Chapter 4 Image Enhancement in the Frequency Domainjan/204584/04-image_enhancement_freq.pdf · Chapter 4 Image Enhancement in the Frequency Domain An periodic signals can be Fourier

Chapter 4Image Enhancement in the Frequency Domain


An periodic signals can beFourier series:

Any periodic signals can beviewed as weighted sumof sinusoidal signals withof sinusoidal signals with different frequencies

Frequency Domain:Frequency Domain: view frequency as an independent variableindependent variable


Chapter 4Image Enhancement in the Frequency DomainImage Enhancement in the Frequency DomainImage Enhancement in the Frequency Domain

Background- From the observation: any periodic function may be written as the sum of sines and consines of varying amplitudes and frequencies

xxf sin)( = xxf sin)(1 =

xxf 3sin31)(2 =

xxf 5sin1)(3 =f5

)(3

xxxxxf 7sin715sin

513sin

31sin)( +++= xxf 7sin

71)(4 =



The more terms of the series we have, the closer of the sum will approach the original function

If f(x) is a function period 2T, then we can writethen we can write

)sincos()(1

0 ++= ∑∞

= Txnb

Txnaaxf

nnn

ππ

)(21

0 = ∫ dxxfT

a

whereT

T

....3,2,1,cos)(1

)(20

== ∫

∫

−

−

ndxT

xnxfT

a

fT

T

Tn

T

π

111

....3,2,1,sin)(1== ∫−

ndxT

xnxfT

bT

Tnπ

xxxxxf 7sin715sin

513sin

31sin)( +++=



)sincos()( 0 ++= ∑∞

Txnb

Txnaaxf nn

ππThe Fourier series expansion of f(x) can

)(1

1

∫

=

df

whereTT

T

nbe expressed in complex form

∑∞

= n dxT

xincxf ,)exp()( π

....3,2,1,cos)(1

)(20

==

=

∫

∫−

ndxxnxfa

dxxfT

a

T

n

T

π∫

∑−∞=

−T

n

dxinf

whereT

)()(1 π

....3,2,1,sin)(1

....3,,,cos)(

== ∫

∫

−

−

ndxT

xnxfT

b

ndxT

xfT

a

T

Tn

Tn

πIf the function is non-periodic, we need

ll f i

∫−=

Tn dxT

xfT

c )exp()(2

TTall frequencies to represent.

[ ]∫∞

∞−+=

h

dxjbxaxf ωωωωω sin)(cos)()(

∫∞

∞−= xdxxfa

where

ωπ

ω ,cos)(1)(θθθ sincos je j +=

∫∞

∞−= xdxxfb ω

πω

π

sin)(1)(



Ti ti l FFourier Tr.

Time, spatial Domain Signals

Frequency Domain SignalsInv Fourier TrSignals SignalsInv Fourier Tr.

1-D, Continuous case

∫∞

j2∫∞−

−= dxexfuF uxj π2)()(Fourier Tr.:

∞

∫∞

∞

= dueuFxf uxj π2)()(Inv. Fourier Tr.:∞−



1-D, Discrete casef 1 1 1 1 1 1 1 1

,

fffff = ][

f=1,1,1,1,-1,-1,-1,-1

FFFFF

fffff N −= 1210

][

],....,,[

whereFFFFF N −

⎤⎡

= 1210 ],....,,[

fxNxui

NF

N

xu ∑

−

=⎥⎦⎤

⎢⎣⎡−=

1

02exp1 π



1-D, Discrete case,

∑−

−=1

/2)(1)(N

NuxjexfuF πFourier Tr : u = 0,…,N-1∑=

=0

)()(x

exfN

uFFourier Tr.:

−1N

Inv. Fourier Tr.: ∑=

=0

/2)()(u

NuxjeuFxf π x = 0,…,N-1

Chapter 4Image Enhancement in the Frequency DomainImage Enhancement in the Frequency Domain

1-D, Discrete caseSuppose

Nx fffff −= 1210 ],....,,[is a sequence of length N. we define DFT to be the sequence

