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0
500
1000
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3500
0 50 100 150 200 2500
500
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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
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)
Frequency Bands
Percentage of image power enclosed in circles (small to large) :
90%, 95%, 98%, 99%, 99.5%, 99.9%
Image Fourier Spectrum
Original High pass Filter
High Frequency Emphasis
High Frequency Emphasis +
Histogram Equalization
High Frequency Emphasis
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