fourier transform - part ii - cs.haifa.ac.ilcs.haifa.ac.il/~dkeren/ip/lecture6.pdf · fourier...
TRANSCRIPT
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Fourier Transform - Part II
Image Processing - Lesson 6
• Discrete Fourier Transform - 1D
• Discrete Fourier Transform - 2D
• Fourier Properties
• Convolution Theorem
• FFT
• Examples
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Discrete Fourier Transform
Move from f(x) (x R) to f(x) (x Z) by sampling at equal intervals.
f(x0), f(x0+∆x), f(x0+2∆x), .... , f(x0+[n-1]∆x),
Given N samples at equal intervals, we redefine f as:
f(x) = f(x0+x∆x) x = 0, 1, 2, ... , N-1
4
3
2
1
4
3
2
1
0 0.25 0.5 0.75 1.0 1.25 0 1 2 3
f(x)
f(x0)
f(x0+∆x)
f(x0+2∆x) f(x0+3∆x)
f(xn) = f(x0 + x∆x)
∈ ∈
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Discrete Fourier Transform
The Discrete Fourier Transform (DFT) is defined as:
u = 0, 1, 2, ..., N-1∑−
=
−
=1
0
2
)(1)(N
x
Niux
exfN
uFπ
∑−
=
=1
0
2
)()(N
u
Niux
euFxfπ
x = 0, 1, 2, ..., N-1
The Inverse Discrete Fourier Transform (IDFT) is defined as:
Matlab: F=fft(f);
Matlab: F=ifft(f);
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4
3
2
1
Discrete Fourier Transform - Example
4
3
2
1
0 0.25 0.5 0.75 1.0 1.25 0 1 2 3
f(x)
f(x0)
f(x0+∆x)
f(x0+2∆x) f(x0+3∆x)
f(x) = f(x0 + x∆x)
F(0) = 1/4 Σ f(x) e = 1/4 Σ f(x) 13
x=0
-2πi0x4
= 1/4(f(0) + f(1) + f(2) + f(3)) = 1/4(2+3+4+4) = 3.25
3
x=0
F(1) = 1/4 Σ f(x) e = 1/4 [2e0+3e-iπ/2+4e−π i 4e-i3π/2] = [-2+i]3
x=0
-2πix4 1
4
F(3) = 1/4 Σ f(x) e = 1/4 [2e0+3e-i3π/2+4e−3πi 4e-i9π/2] = [-2-i]3
x=0
-6πix4 1
4
F(2) = 1/4 Σ f(x) e = 1/4 [2e0+3e-iπ+4e−2πi 4e-3πi] = [-1-0i]=3
x=0
-4πix4 -1
4-14
Fourier Spectrum:
|F(0)| = 3.25
|F(1)| = [(-1/2)2 + (1/4)2]0.5
|F(2)| = [(-1/4)2 + (0)2]0.5
|F(3)| = [(-1/2)2 + (-1/4)2]0.5
![Page 5: Fourier Transform - Part II - cs.haifa.ac.ilcs.haifa.ac.il/~dkeren/ip/lecture6.pdf · Fourier Transform - Part II Image ... The Discrete Fourier Transform (DFT) ... Image using Matlab](https://reader030.vdocuments.mx/reader030/viewer/2022020204/5adcf77c7f8b9aa5088c30fb/html5/thumbnails/5.jpg)
Discrete Fourier Transform - 2D
Image f(x,y) x = 0,1,...,N-1 y= 0,1,...,M-1
The Discrete Fourier Transform (DFT) is defined as:
The Inverse Discrete Fourier Transform (IDFT) is defined as:
x = 0, 1, 2, ..., N-1∑−
=∑
−
=
+=
1
0
1
0
)(2),(),(
N
u
Mvy
NuxM
v
ievuFyxf
π
y = 0, 1, 2, ..., M-1
u = 0, 1, 2, ..., N-1v = 0, 1, 2, ..., M-1
∑−
=
−
=∑
+−=
1
0
1
0
)(2),(1),(F
N
x
M
y
Mvy
Nuxi
eyxfMN
vuπ
Matlab: F=fft2(f);
Matlab: F=ifft2(f);
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u=0, v=0 u=1, v=0 u=2, v=0u=-2, v=0 u=-1, v=0
u=0, v=1 u=1, v=1 u=2, v=1u=-2, v=1 u=-1, v=1
u=0, v=2 u=1, v=2 u=2, v=2u=-2, v=2 u=-1, v=2
u=0, v=-1 u=1, v=-1 u=2, v=-1u=-2, v=-1 u=-1, v=-1
u=0, v=-2 u=1, v=-2 u=2, v=-2u=-2, v=-2 u=-1, v=-2
U
V
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v
u
Fourier Transform - Image
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Visualizing the Fourier Transform Image using Matlab Routines
• F(u,v) is a Fourier transform of f(x,y) and it has complex entries.