Nu FFFFF −= 1210 ],....,,[ℑfF

N

fxuiF

where

∑−

⎥⎤

⎢⎡1

2exp1 π ⎤⎡

ℑ=where

fF

xx

u fN

iN

F ∑=

⎥⎦⎢⎣−=

0

2exp π⎥⎦⎤

⎢⎣⎡−=ℑ 2exp1

, Nmni

Nnm π

As a matrix multiplication: ⎟⎠⎞

⎜⎝⎛ −+−=ℑ )2sin()2cos(1

, Nmnii

Nmni

Nnm ππ

1-D Discrete Fourier Transforms

1 D Discrete case S f [1 2 3 4] th t N 41-D, Discrete case

Nx fffff −= 1210 ],....,,[

Suppose f=[1,2,3,4] so that N=4

⎥⎤

⎢⎡ 1111

NNu

fxuiF

FFFFF

∑−

−

⎥⎤

⎢⎡−=

=1

1210

2exp1],....,,[

π ⎥⎥⎥⎥

⎦⎢⎢⎢⎢

⎣

−−−−

=ℑ

iii

iinm

11111

11,

xx

u fN

iN

F ∑=

⎥⎦⎢⎣0

2exp π

ℑ= fF⎥⎥⎤

⎢⎢⎡

⎥⎥⎤

⎢⎢⎡

−−

⎥⎦

⎢⎣ −−

ii

ii

21

111111

1

11

⎥⎦⎤

⎢⎣⎡−=ℑ 2exp1

, Nmni

N

where

nm π ⎥⎥⎥

⎦⎢⎢⎢

⎣⎥⎥⎥

⎦⎢⎢⎢

⎣ −−−−

=

iii

iiF

432

11111

1141

⎟⎠⎞

⎜⎝⎛ −+−=ℑ

⎦⎣

)2sin()2cos(1, N

mniiNmni

N

NN

nm ππ

⎥⎥⎥⎤

⎢⎢⎢⎡

+−=

iF

222

10

41

⎥⎥

⎦⎢⎢

⎣ −−−

i2224

1-D Discrete Fourier Transforms

1 D Discrete case using Matlab>>f=[1,2,3,4]

ff (f ’)

1-D, Discrete case using Matlab

>> fft(f ’)ans =10.0000 -2.0000 + 2.0000i-2.0000 2 0000 2 0000i-2.0000 - 2.0000i

>> ifft(ans)ans =

12334



Sines and Consines have variousFrom Euler’s formular

sincos jj θθθ +

Sines and Consines have various frequencies, The domain (u) calledfrequency domain

)]/2sin()/2)[cos((1)(

sincos1

NuxjNuxxfuF

jeM

j

ππ

θθθ +=

∑−

f q y

F( ) b i

)]/2sin()/2)[cos(()(0

NuxjNuxxfN

uFx

ππ −= ∑=

F(u) can be written as)()()( ujeuFuF φ−=or)()()( ujIuRuF += )()(or)()()( j

22

where

⎟⎟⎞

⎜⎜⎛− )(tan)( 1 uIuφ22 )()()( uIuRuF += ⎟⎟

⎠⎜⎜⎝

=)(

tan)(uR

uφ

Chapter 4Example of 1 D Fourier Transforms

Chapter 4Example of 1 D Fourier Transforms Example of 1-D Fourier Transforms Example of 1-D Fourier Transforms

Notice that the longerNotice that the longerthe time domain signal,The shorter its Fouriertransform

(Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2nd Edition.

Chapter 4Relation Between Δx and Δu

Chapter 4Relation Between Δx and Δu

F i l f( ) ith N i t l t ti l l ti Δx bFor a signal f(x) with N points, let spatial resolution Δx be space between samples in f(x) and let frequency resolution Δu be space between frequencies components in F(u) we havebetween frequencies components in F(u), we have

u =Δ1)()( 0 xxxfxf Δ+=

Δ

For x = 0,1,2,... N-1

xNu

Δ=Δ

E l f i l f( ) ith li i d 0 5 100 i t)()( uuFuF Δ=

Δ

u = 0,1,2,…N-1Example: for a signal f(x) with sampling period 0.5 sec, 100 point, we will get frequency resolution equal to

1 Hz02.05.0100

1=

×=Δu

This means that in F(u) we can distinguish 2 frequencies that areThis means that in F(u) we can distinguish 2 frequencies that are apart by 0.02 Hertz or more.

