )F u
u
1( )* ( )M
F u comb u
x
( ) ( )Mf x comb x
WW
M
W
1
M1
2WM
No aliasing if
Bahadir K. Gunturk EE 7730 - Image Analysis I 28
Sampling
u
1( )* ( )M
F u comb u
x
( ) ( )Mf x comb x
M
W
1
M
If there is no aliasing, the original signal can be recovered from its samples by low-pass filtering.
1
2M
Bahadir K. Gunturk EE 7730 - Image Analysis I 29
Sampling
x
( )f x
u
( )F u
u
1( )* ( )M
F u comb u
( ) ( )Mf x comb x
WW
W
1
MAliased
Bahadir K. Gunturk EE 7730 - Image Analysis I 30
Sampling
x
( )f x
u
( )F u
u ( )* ( ) ( )Mf x h x comb x
WW
1M
Anti-aliasing filter
uWW
( )* ( )f x h x
1
2M
Bahadir K. Gunturk EE 7730 - Image Analysis I 31
Sampling
u ( )* ( ) ( )Mf x h x comb x
1
M
u( ) ( )Mf x comb x
W
1
M
■ Without anti-aliasing filter:
■ With anti-aliasing filter:
Bahadir K. Gunturk EE 7730 - Image Analysis I 32
Anti-Aliasing
a=imread(‘barbara.tif’);
Bahadir K. Gunturk EE 7730 - Image Analysis I 33
Anti-Aliasing
a=imread(‘barbara.tif’);b=imresize(a,0.25);c=imresize(b,4);
Bahadir K. Gunturk EE 7730 - Image Analysis I 34
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 EE 7730 - Image Analysis I 35
Sampling
x
y
u
v
uW
vW
x
y
( , )f x y ( , )F u v
M
N
, ( , )M Ncomb x y
u
v
1
M
1
N
1 1,
( , )M N
comb u v
Bahadir K. Gunturk EE 7730 - Image Analysis I 36
Sampling
u
v
uW
vW
,( , ) ( , )M Nf x y comb x y
1
M
1
N
12 uW
MNo aliasing if and
12 vW
N
Bahadir K. Gunturk EE 7730 - Image Analysis I 37
Interpolation
u
v
1
M
1
N
1 1, for and v
( , ) 2 20, otherwise
MN uH u v M N
1
2N
1
2M
Ideal reconstruction filter:
Bahadir K. Gunturk EE 7730 - Image Analysis I 38
Ideal Reconstruction Filter
1 1
2 22 ( ) 2 ( )
1 1
2 2
( , ) ( , )N M
j ux vy j ux vy
N M
h x y H u v e dudv MNe dudv
11
222 2
1 1
2 2
1 11 12 22 2
2 22 21 1
2 2
sin sin
NMj ux j vy
M N
j y j yj x j xN NM M
Me du Ne dv
M e e N e ej x j y
x yM N
x yM N
1sin( )
2jx jxx e e
j
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