Digital image Processing

1. Frequency Domain : 1
Frequency DomainFrequency Domain

2. Frequency Domain : 2
Fourier Series and TransformFourier Series and Transform

3. Frequency Domain : 3
Fourier Transform of ContinuousFourier Transform of Continuous
VariableVariable
2
( ) ( ) j t
F f t e dtπµ
µ
∞
−
−∞
= ∫
{ }1 2
( ) ( ) ( ) j t
F f t F e dπµ
µ µ µ
∞
−
−∞
ℑ = = ∫
2
( ) ( ) j t
f t e dtπµ
µ
∞
−
−∞
ℑ = ∫
( ) ( )[cos(2 ) sin(2 )]F f t t j t dtµ πµ πµ
∞
−∞
= −∫

4. Frequency Domain : 4
Discrete Fourier Transform (DFT)Discrete Fourier Transform (DFT)
1
2 /
0
( ) ( ) 1,2,3,..., 1
M
j ux M
x
F u f x e u Mπ
−
−
=
= = −∑
1
2 /
0
1
( ) ( ) 1,2,3,..., 1
M
j ux M
u
f t F u e u M
M
π
−
=
= = −∑

5. Frequency Domain : 5
Fourier Transform: VisualizationFourier Transform: Visualization

6. Frequency Domain : 6
2-D Discrete Fourier Transform2-D Discrete Fourier Transform
1 1
2 ( / / )
0 0
( , ) ( , )
M N
j ux M vy N
x y
F u v f x y e π
− −
− +
= =
= ∑ ∑
1 1
2 ( / / )
0 0
1
( , ) ( , )
M N
j ux M vy N
u v
f x y F u v e
MN
π
− −
+
= =
= ∑ ∑

7. Frequency Domain : 7
2-D Fourier Transform: Visualization2-D Fourier Transform: Visualization

8. Frequency Domain : 8
2-D Fourier Transform:2-D Fourier Transform:
ImplementationImplementation

9. Frequency Domain : 9
2-D Fourier Transform:2-D Fourier Transform:
ImplementationImplementation

10. Frequency Domain : 10
Basic Steps of Filtering in FrequencyBasic Steps of Filtering in Frequency
DomainDomain
1. Multiply input f(x,y) by (-1)x+y
to center transform
2. Compute DFT of image, F(u,v)
3. Multiply F(u,v) by filter function H(u,v) to get G(u,v)
4. Compute inverse DFT of G(u,v) to get g(x,y)
5. Multiply g(x,y) by (-1)x+y
to get filtered image

11. Frequency Domain : 11
Image Characteristics in FrequencyImage Characteristics in Frequency
DomainDomain
Low frequencies responsible for general appearance of image over
smooth areas
High frequencies responsible for detail (e.g., edges and noise)
Intuitively, modifying different frequency coefficients affects different
characteristics of an image

12. Frequency Domain : 12
Example: DC component removalExample: DC component removal
Suppose we remove the DC component from the Fourier transform
of an image

13. Frequency Domain : 13
Why does it look like that?Why does it look like that?
DC component characterizes the mean of the image intensities

14. Frequency Domain : 14
Examples of Frequency DomainExamples of Frequency Domain
FilteringFiltering

15. Frequency Domain : 15
Correspondence between Filtering inCorrespondence between Filtering in
Spatial and Frequency DomainsSpatial and Frequency Domains
Basic spatial filtering is essentially 2D discrete convolution
between an image f and filter function h
Convolution in spatial domain becomes multiplication in
frequency domain
( , ) ( , ) ( , )g x y f x y h x y= ∗
( , ) ( , ) ( , )G u v F v v H u v=

16. Frequency Domain : 16
Correspondence between Filtering inCorrespondence between Filtering in
Spatial and Frequency DomainsSpatial and Frequency Domains
What does this mean?
Given a filter in frequency domain
 Corresponding filter in spatial domain can be obtained by
taking inverse Fourier transform
Given a filter in spatial domain,
 Corresponding filter in frequency domain can be obtained
by taking Fourier transform

17. Frequency Domain : 17
Correspondence between Filtering inCorrespondence between Filtering in
Spatial and Frequency DomainsSpatial and Frequency Domains