Chapter 42-Dimensional Discrete Fourier Transform

Chapter 42-Dimensional Discrete Fourier Transform

F i f i M N i l

1 1M N2-D DFT

For an image of size MxN pixels

∑∑−

=

−

=

+−=1

0

1

0

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

x

N

y

NvyMuxjeyxfMN

vuF π

= =0 0x y

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

2-D IDFT

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

∑∑− −

+=1

0

1

0

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

NvyMuxjevuFyxf π

0 M 1= =0 0u v x = 0 ,…, M-1y = 0 ,…, N-1

Chapter 42 Di i l Di t F i T f ( t )

Chapter 42 Di i l Di t F i T f ( t )2-Dimensional Discrete Fourier Transform (cont.)2-Dimensional Discrete Fourier Transform (cont.)

F(u,v) can be written asF(u,v) can be written as),(),(),( vujevuFvuF φ−=or),(),(),( vujIvuRvuF +=

22 )()()( vuIvuRvuF +

where

⎟⎟⎞

⎜⎜⎛− ),(tan)( 1 vuIvuφ),(),(),( vuIvuRvuF += ⎟⎟

⎠⎜⎜⎝

=),()(tan),(

vuRvuφ

For the purpose of viewing, we usually display only theMagnitude part of F(u,v)ag ude pa o (u,v)

Chapter 4Relation Between Spatial and Frequency

l i

Chapter 4Relation Between Spatial and Frequency

l iResolutionsResolutions

u =Δ1

Nv

Δ=Δ

1xMΔ yNΔ

whereΔx = spatial resolution in x directionΔy = spatial resolution in y direction

( Δx and Δy are pixel width and height. )

Δu = frequency resolution in x directionΔv = frequency resolution in y directionN,M = image width and height

Displaying transforms

• Having obtained the Fourier transform F(u) of a signal f(x), what does it look like?

• Having obtained the Fourier transform F(u,v) of an image f(x,y), what does it look like?g f( y)

• F(u,v) are complex number, we can’t view them directly, but we view their magnitudedirectly, but we view their magnitude |F(u,v)| . We have 3 approaches– 1. Find the maximum value m of |F(u v)| ( DC term), and1. Find the maximum value m of |F(u,v)| ( DC term), and

use imshow to view |F(u,v)| /m– 2. use mat2gray to view |F(u,v)| directlyg y | ( , )| y– 3. use imshow to view log(1+ |F(u,v)| )

Chapter 4Periodicity of 1-D DFT


∑−

−=1

0

/2)(1)(M

x

NuxjexfN

uF πFrom DFT:=0xN

0 N 2N-N 0 N 2N-N

DFT repeats itself every N points (Period = N) but we usuallyWe display only in this range

DFT repeats itself every N points (Period N) but we usually display it for n = 0 ,…, N-1

Chapter 4Conventional Display for 1-D DFT

Chapter 4Conventional Display for 1-D DFT

)(uFDFTf(x)

0 N-1

Time Domain Signal

0 N-1

g

L f

High frequencyarea

Low frequencyarea

The graph F(u) is notThe graph F(u) is not easy to understand !

Chapter 4Conventional Display for DFT : FFT Shift

Chapter 4Conventional Display for DFT : FFT Shift

)(uF )(uFFFT Shift: Shift center of thegraph F(u) to 0 to get betterDisplay which is easier to

)(uF0 N 1

Display which is easier to understand.