F = fft2(f);• In order to display the Fourier Spectrum |F(u,v)|
– Cyclically rotate the image so that F(0,0) is in the center:
F = fftshift(F);– Reduce dynamic range of |F(u,v)| by displaying
the log:
D = log(1+abs(F));
Example:
|F(u)| = 100 4 2 1 0 0 1 2 4
|F(u)| = 0 1 2 4 100 4 2 1 0Cyclic
|F(u)|/10 = 0 0 0 0 10 0 0 0 0
log(1+|F(u)|) = 0 0.69 1.01 1.61 4.62 1.61 1.01 0.69 0
0 1 2 4 10 4 2 1 0
Display in Range([0..10]):
log(1+|F(u)|)/0.462 =
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Original
Visualizing the Fourier Image - Example
|F(u,v)|
|F(0,0)| at center log(1 + |F(u,v)|)
![Page 10: Fourier Transform - Part II - cs.haifa.ac.ilcs.haifa.ac.il/~dkeren/ip/lecture6.pdf · Fourier Transform - Part II Image ... The Discrete Fourier Transform (DFT) ... Image using Matlab](https://reader030.vdocuments.mx/reader030/viewer/2022020204/5adcf77c7f8b9aa5088c30fb/html5/thumbnails/10.jpg)
Original
Shifted Fourier Image Shifted Log Fourier Image = log(1+ |F(u,v)|)
Fourier Image = |F(u,v)|
![Page 11: Fourier Transform - Part II - cs.haifa.ac.ilcs.haifa.ac.il/~dkeren/ip/lecture6.pdf · Fourier Transform - Part II Image ... The Discrete Fourier Transform (DFT) ... Image using Matlab](https://reader030.vdocuments.mx/reader030/viewer/2022020204/5adcf77c7f8b9aa5088c30fb/html5/thumbnails/11.jpg)
Properties of The Fourier Transform
Distributive (addition)
LinearityF[a f(x,y)] = aF [f(x,y)]
a f(x,y) a F(u,v)
CyclicF(u,v) = F(u+N,v) = F(u,v+N) = F(u+N,v+N)
F(x,y) = F(x+N,y+N)
SymmetricF(u,v) = F*(-u,-v)
thus:
|F(u,v)| = |F (-u,-v)| Fourier Spectrum is symmetric
DC (Average)1MF(0,0) = Σ Σ f(x) e0
N-1
x=0
1N
M-1
y=0
if f(x) is real:
F[ f1(x,y) + f2(x,y)] = [f1(x,y)] + F [f2(x,y)]F~ F
~F~
F~
F~
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x
f(x)
ω
|F(ω)|
x
g(x)
ω
|G(ω)|
x
f+g
ω
|F(ω)+G(ω)|
Distributive: { } { } { }gFfFgfF ~~~ +=+
![Page 13: Fourier Transform - Part II - cs.haifa.ac.ilcs.haifa.ac.il/~dkeren/ip/lecture6.pdf · Fourier Transform - Part II Image ... The Discrete Fourier Transform (DFT) ... Image using Matlab](https://reader030.vdocuments.mx/reader030/viewer/2022020204/5adcf77c7f8b9aa5088c30fb/html5/thumbnails/13.jpg)
1 cycle
0 N/2 N-1N
-N/2
Cyclic and Symmetry of the Fourier Transform - 1D Example
|F(u)|
