07 frequency domain dip
DESCRIPTION
Digital image ProcessingTRANSCRIPT
Frequency Domain : 1
Frequency DomainFrequency Domain
Frequency Domain : 2
Fourier Series and TransformFourier Series and Transform
Frequency Domain : 3
Fourier Transform of Continuous VariableFourier Transform of Continuous Variable
2( ) ( ) j tF f t e dt
1 2( ) ( ) ( ) j tF f t F e d
2( ) ( ) j tf t e dt
( ) ( )[cos(2 ) sin(2 )]F f t t j t dt
Frequency Domain : 4
Discrete Fourier Transform (DFT) Discrete Fourier Transform (DFT)
12 /
0
( ) ( ) 1, 2,3,..., 1M
j ux M
x
F u f x e u M
12 /
0
1( ) ( ) 1, 2,3,..., 1
Mj ux M
u
f t F u e u MM
Frequency Domain : 5
Fourier Transform: Visualization Fourier Transform: Visualization
Frequency Domain : 6
2-D Discrete Fourier Transform2-D Discrete Fourier Transform
1 12 ( / / )
0 0
( , ) ( , )M N
j ux M vy N
x y
F u v f x y e
1 12 ( / / )
0 0
1( , ) ( , )
M Nj ux M vy N
u v
f x y F u v eMN
Frequency Domain : 7
2-D Fourier Transform: Visualization2-D Fourier Transform: Visualization
Frequency Domain : 8
2-D Fourier Transform: Implementation2-D Fourier Transform: Implementation
Frequency Domain : 9
2-D Fourier Transform: Implementation2-D Fourier Transform: Implementation
Frequency Domain : 10
Basic Steps of Filtering in Frequency DomainBasic Steps of Filtering in Frequency Domain
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
Frequency Domain : 11
Image Characteristics in Frequency DomainImage Characteristics in Frequency Domain
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
Frequency Domain : 12
Example: DC component removalExample: DC component removal
Suppose we remove the DC component from the Fourier transform of an image
Frequency Domain : 13
Why does it look like that?Why does it look like that?
DC component characterizes the mean of the image intensities
Frequency Domain : 14
Examples of Frequency Domain FilteringExamples of Frequency Domain Filtering
Frequency Domain : 15
Correspondence between Filtering in Correspondence 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
Frequency Domain : 16
Correspondence between Filtering in Correspondence 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
Frequency Domain : 17
Correspondence between Filtering in Correspondence between Filtering in Spatial and Frequency DomainsSpatial and Frequency Domains