Digital Image Processing 10/19/2010 1
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Digital Image Processing
10/19/2010 1
Digital Image Processing
� Image Enhancement in Frequency Domain
Basic Properties of Fourier Transforms
10/19/2010 2
Basic Properties of Fourier Transforms
Fourier Transform: Frequency components of a signal
10/19/2010 3
Applications of Fourier transforms
� 1-D Fourier transforms are used in Signal Processing
� 2-D Fourier transforms are used in Image Processing
Applications of Fourier transforms in Image processing:
10/19/2010 4
Applications of Fourier transforms in Image processing:
– Image enhancement,
– Image restoration,
– Image encoding / decoding,
– Image description
Discrete Fourier Transforms (DFT)
1-D DFT for M samples is given as
The inverse Fourier transform in 1-D is given as
1-..,0,1,2,....for )(1
)( /21
0
MuexfM
uF MuxjM
x
== −−
=∑ π
1-..,0,1,2,....for )()( /21
MxeuFxf MuxjM
==∑−
π
10/19/2010 5
Since
Real Imaginary
1-..,0,1,2,....for )()(0
MxeuFxfu
==∑=
,sincos θθθ je j +=
)()()(
)]./2sin()/2[cos()(1
)(1
0
ujIuRuF
MuxjMuxxfM
uFM
x
+=
−= ∑−
=
ππ
2-D DFT
2-D DFT for a total of MxN samples is given as
1-..,0,1,2,.... vand 1-..,0,1,2,....for
),(1
),(1
0
)//(21
0
NMu
eyxfMN
vuFM
x
NvyMuxjN
y
==
= ∑∑−
=
+−−
=
π
10/19/2010 6
2-D inverse DFT is given as
1-..,0,1,2,....y and 1-..,0,1,2,....for
),(),(1
0
)//(21
0
NMx
evuFyxfM
u
NvyMuxjN
v
==
=∑∑−
=
+−
=
π
2-D DFT
10/19/2010 7
Properties of Fourier Transform
1. Average value
F(0,0) gives the average intensity value of an image
2. Modulation
Therefore a modulation in spatial domain will be equivalent to a
∑∑−
=
−
=
=1
0
1
0
.),(1
)0,0(M
x
N
y
yxfMN
F
10/19/2010 8
Therefore a modulation in spatial domain will be equivalent to a
translation in Fourier domain
Using this property
� Translation
Similar to the above case
),(),( 00
)//(2 00 vvuuFeyxfNyvMxuj −−⇔+π
)//(2
0000),(),(NvyMuxj
evuFyyxxf+−⇔−− π
)2/,2/()1)(,( )( NvMuFyxf yx −−⇔− +
Translation property of 2-D DFT
For a function f(x,y), the DFT is given as:
If instead of f(x,y), we put
The DFT will be equal to
∑∑−
=
+−−
=
=1
0
)//(21
0
),(1
),(M
x
NvyMuxjN
y
eyxfMN
vuF π
)//(2 00),(NxvMxuj
eyxf+π
10/19/2010 9
The DFT will be equal to
Therefore the Fourier transform pair has the following translation property
Similarly
),(),(1
00
1
0
}/)(/){(21
0
00 vvuuFeyxfMN
M
x
NyvvMxuujN
y
−−=∑∑−
=
−+−−−
=
π
),(),( 00
)//(2 00 vvuuFeyxfNyvMxuj −−⇔+π
)//(2
0000),(),(NvyMuxj
evuFyyxxf+−⇔−− π
Shifting the origin to the center
If we put ,2/ and ,2/ 00 NvMu ==
)(
)()//(2
)1(
00
yx
yxjNxvMxujee
+
++
−=
= ππ
10/19/2010 10
This means that
To transform the origin to the center in the transformed
image the input is always multiplied by the factor
)2/,2/()1)(,( )( NvMuFyxf yx −−⇔− +
)()1( yx+−
Shifting the origin to the center
10/19/2010 11
Properties of Fourier Transform
4. Periodicity property
The Fourier transform is periodic and obeys the following
periodicity property
The inverse transform is also periodic
),,(),(),(),( NvMuFNvuFvMuFvuF ++=+=+=
10/19/2010 12
5. Symmetry Properties
If f(x,y) is real, the Fourier Transform is conjugate symmetric, i.e.,
).,(),(and,),(),(
),,(),(),,(),( ),,(),( *
vuvuvuFvuF
vuIvuIvuRvuRvuFvuF
−−−=−−=
−−−=−−=−−=
φφ
.),(),(),(),( NyMxfNyxfyMxfyxf ++=+=+=
Properties of Fourier Transform
6. Distributive property:
Fourier transform is distributive over addition but not over
multiplication
7. Linearity and scaling
[ ] [ ] [ ][ ] [ ] [ ],),(.),(),().,(
,),(),(),(),(
2121
2121
yxfyxfyxfyxf
yxfyxfyxfyxf
ℑℑ≠ℑ
ℑ+ℑ=+ℑ
10/19/2010 13
7. Linearity and scaling
Linearity:
Scaling:
� The direction of amplitude change in spatial domain and the
amplitude change in the frequency domain are orthogonal (see the
examples)
),(),(),(),( 2121 vubFvuaFyxbfyxaf +⇔+
)/,/(1
),( bvauFab
byaxf ⇔
Reciprocality of lengths in transform pair (1-D)
10/19/2010 14
Reciprocality of lengths due to scaling property
10/19/2010 15
DFT Examples
10/19/2010 16
Three sinusoidal patterns and their sum with their Fourier Transforms
DFT Examples
10/19/2010 17
Properties of Fourier Transform
8. Rotation
Representing the image in polar coordinates, i.e.
.sin,cos,sin,cos ϕϕθθ wvwuryrx ====
).,(),(),,(),( ϕθ wFvuFrfyxf →→
10/19/2010 18
The rotation property states that,
If the image is rotated in spatial domain by a fixed angle, the
Fourier transform is also rotated at the same angle.
).,(),( 00 θϕθθ +⇔+ wFrf
DFT Examples
Sinusoidal lines and its DFT
Sinusoidal lines with non-
10/19/2010 19
Sinusoidal lines with non-
sinusoidal brightness profile
and its DFT
Display defects (lines on
smaller angles are shown in
steps) complicates the DFT
Rotation Examples
10/19/2010 20
Rotation of spatial domain images and their
corresponding Fourier Transforms
Properties of Fourier Transform
9. Separability
The 2D Fourier transform can be performed as a series of 1D
DFT (complex exponential is separable)
∑∑ ∑−
=
−−
=
−−
=
− ==1
0
/21
0
/21
0
/2 ),(1
),(11
),(M
x
MuxjM
x
NvyjN
y
Muxj evxFM
eyxfN
eM
vuF πππ
10/19/2010 21
where .),(1
),( /21
0
NvyjN
y
eyxfN
vxF π−−
=∑=
Properties of Fourier Transform
10.Derivatives and Laplacian
Considering 1-D, it can be shown that
),()()(
uFjudx
xfd n
n
n
=
ℑ
10/19/2010 22
Implementing the above equation for 2-D Laplacian
).,()(
),()(),()(),(),(
22
22
2
2
2
2
vuFvu
vuFjvvuFjuy
yxf
x
yxf
+−=
+=
∂∂
+∂
∂ℑ
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