![Page 14: Fourier Transform - Part II - cs.haifa.ac.ilcs.haifa.ac.il/~dkeren/ip/lecture6.pdf · Fourier Transform - Part II Image ... The Discrete Fourier Transform (DFT) ... Image using Matlab](https://reader030.vdocuments.mx/reader030/viewer/2022020204/5adcf77c7f8b9aa5088c30fb/html5/thumbnails/14.jpg)
Translation
Image Transformations
f(x-x0,y-y0) F(u,v)e-2πi(ux0+vy0)
N
f(x,y)e F(u-u0,v-v0)2πi(u0x+v0y)
N
The Fourier Spectrum remains unchanged under translation:
|F(u,v)| = |F(u,v)e |-2πi(ux0+vy0)
N
Rotation
Rotation of f(x,y) by θ
Rotation of F(u,v)by θ
Scale
f(ax,by) F(u/a,v/b)1|ab|
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x
f(x)
ω
|F(ω)|
x
f(x)
ω
|F(ω)|
Change of Scale- 1D:
x
f(x)
ω
|F(ω)|
( ){ } ( ) ( ){ }
==
aF
a1axfF~thenFxfF~if ωω
![Page 16: Fourier Transform - Part II - cs.haifa.ac.ilcs.haifa.ac.il/~dkeren/ip/lecture6.pdf · Fourier Transform - Part II Image ... The Discrete Fourier Transform (DFT) ... Image using Matlab](https://reader030.vdocuments.mx/reader030/viewer/2022020204/5adcf77c7f8b9aa5088c30fb/html5/thumbnails/16.jpg)
x
f(x)
ω
F(ω)
x
f(2x)
ω
0.5 F(ω/2)
Change of Scale
x
f(x/2)
ω
2 F(2ω)
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2D Image 2D Image - Rotated
Fourier Spectrum Fourier Spectrum
Example - Rotation
![Page 18: Fourier Transform - Part II - cs.haifa.ac.ilcs.haifa.ac.il/~dkeren/ip/lecture6.pdf · Fourier Transform - Part II Image ... The Discrete Fourier Transform (DFT) ... Image using Matlab](https://reader030.vdocuments.mx/reader030/viewer/2022020204/5adcf77c7f8b9aa5088c30fb/html5/thumbnails/18.jpg)
Image Domain Frequency Domain
Fourier Transform Examples
![Page 19: Fourier Transform - Part II - cs.haifa.ac.ilcs.haifa.ac.il/~dkeren/ip/lecture6.pdf · Fourier Transform - Part II Image ... The Discrete Fourier Transform (DFT) ... Image using Matlab](https://reader030.vdocuments.mx/reader030/viewer/2022020204/5adcf77c7f8b9aa5088c30fb/html5/thumbnails/19.jpg)
Separabitity
F(u,v) = Σ Σ f(x,y) e-2πi(ux+vy)/n
N-1
x=0
1N
N-1
y=0
F(u,v) = Σ (Σ f(x,y) e-2π ivy/n) e-2π iux/nN-1
x=0
1N
N-1
y=0
F(u,v) = Σ F(x,v) e-2π iux/n
N-1
x=0
1N
Thus, to perform a 2D Fourier Transform is equivalent to performing 2 1D transforms:
1) Perform 1D transform on EACH column of image f(x,y).Obtain F(x,v).
2) Perform 1D transform on EACH row of F(x,v).
Higher Dimensions:Fourier in any dimension can be performed by applying 1D transform on each dimension.
