imageprocessing7-frequencyfiltering

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Course Website:http://www.comp.dit.ie/bmacname

Digital Image Processing

Image Enhancement:

Filtering in the Frequency Domain
http://www.comp.dit.ie/bmacnameehttp://www.comp.dit.ie/bmacnamee


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54Contents

In this lecture we will look at imageenhancement in the frequency domain

Jean Baptiste Joseph Fourier

The Fourier series & the Fourier transform

Image Processing in the frequency domain

Image smoothing

Image sharpening

Fast Fourier Transform


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54Jean Baptiste Joseph Fourier

Fourier was born in Auxerre,France in 1768

Most famous for his work La ThorieAnalitique de la Chaleurpublished in

1822 Translated into English in 1878: The

Analytic Theory of Heat

Nobody paid much attention when the work was

first publishedOne of the most important mathematical theories inmodern engineering


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4

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54The Big Idea

=

Any function that periodically repeats itself can be

expressed as a sum of sines and cosines of

different frequencies each multiplied by a different

coefficienta Fourier seriesImagestakenfromG

onzalez&Woods,Digital

ImageProcessing(2002)


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6

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54The Big Idea (cont)

Frequencydomain signal

processing

example in Excel


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7

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54The Discrete Fourier Transform (DFT)

The Discrete FourierTransformoff(x, y), forx=0, 1, 2M-1 andy= 0,1,2N-1, denoted by

F(u, v),is given by the equation:

for u= 0, 1, 2M-1 and v= 0, 1, 2N-1.

1

0

1

0

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

x

N

y

NvyMuxjeyxfvuF


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54DFT & Images

ImagestakenfromG

onzalez&Woods,DigitalImageProcessing(2002)


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54DFT & Images

ImagestakenfromG



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54DFT & Images (cont)

ImagestakenfromG


DFT

Scanning electron microscope

image of an integrated circuitmagnified ~2500 times

Fourier spectrum of the image


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ImagestakenfromG



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ImagestakenfromG



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14

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54The Inverse DFT

It is really important to note that the Fouriertransform is completely reversible

The inverse DFT is given by:

forx= 0, 1, 2M-1 andy= 0, 1, 2N-1

1

0

1

0

)//(2),(1),(M

u

N

v

NvyMuxjevuFMN

yxf


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54Some Basic Frequency Domain Filters

ImagestakenfromGonzalez&Woods,Digita

lImageProcessing(2002)

Low Pass Filter

High Pass Filter


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22

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54Ideal Low Pass Filter (cont)

Above we show an image, its Fourier spectrum

and a series of ideal low pass filters of radius 5,

15, 30, 80 and 230 superimposed on top of it


lImageProcessing(2002

)


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Imagestakenfrom

Gonzalez&Woods,Digita

lImageProcessing(2002

)


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Imagestakenfrom

Gonzalez&Woods,DigitalImageProcessing(2002

)

Result of filteringwith ideal low pass

filter of radius 15

28

B h L Fil


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54Butterworth Lowpass Filters

The transfer function of a Butterworth lowpassfilter of order nwith cutoff frequency at distance

D0from the origin is defined as:

Imagestakenfrom


)

nDvuDvuH 20 ]/),([1

1

),(

29

B tt th L Filt ( t )


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54Butterworth Lowpass Filter (cont)

Imagestakenfrom


)

Original

image

Result of filteringwith Butterworth filter

of order 2 and cutoff

radius 5

Result of filtering

with Butterworth

filter of order 2 and

cutoff radius 30

Result of filtering

with Butterworth

filter of order 2 and

cutoff radius 230

Result of filtering with

Butterworth filter of

order 2 and cutoff

radius 80

Result of filtering with

Butterworth filter of

order 2 and cutoff

radius 15


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G i L Filt ( t )


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54Gaussian Lowpass Filters (cont)

Imagestakenfrom

Gonzalez&Woods,DigitalImageProcessing(2002)

Original

image

Result of filteringwith Gaussian filter

with cutoff radius 5

Result of filtering

with Gaussian filter

with cutoff radius 30

Result of filtering

with Gaussian filter

with cutoff radius

230

Result of filtering

with Gaussian

filter with cutoff

radius 85

Result of filteringwith Gaussian

filter with cutoff

radius 15


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of Lowpass Filtering Examples (cont )


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54Lowpass Filtering Examples (cont)

Different lowpass Gaussian filters used to

remove blemishes in a photograph

Imagestakenfrom



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of Ideal High Pass Filters (cont )


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54Ideal High Pass Filters (cont)

Results of ideal

high pass filtering

withD0= 15

Results of ideal

high pass filtering

withD0= 30

Results of ideal

high pass filtering

withD0= 80Imagestakenfrom


2)


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of Butterworth High Pass Filters (cont )


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54Butterworth High Pass Filters (cont)

Results of

Butterworth

high pass

filtering of

order 2 withD0= 15

Results of

Butterworth

high pass

filtering of

order 2 withD0= 80

Results of Butterworth high pass

filtering of order 2 with D0= 30Imagestakenfrom


2)

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of Gaussian High Pass Filters


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54Gaussian High Pass Filters

The Gaussian high pass filter is given as:

whereD0is the cut off distance as before

20

2 2/),(1),(

DvuDevuH

Imagestakenfrom


2)


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of Highpass Filter Comparison


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54Highpass Filter Comparison

Results of ideal

high pass filtering

withD0= 15

Imagestakenfrom


2)

48



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Results of Butterworth

high pass filtering of order2 withD0= 15

Imagestakenfrom


2)

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54g p p

Results of Gaussianhigh pass filtering with

D0= 15

Imagestakenfrom


2)


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54g p p

Results of idealhigh pass filtering

withD0= 15

Imagestakenfrom



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54g p p

Results of Gaussianhigh pass filtering with

D0= 15

ImagestakenfromGonzalez&Woods,Dig

italImageProcessing(2002)


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54Highpass Filtering Example


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54Laplacian In The Frequency Domain


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Laplacianinthe

frequencydoma

in

2-D

imageofLaplacian

inthefrequency

domain

Inverse

DFTof

Laplacianinthe

frequenc

ydomain

Zoomed section o

the image on the

left compared to

spatial filterImagestakenfromGonzalez&Woods,Dig

italImageProcessing(20

02)

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54Frequency Domain Laplacian Example


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Original

image

Laplacian

filtered

image

Laplacian

image scaled

Enhanced

image

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54Fast Fourier Transform


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The reason that Fourier based techniques have

become so popular is the development of theFast Fourier Transform (FFT) algorithm

Allows the Fourier transform to be carried out in

a reasonable amount of timeReduces the amount of time required to perform

a Fourier transform by a factor of 100600

times!

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Frequency Domain Filtering & SpatiaDomain Filtering


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Domain Filtering

Similar jobs can be done in the spatial and

frequency domains

Filtering in the spatial domain can be easier to

understand

Filtering in the frequency domain can be muchfasterespecially for large images

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54Summary


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In this lecture we examined image enhancement

in the frequency domain The Fourier series & the Fourier transform

Image Processing in the frequency domain

Image smoothing Image sharpening

Fast Fourier Transform

Next time we will begin to examine image

restoration using the spatial and frequencybased techniques we have been looking at

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Imteresting Application Of FrequencyDomain Filtering


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Imteresting Application Of FrequencyDomain Filtering


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54Questions?


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imageprocessing7-frequencyfiltering

Documents

frequency domainhttp

reversiblethe inverse

timesfourier spectrum

course website

sum of sines

chaleurpublished in1822

denoted byfu