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