Embed Size (px)
Citation preview
The Fourier Transform
Jean Baptiste Joseph Fourier
0 50 100 150 200 250
0
500
1000
1500
2000
2500
3000
3500
0 50 100 150 200 2500
500
1000
1500
2000
2500
3000
3500
Original histogram Equalized histogram
Image Operations in Different Domains
1) Gray value (histogram) domain
2) Spatial (image) domain
3) Frequency (Fourier) domain
- Histogram stretching, equalization, specification, etc...
- Average filter, median filter, gradient, laplacian, etc…
Original image Gradient magnitude22yx fff
f
Blurry Image Laplacian
+ =
Sharpened Image
Noisy image(Salt & Pepper noise)
3 X 3 Average 5 X 5 Average
7 X 7 Average Median
=
3 sin(x) A
+ 1 sin(3x) B A+B
+ 0.8 sin(5x) CA+B+C
+ 0.4 sin(7x) DA+B+C+D
A sum of sines and cosines
sin(x) A
Higher frequencies dueto sharp image variations
(e.g., edges, noise, etc.)
The Continuous Fourier Transform
u
iuxduxf 2F(u)e)(
)2sin()2cos2 uxiux(iux e
Complex Numbers
Real
ImaginaryZ=(a,b)
a
b
|Z|
)(conjugate
(phase)
spectrum)(Fourier
vector)unit (a
i
i
eZibaZZ
abtg
baZ
ie
ii
*
1
22
2
)/(
sincos
)1(1
ieZ
iba
ZiZZ
)Im()Re(
x
ux2cos
– The wavelength is 1/u .
– The frequency is u .
1
The 1D Basis Functions iuxe 2
)2sin()Im(
)2cos()Re(2
2
uxe
uxeiux
iux
1/u
The Fourier Transform
))(( xf
x
iuxdxuF 2f(x)e)(
1D Continuous Fourier Transform:
))((1 uFThe InverseFourier Transform
The Continuous Fourier Transform
x y
uxi dxdyvuF vy)(2y)ef(x,),(
2D Continuous Fourier Transform:
u v
uxi dudvyxf vy)(2v)eF(u,),(
u
iuxduxf 2F(u)e)(
The Inverse Transform
The Transform
The wavelength is . The direction is u/v .22/1 vu
The 2D Basis Functions
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
)(2 vyuxie
Discrete Functions
0 1 2 3 ... N-1
f(x)
f(x0)
f(x0+x)
f(x0+2x) f(x0+3x)
f(n) = f(x0 + nx)
x0 x0+x x0+2x x0+3x
The discrete function f:{ f(0), f(1), f(2), … , f(N-1) }
1
0
2
)()(N
u
N
iux
euFxf
1
0
2
)(1
)(N
x
N
iux
exfN
uF
(u = 0,..., N-1)
(x = 0,..., N-1)
1D Discrete Fourier Transform:
The Discrete Fourier Transform
1
0
1
0
)(2),(
11),(
N
x
M
y
Mvy
Nux
ieyxf
MNvuF
2D Discrete Fourier Transform:
1
0
1
0
)(2),(),(
N
u
M
v
Mvy
Nux
ievuFyxf
(x = 0,..., N-1; y = 0,…,M-1)
(u = 0,..., N-1; v = 0,…,M-1)
Fourier spectrum log(1 + |F(u,v)|) Image f
The Fourier Image
Fourier spectrum |F(u,v)|
Frequency Bands
Percentage of image power enclosed in circles (small to large) :
90%, 95%, 98%, 99%, 99.5%, 99.9%
Image Fourier Spectrum
Low pass Filtering
90% 95%
98% 99%
99.5% 99.9%
Noise Removal
Noisy image
Fourier Spectrum Noise-cleaned image
High Pass Filtering
Original High Pass Filtered
High Frequency Emphasis
+Original High Pass Filtered
High Frequency EmphasisOriginal High Frequency Emphasis
OriginalHigh Frequency Emphasis
Original High pass Filter
High Frequency Emphasis
High Frequency Emphasis +
Histogram Equalization
High Frequency Emphasis
2D Image 2D Image - Rotated
Fourier Spectrum Fourier Spectrum
Rotation
Image Domain
Frequency Domain
Fourier Transform -- Examples
Image Fourier spectrum
Fourier Transform -- Examples
