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18-11-2015

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Unit 6 (continued) Image Enhancement in the

Frequency Domain

Background: Fourier Series

Any periodic signals can be

viewed as weighted sum

of sinusoidal signals with

different frequencies

Fourier series:

Frequency Domain:

view frequency as an

independent variable

(Images from Rafael C. Gonzalez and Richard E.

Wood, Digital Image Processing, 2nd Edition.

Fourier Tr. and Frequency Domain

Time, spatial

Domain

Signals

Frequency

Domain

Signals

Fourier Tr.

Inv Fourier Tr.

1-D, Continuous case

dxexfuF uxj 2)()(Fourier Tr.:

dueuFxf uxj 2)()(Inv. Fourier Tr.:

Fourier Tr. and Frequency Domain (cont.)

1-D, Discrete case

1

0

/2)(1

)(M

x

MuxjexfM

uF Fourier Tr.:

Inv. Fourier Tr.:

1

0

/2)()(M

u

MuxjeuFxf

u = 0,…,M-1

x = 0,…,M-1

F(u) can be written as

)()()( ujeuFuF or )()()( ujIuRuF

22 )()()( uIuRuF

where

)(

)(tan)( 1

uR

uIu

Example of 1-D Fourier Transforms

Notice that the longer

the time domain signal,

The shorter its Fourier

transform



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Relation Between Dx and Du

For a signal f(x) with M points, let spatial resolution Dx be space

between samples in f(x) and let frequency resolution Du be space

between frequencies components in F(u), we have

xMu

DD

1

Example: for a signal f(x) with sampling period 0.5 sec, 100 point,

we will get frequency resolution equal to

Hz02.05.0100

1

Du

This means that in F(u) we can distinguish 2 frequencies that are

apart by 0.02 Hertz or more.

2-Dimensional Discrete Fourier Transform

1

0

1

0

)//(2),(1

),(M

x

N

y

NvyMuxjeyxfMN

vuF

2-D IDFT

1

0

1

0

)//(2),(),(M

u

N

v

NvyMuxjevuFyxf

2-D DFT

u = frequency in x direction, u = 0 ,…, M-1

v = frequency in y direction, v = 0 ,…, N-1

x = 0 ,…, M-1

y = 0 ,…, N-1

For an image of size MxN pixels

F(u,v) can be written as

),(),(),( vujevuFvuF or ),(),(),( vujIvuRvuF

22 ),(),(),( vuIvuRvuF

where

),(

),(tan),( 1

vuR

vuIvu

2-Dimensional Discrete Fourier Transform (cont.)

For the purpose of viewing, we usually display only the

Magnitude part of F(u,v)



2-D DFT Properties



2-D DFT Properties (cont.)




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Computational Advantage of FFT Compared to DFT

Relation Between Spatial and Frequency Resolutions

xMu

DD

1

yNv

DD

1

where

Dx = spatial resolution in x direction

Dy = spatial resolution in y direction

Du = frequency resolution in x direction

Dv = frequency resolution in y direction

N,M = image width and height

( Dx and Dy are pixel width and height. )

How to Perform 2-D DFT by Using 1-D DFT

f(x,y)

1-D

DFT

by row F(u,y)

1-D DFT

by column

F(u,v)

How to Perform 2-D DFT by Using 1-D DFT (cont.)

f(x,y)

1-D

DFT

by row F(x,v)

1-D DFT

by column

F(u,v)

Alternative method

Periodicity of 1-D DFT

0 N 2N -N

DFT repeats itself every N points (Period = N) but we usually

display it for n = 0 ,…, N-1

We display only in this range

1

0

/2)(1

)(M

x

MuxjexfM

uF From DFT:

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Conventional Display for 1-D DFT

0 N-1

Time Domain Signal

DFT

f(x)

)(uF

0 N-1

Low frequency

area

High frequency

area

The graph F(u) is not

easy to understand !

)(uF

0 -N/2 N/2-1

)(uF

0 N-1

Conventional Display for DFT : FFT Shift

FFT Shift: Shift center of the

graph F(u) to 0 to get better

Display which is easier to

understand.

High frequency area

Low frequency area

Periodicity of 2-D DFT

For an image of size NxM

pixels, its 2-D DFT repeats

itself every N points in x-

direction and every M points

in y-direction.

We display only

in this range

1

0

1

0

)//(2),(1

),(M

x

N

y

NvyMuxjeyxfMN

vuF

0 N 2N -N

0

M

2M

-M

2-D DFT:

g(x,y)



Conventional Display for 2-D DFT

High frequency area

Low frequency area

F(u,v) has low frequency areas

at corners of the image while high

frequency areas are at the center

of the image which is inconvenient

to interpret.