![Page 20: Fourier Transform - Part II - cs.haifa.ac.ilcs.haifa.ac.il/~dkeren/ip/lecture6.pdf · Fourier Transform - Part II Image ... The Discrete Fourier Transform (DFT) ... Image using Matlab](https://reader030.vdocuments.mx/reader030/viewer/2022020204/5adcf77c7f8b9aa5088c30fb/html5/thumbnails/20.jpg)
Image Domain Frequency Domain
Fourier Transform Examples
![Page 21: Fourier Transform - Part II - cs.haifa.ac.ilcs.haifa.ac.il/~dkeren/ip/lecture6.pdf · Fourier Transform - Part II Image ... The Discrete Fourier Transform (DFT) ... Image using Matlab](https://reader030.vdocuments.mx/reader030/viewer/2022020204/5adcf77c7f8b9aa5088c30fb/html5/thumbnails/21.jpg)
Image Domain Frequency Domain
Fourier Transform Examples
![Page 22: Fourier Transform - Part II - cs.haifa.ac.ilcs.haifa.ac.il/~dkeren/ip/lecture6.pdf · Fourier Transform - Part II Image ... The Discrete Fourier Transform (DFT) ... Image using Matlab](https://reader030.vdocuments.mx/reader030/viewer/2022020204/5adcf77c7f8b9aa5088c30fb/html5/thumbnails/22.jpg)
Linear Systems and Responses
SpatialDomain
FrequencyDomain
Input f F
Output g G
ImpulseResponse h
Freq.Response H
Relationship g=f*h G=FH
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The Convolution Theorem
g = f * h g = f h
implies implies
G = F H G = F * H
Convolution in one domain is multiplication in the other and vice
versa
![Page 24: Fourier Transform - Part II - cs.haifa.ac.ilcs.haifa.ac.il/~dkeren/ip/lecture6.pdf · Fourier Transform - Part II Image ... The Discrete Fourier Transform (DFT) ... Image using Matlab](https://reader030.vdocuments.mx/reader030/viewer/2022020204/5adcf77c7f8b9aa5088c30fb/html5/thumbnails/24.jpg)
The Convolution Theorem
Convolution in one domain is multiplication in the other and vice
versa
( ) ( ){ } ( ){ } ( ){ }xgFxfFxgxfF ~~~ =∗
and likewise
( ) ( ){ } ( ){ } ( ){ }xgFxfFxgxfF ~~~ ∗=
f(x,y) * g(x,y) F(u,v) G(u,v)
f(x,y) g(x,y) F(u,v) * G(u,v)
![Page 25: Fourier Transform - Part II - cs.haifa.ac.ilcs.haifa.ac.il/~dkeren/ip/lecture6.pdf · Fourier Transform - Part II Image ... The Discrete Fourier Transform (DFT) ... Image using Matlab](https://reader030.vdocuments.mx/reader030/viewer/2022020204/5adcf77c7f8b9aa5088c30fb/html5/thumbnails/25.jpg)
f(x)
-0.5 0.5
x
h(x)Example:
-1 1
f(x)
-0.5 0.5
*h(x) =
F(ω) F(ω)
H(ω) = .
=
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f(x,y) g(x,y) f * g (x,y)
F(u,v) G(u,v) F(u,v) G(u,v)
F[F(u,v) G(u,v)]
Convolution Theorem -2D Example
F-1ℑ
1−ℑ
![Page 27: Fourier Transform - Part II - cs.haifa.ac.ilcs.haifa.ac.il/~dkeren/ip/lecture6.pdf · Fourier Transform - Part II Image ... The Discrete Fourier Transform (DFT) ... Image using Matlab](https://reader030.vdocuments.mx/reader030/viewer/2022020204/5adcf77c7f8b9aa5088c30fb/html5/thumbnails/27.jpg)
Sampling the Image
x
f(x)
ω
|F(ω)|
F̂
x
g(x)
ω
F̂|G(ω)|
ω
T 1/T
x
*
F̂
-1/2T 1/2T
![Page 28: Fourier Transform - Part II - cs.haifa.ac.ilcs.haifa.ac.il/~dkeren/ip/lecture6.pdf · Fourier Transform - Part II Image ... The Discrete Fourier Transform (DFT) ... Image using Matlab](https://reader030.vdocuments.mx/reader030/viewer/2022020204/5adcf77c7f8b9aa5088c30fb/html5/thumbnails/28.jpg)
Undersampling the Image
x
g(x)
F̂|G(ω)|
ω
T 1/T
*
ω
-1/2T 1/2Tx
F̂
x
f(x)
ω
|F(ω)|
F̂
![Page 29: Fourier Transform - Part II - cs.haifa.ac.ilcs.haifa.ac.il/~dkeren/ip/lecture6.pdf · Fourier Transform - Part II Image ... The Discrete Fourier Transform (DFT) ... Image using Matlab](https://reader030.vdocuments.mx/reader030/viewer/2022020204/5adcf77c7f8b9aa5088c30fb/html5/thumbnails/29.jpg)
Critical Sampling• If the maximal frequency of f(x) is ωmax , it
is clear from the above replicas that ωmax should be smaller that 1/2T.
• Alternatively:
• Nyquist Theorem: If the maximal frequency of f(x) is ωmax the sampling rate should be larger than 2ωmax in order to fully reconstruct f(x) from its samples.
