EE 7730 2D Fourier Transform. Bahadir K. Gunturk EE 7730 - Image Analysis I 2 Summary of Lecture 2...
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Transcript of EE 7730 2D Fourier Transform. Bahadir K. Gunturk EE 7730 - Image Analysis I 2 Summary of Lecture 2...
![Page 1: 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.](https://reader030.fdocuments.in/reader030/viewer/2022020111/56649e8f5503460f94b93c7f/html5/thumbnails/1.jpg)
EE 7730
2D Fourier Transform
![Page 2: 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.](https://reader030.fdocuments.in/reader030/viewer/2022020111/56649e8f5503460f94b93c7f/html5/thumbnails/2.jpg)
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.
![Page 3: 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.](https://reader030.fdocuments.in/reader030/viewer/2022020111/56649e8f5503460f94b93c7f/html5/thumbnails/3.jpg)
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)
![Page 4: 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.](https://reader030.fdocuments.in/reader030/viewer/2022020111/56649e8f5503460f94b93c7f/html5/thumbnails/4.jpg)
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.
![Page 5: 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.](https://reader030.fdocuments.in/reader030/viewer/2022020111/56649e8f5503460f94b93c7f/html5/thumbnails/5.jpg)
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)
![Page 6: 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.](https://reader030.fdocuments.in/reader030/viewer/2022020111/56649e8f5503460f94b93c7f/html5/thumbnails/6.jpg)
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”
![Page 7: 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.](https://reader030.fdocuments.in/reader030/viewer/2022020111/56649e8f5503460f94b93c7f/html5/thumbnails/7.jpg)
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.
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Bahadir K. Gunturk EE 7730 - Image Analysis I 8
Examples
Magnitude: “how much” of each componentPhase: “where” the frequency component in the image
![Page 9: 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.](https://reader030.fdocuments.in/reader030/viewer/2022020111/56649e8f5503460f94b93c7f/html5/thumbnails/9.jpg)
Bahadir K. Gunturk EE 7730 - Image Analysis I 9
Examples
![Page 10: 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.](https://reader030.fdocuments.in/reader030/viewer/2022020111/56649e8f5503460f94b93c7f/html5/thumbnails/10.jpg)
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
![Page 11: 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.](https://reader030.fdocuments.in/reader030/viewer/2022020111/56649e8f5503460f94b93c7f/html5/thumbnails/11.jpg)
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.
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Bahadir K. Gunturk EE 7730 - Image Analysis I 12
Fourier Transform: Properties■ Energy conservation
2 2( , ) ( , )f x y dxdy F u v dudv
![Page 13: 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.](https://reader030.fdocuments.in/reader030/viewer/2022020111/56649e8f5503460f94b93c7f/html5/thumbnails/13.jpg)
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
![Page 14: 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.](https://reader030.fdocuments.in/reader030/viewer/2022020111/56649e8f5503460f94b93c7f/html5/thumbnails/14.jpg)
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
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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
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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
![Page 17: 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.](https://reader030.fdocuments.in/reader030/viewer/2022020111/56649e8f5503460f94b93c7f/html5/thumbnails/17.jpg)
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.
![Page 18: 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.](https://reader030.fdocuments.in/reader030/viewer/2022020111/56649e8f5503460f94b93c7f/html5/thumbnails/18.jpg)
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
![Page 19: 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.](https://reader030.fdocuments.in/reader030/viewer/2022020111/56649e8f5503460f94b93c7f/html5/thumbnails/19.jpg)
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
![Page 20: 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.](https://reader030.fdocuments.in/reader030/viewer/2022020111/56649e8f5503460f94b93c7f/html5/thumbnails/20.jpg)
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
![Page 21: 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.](https://reader030.fdocuments.in/reader030/viewer/2022020111/56649e8f5503460f94b93c7f/html5/thumbnails/21.jpg)
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
![Page 22: 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.](https://reader030.fdocuments.in/reader030/viewer/2022020111/56649e8f5503460f94b93c7f/html5/thumbnails/22.jpg)
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:
![Page 23: 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.](https://reader030.fdocuments.in/reader030/viewer/2022020111/56649e8f5503460f94b93c7f/html5/thumbnails/23.jpg)
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
![Page 24: 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.](https://reader030.fdocuments.in/reader030/viewer/2022020111/56649e8f5503460f94b93c7f/html5/thumbnails/24.jpg)
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:
![Page 25: 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.](https://reader030.fdocuments.in/reader030/viewer/2022020111/56649e8f5503460f94b93c7f/html5/thumbnails/25.jpg)
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
![Page 26: 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.](https://reader030.fdocuments.in/reader030/viewer/2022020111/56649e8f5503460f94b93c7f/html5/thumbnails/26.jpg)
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
![Page 27: 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.](https://reader030.fdocuments.in/reader030/viewer/2022020111/56649e8f5503460f94b93c7f/html5/thumbnails/27.jpg)
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
![Page 28: 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.](https://reader030.fdocuments.in/reader030/viewer/2022020111/56649e8f5503460f94b93c7f/html5/thumbnails/28.jpg)
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
![Page 29: 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.](https://reader030.fdocuments.in/reader030/viewer/2022020111/56649e8f5503460f94b93c7f/html5/thumbnails/29.jpg)
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
![Page 30: 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.](https://reader030.fdocuments.in/reader030/viewer/2022020111/56649e8f5503460f94b93c7f/html5/thumbnails/30.jpg)
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
![Page 31: 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.](https://reader030.fdocuments.in/reader030/viewer/2022020111/56649e8f5503460f94b93c7f/html5/thumbnails/31.jpg)
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:
![Page 32: 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.](https://reader030.fdocuments.in/reader030/viewer/2022020111/56649e8f5503460f94b93c7f/html5/thumbnails/32.jpg)
Bahadir K. Gunturk EE 7730 - Image Analysis I 32
Anti-Aliasing
a=imread(‘barbara.tif’);
![Page 33: 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.](https://reader030.fdocuments.in/reader030/viewer/2022020111/56649e8f5503460f94b93c7f/html5/thumbnails/33.jpg)
Bahadir K. Gunturk EE 7730 - Image Analysis I 33
Anti-Aliasing
a=imread(‘barbara.tif’);b=imresize(a,0.25);c=imresize(b,4);
![Page 34: 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.](https://reader030.fdocuments.in/reader030/viewer/2022020111/56649e8f5503460f94b93c7f/html5/thumbnails/34.jpg)
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));
![Page 35: 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.](https://reader030.fdocuments.in/reader030/viewer/2022020111/56649e8f5503460f94b93c7f/html5/thumbnails/35.jpg)
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
![Page 36: 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.](https://reader030.fdocuments.in/reader030/viewer/2022020111/56649e8f5503460f94b93c7f/html5/thumbnails/36.jpg)
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
![Page 37: 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.](https://reader030.fdocuments.in/reader030/viewer/2022020111/56649e8f5503460f94b93c7f/html5/thumbnails/37.jpg)
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:
![Page 38: 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.](https://reader030.fdocuments.in/reader030/viewer/2022020111/56649e8f5503460f94b93c7f/html5/thumbnails/38.jpg)
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