2-D FFT Shift : Better Display of 2-D DFT

2D FFTSHIFT

2-D FFT Shift is a MATLAB function: Shift the zero frequency

of F(u,v) to the center of an image.

High frequency area Low frequency area (Images from Rafael C. Gonzalez and Richard E.


Original display

of 2D DFT

0 N 2N -N

0

M

2M

-M



Display of 2D DFT

After FFT Shift

2-D FFT Shift (cont.) : How it works

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Example of 2-D DFT

Notice that the longer the time domain signal,

The shorter its Fourier transform (Images from Rafael C. Gonzalez and Richard E.


Example of 2-D DFT

Notice that direction of an

object in spatial image and

Its Fourier transform are

orthogonal to each other.



Example of 2-D DFT

Original image

2D DFT

2D FFT Shift

Example of 2-D DFT

Original image

2D DFT

2D FFT Shift

Basic Concept of Filtering in the Frequency Domain

From Fourier Transform Property:

),(),(),(),(),(),( vuGvuHvuFyxhyxfyxg

We can perform filtering process by using



Multiplication in the frequency domain

is easier than convolution in the spatial

Domain.

Filtering in the Frequency Domain with FFT shift

f(x,y)

2D FFT

FFT shift

F(u,v)

FFT shift

2D IFFT X

H(u,v)

(User defined)

G(u,v)

g(x,y)

In this case, F(u,v) and H(u,v) must have the same size and

have the zero frequency at the center.

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0 20 40 60 80 100 120

0

0.5

1

0 20 40 60 80 100 120

0

0.5

1

0 20 40 60 80 100 1200

20

40

Multiplication in Freq. Domain = Circular Convolution

f(x) DFT F(u) G(u) = F(u)H(u)

h(x) DFT H(u) g(x) IDFT

Multiplication of

DFTs of 2 signals

is equivalent to

perform circular

convolution

in the spatial domain.

f(x)

h(x)

g(x)

“Wrap around” effect

H(u,v)

Gaussian

Lowpass

Filter with D0 = 5

Original image

Filtered image

(obtained using circular convolution)

Incorrect areas at image rims

Multiplication in Freq. Domain = Circular Convolution

Linear Convolution by using Circular Convolution and Zero Padding

f(x) DFT F(u) G(u) = F(u)H(u)

h(x) DFT H(u)

g(x)

IDFT

Zero padding

Zero padding

Concatenation

0 50 100 150 200 250

0

0.5

1

0 50 100 150 200 250

0

0.5

1

0 50 100 150 200 2500

20

40

Padding zeros Before DFT

Keep only this part







Zero padding area in the spatial

Domain of the mask image (the ideal lowpass filter)

Filtered image

Only this area is kept.

Filtering in the Frequency Domain : Example

In this example, we set F(0,0) to zero

which means that the zero frequency

component is removed.

Note: Zero frequency = average

intensity of an image (Images from Rafael C. Gonzalez and Richard E.


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Filtering in the Frequency Domain : Example



Highpass Filter

Lowpass Filter

Filtering in the Frequency Domain : Example (cont.)



Result of Sharpening Filter

Filter Masks and Their Fourier Transforms



Ideal Lowpass Filter

0

0

),( 0

),( 1),(

DvuD

DvuDvuH

where D(u,v) = Distance from (u,v) to the center of the mask.

Ideal LPF Filter Transfer function



Examples of Ideal Lowpass Filters



The smaller D0, the more high frequency components are removed.

Results of Ideal Lowpass Filters

Ringing effect can be

obviously seen!



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How ringing effect happens

Ideal Lowpass Filter

with D0 = 5

-20

0

20

-20

0

20

0

0.2

0.4

0.6

0.8

1

Surface Plot

Abrupt change in the amplitude

0

0

),( 0

),( 1),(

DvuD

DvuDvuH

How ringing effect happens (cont.)

-20

0

20

-20

0

20

0

5

10

15

x 10-3

Spatial Response of Ideal

Lowpass Filter with D0 = 5

Surface Plot

Ripples that cause ringing effect

How ringing effect happens (cont.)



Butterworth Lowpass Filter

NDvuD

vuH2

0/),(1

1),(

Transfer function

Where D0 = Cut off frequency, N = filter order.



Results of Butterworth Lowpass Filters

There is less ringing

effect compared to

those of ideal lowpass

filters!



Spatial Masks of the Butterworth Lowpass Filters

Some ripples can be seen. (Images from Rafael C. Gonzalez and Richard E.


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Gaussian Lowpass Filter

20

2 2/),(),(

DvuDevuH

Transfer function

Where D0 = spread factor.