• If the sampling rate is smaller than 2ωmax overlapping replicas produce aliasing.
max21
ω>T
ω
|F(ω)|
-1/2T 1/2T
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Optimal Interpolation
• It is possible to fully reconstruct f(x) from its samples:
ω
|F(ω)|
x
f(x)F-1^
-1/2T 1/2T
ω
|F(ω)|
x
f(x)F-1^
-1/2T 1/2T
ω
|F(ω)|
x
f(x)F-1^
-1/2T 1/2T
1
*
![Page 31: Fourier Transform - Part II - cs.haifa.ac.ilcs.haifa.ac.il/~dkeren/ip/lecture6.pdf · Fourier Transform - Part II Image ... The Discrete Fourier Transform (DFT) ... Image using Matlab](https://reader030.vdocuments.mx/reader030/viewer/2022020204/5adcf77c7f8b9aa5088c30fb/html5/thumbnails/31.jpg)
Note, that only N/2 different transform values are obtainedfor the N/2 point transforms.
Fast Fourier Transform - FFT
u = 0, 1, 2, ..., N-1∑−
=
−
=1
0
2
)(1)(N
x
Niux
exfN
uFπ
O(n2) operations
∑∑−
=
−
=
+−−
++=12/
0
12/
0
)12(222
)12(1)2(1)(N
x
N
x
Nxiu
Nxiu
exfN
exfN
uFππ
The Fourier transform of N inputs, can be performed as2 Fourier Transforms of N/2 inputs each + one complex multiplication and addition for each value i.e. O(N).
even x odd x
−−
∑ ++∑=−
=
−−
=
12/
0
212/
0
2/2
2/2
2/)12(
2/1)2(1
21 N
x
NiuN
x
Niux
Niux
Nexf
Neexf
πππ
Fourier Transform ofof N/2 even points
Fourier Transform ofof N/2 odd points
0 1 2 3 4 5 6 7 0 2 4 6 1 3 5 7
All sampling pointsOdd
sampling pointsEven
sampling points
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For u = 0, 1, 2, ..., N/2-1
−−−
∑ ++∑=−
=
−
=
12/
0
12/
0
2/22
2/2
2/)12(
2/1)2(1
21)(
N
x
N
xN
Niux
Niu
Niux
Nexf
NeexfuF
πππ
−
+= )()(21)(
2/2/
2
uFeuFuF o
N
e
NNN
iuπ
For u' = u+N/2 :N
iuiN
iuN
NuiN
iu
eeeeeπ
ππππ 22)2/(2'2 −
−=−−
=
+−
=
−
obtain :
Thus: only one complex multiplication is needed for two terms.
−
+= )()(21)(
2/
2
2/uFeuFuF o
NN
iue
NN
π
−
−=+ )()(21)(
2/
2
2/2 uFeuFuF oN
Niu
eN
NN
π
)u(e2/NFCalculating and is done recursively by
calculating and .)u(o
2/NF)u(e
4/NF )u(o4/NF
![Page 33: Fourier Transform - Part II - cs.haifa.ac.ilcs.haifa.ac.il/~dkeren/ip/lecture6.pdf · Fourier Transform - Part II Image ... The Discrete Fourier Transform (DFT) ... Image using Matlab](https://reader030.vdocuments.mx/reader030/viewer/2022020204/5adcf77c7f8b9aa5088c30fb/html5/thumbnails/33.jpg)
F(0) F(1) F(2) F(3) F(4) F(5) F(6) F(7)
F(0) F(2) F(4) F(6) F(1) F(3) F(5) F(7)
F(0) F(4) F(2) F(6) F(1) F(5) F(3) F(7)
F(0) F(1) F(2) F(3) F(4) F(5) F(6) F(7)
2-pointtransform
4-pointtransform
FFT
FFT : O(nlog(n)) operations
FFT of NxN Image: O(n2log(n)) operations
![Page 34: Fourier Transform - Part II - cs.haifa.ac.ilcs.haifa.ac.il/~dkeren/ip/lecture6.pdf · Fourier Transform - Part II Image ... The Discrete Fourier Transform (DFT) ... Image using Matlab](https://reader030.vdocuments.mx/reader030/viewer/2022020204/5adcf77c7f8b9aa5088c30fb/html5/thumbnails/34.jpg)
FilteredImage
Image Transform
Filtered
FFT
FFT-1NewImage
Frequency Enhancement