Note: the Gaussian filter is the only filter that has no ripple and

hence no ringing effect.



Gaussian Lowpass Filter (cont.)

-20

0

20

-20

0

20

0.2

0.4

0.6

0.8

1

Gaussian lowpass

filter with D0 = 5

-20

0

20

-20

0

20

0

0.01

0.02

0.03

Spatial respones of the

Gaussian lowpass filter

with D0 = 5

Gaussian shape

20

2 2/),(),(

DvuDevuH

Results of Gaussian Lowpass Filters

No ringing effect!



Application of Gaussian Lowpass Filters

The GLPF can be used to remove jagged edges

and “repair” broken characters.

Better Looking Original image



Application of Gaussian Lowpass Filters (cont.)

Remove wrinkles

Softer-Looking

Original image



Application of Gaussian Lowpass Filters (cont.)

Remove artifact lines: this is a simple but crude way to do it!

Filtered image Original image : The gulf of Mexico and

Florida from NOAA satellite. (Images from Rafael C. Gonzalez and Richard E.


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Highpass Filters

Hhp = 1 - Hlp



Ideal Highpass Filters

0

0

),( 1

),( 0),(

DvuD

DvuDvuH

where D(u,v) = Distance from (u,v) to the center of the mask.

Ideal HPF Filter Transfer function

Butterworth Highpass Filters



NvuDD

vuH2

0 ),(/1

1),(

Transfer function

Where D0 = Cut off frequency, N = filter order.

Gaussian Highpass Filters



20

2 2/),(1),(

DvuDevuH

Transfer function

Where D0 = spread factor.

Gaussian Highpass Filters (cont.)

20

2 2/),(1),(

DvuDevuH

1020

3040

5060

20

40

60

0

0.2

0.4

0.6

0.8

1

1020

3040

5060

20

40

60

0

1000

2000

3000

Gaussian highpass

filter with D0 = 5

Spatial respones of the

Gaussian highpass filter

with D0 = 5



Spatial Responses of Highpass Filters

Ripples

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Results of Ideal Highpass Filters

Ringing effect can be

obviously seen!



Results of Butterworth Highpass Filters



Results of Gaussian Highpass Filters



Laplacian Filter in the Frequency Domain

( )()(

uFjudx

xfd n

n

n

From Fourier Tr. Property:

Then for Laplacian operator

( ),(22

2

2

2

22 vuFvu

y

f

x

ff

Surface plot

( 222 vu

We get

Image of –(u2+v2)

Laplacian Filter in the Frequency Domain (cont.)

Spatial response of –(u2+v2) Cross section

Laplacian mask in Chapter 3



Sharpening Filtering in the Frequency Domain

),(),(),( yxfyxfyxf lphp

),(),(),( yxfyxAfyxf lphb

Spatial Domain

Frequency Domain Filter

),(),(),()1(),( yxfyxfyxfAyxf lphb

),(),()1(),( yxfyxfAyxf hphb

),(1),( vuHvuH lphp

),()1(),( vuHAvuH hphb

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Sharpening Filtering in the Frequency Domain (cont.)

p P2

P2 PP 2



Sharpening Filtering in the Frequency Domain (cont.)

),(),()1(),( yxfyxfAyxf hphb

Pfhp

2f

A = 2 A = 2.7



High Frequency Emphasis Filtering

),(),( vubHavuH hphfe

Original Butterworth

highpass

filtered image

High freq. emphasis

filtered image After

Hist Eq.

a = 0.5, b = 2

Homomorphic Filtering:

• If the image model is based on illumination-reflectance, then frequency

domain procedures are not as easy to perform.

• The main reason is that illumination and reflectance components of the model

are not separable.

• To be able to improve appearance of an image by simultaneous brightness

range compression and contrast enhancement it is necessary to separate the

two components.

Homomorphic Filtering

An image can be expressed as

),(),(),( yxryxiyxf

i(x,y) = illumination component

r(x,y) = reflectance component

To accomplish separability, first map the model to natural

log domain and then take the Fourier transform of it.

• The illumination component of an image is generally characterized by slow

spatial variation.

• The reflectance component of an image tends to vary abruptly.

• These characteristics lead to associating the low frequencies of the Fourier

transform of the natural log of an image with illumination and high frequencies

with reflectance.

• a good deal of control can be gained over the illumination and reflectance

components with a homomorphic filter.

• For homomorphic filter to be effective it needs to affect the low- and high-

frequency components of the Fourier transform in different way.

• To compress the dynamic range of an image, the low frequency components

ought to be attenuated to some degree.

• On the other hand, to enhance the contrast, the high frequency components of

the Fourier transform ought to be magnified.


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More details in the room can be seen!



Correlation Application: Object Detection

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