)(uF0 N-1

High frequency area

0-N/2 N/2-1Low frequency area



∑∑− −

+−=1 1

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

NvyMuxjeyxfMN

vuF π2-D DFT: ∑∑= =0 0x yMN

-M

g(x y)

0

g(x,y)

For an image of size NxMpixels, its 2-D DFT repeats

0

itself every N points in x-direction and every M points in y direction

M

in y-direction.

We display only 2M in this range

0 N 2N-N

2M


Chapter 4C ti l Di l f 2 D DFT

Chapter 4C ti l Di l f 2 D DFTConventional Display for 2-D DFTConventional Display for 2-D DFT

F(u,v) has low frequency areasat corners of the image while highat corners of the image while highfrequency areas are at the centerof the image which is inconvenientgto interpret.

High frequency area

Low frequency area

Chapter 42 D FFT Shift : Better Display of 2 D DFT

Chapter 42 D FFT Shift : Better Display of 2 D DFT

2-D FFT Shift is a MATLAB function: Shift the zero frequency of F(u v) to the center of an image

2-D FFT Shift : Better Display of 2-D DFT2-D FFT Shift : Better Display of 2-D DFT

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

2D FFTSHIFT2D FFTSHIFT

High frequency area Low frequency area(Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2nd Edition.

Chapter 42 D FFT Shift (cont ) : How it works

Chapter 42 D FFT Shift (cont ) : How it works

M

2-D FFT Shift (cont.) : How it works2-D FFT Shift (cont.) : How it works

-M

0

Display of 2D DFTAfter FFT Shift

MS

Original displayof 2D DFT

2Mof 2D DFT

0 N 2N-N

Chapter 4Example of DFT


• Fourier transforms in MatlabThe relevant Matlab functions for us are:The relevant Matlab functions for us are:

– fft which takes the DFT of a vector,iff hi h k h i DFT f– ifft which takes the inverse DFT of a vector,

– fft2 which takes the DFT of a matrix,– ifft2 which takes the inverse DFT of a matrix,– fftshift which shifts a transformfftshift which shifts a transform.



• Example 1Example 1

Note that the DC coecient is indeed the sum of all theNote that the DC coecient is indeed the sum of all the matrix values.



• Example 2Example 2



E l 3• Example 3

Chapter 4Fourier transforms of images


• Example 1>> a=[zeros(256 128) ones(256 128)];>> a=[zeros(256,128) ones(256,128)];>> af=fftshift(fft2(a));>> fm = max(af(:));>> imshow(im2uint8(af/fm));( ( ))



• Example 2>> a=zeros(256,256);>> a(78:178,78:178)=1;>> imshow(a)( )>> af=fftshift(fft2(a));>> figure,fftshow(af,'abs')g , ( , )



• Example 3>> [x,y]=meshgrid(1:256,1:256);

b ( 329)&( 182)&( 67)&( 73)>> b=(x+y<329)&(x+y>182)&(x-y>-67)&(x-y<73);>> imshow(b)>> bf=fftshift(fft2(b));>> bf fftshift(fft2(b));>> fm = max(bf(:));>> imshow(im2uint8(bf/fm));



E l 4• Example 4>> [x,y]=meshgrid(-128:217,-128:127);>> z=sqrt(x.^2+y.^2);>> c=(z<15);>> imshow(c);>> cf=fft2shift(fft2(c));>> figure,fftshow(af,'abs')

Chapter 4Example of 2 D DFT

Chapter 4Example of 2 D DFTExample of 2-D DFTExample of 2-D DFT

Notice that the longer the time domain signalNotice that the longer the time domain signal,The shorter its Fourier transform

Chapter 4Example of 2-D DFT

Chapter 4Example of 2-D DFTpp

Notice that direction of an object in spatial image andobject in spatial image andIts Fourier transform are orthogonal to each other.



2D DFT

Example of 2 D DFTExample of 2 D DFT

2D DFT

Original image

2D FFT Shift


f = zeros(30,30);f(5:24 13:17) = 1;f(5:24,13:17) 1;figure;imshow(f,'notruesize');F = fft2(f);( )F2 = log(abs(F)+1);figure;imshow(F2,[-1 5],'notruesize'); F3 l ( b (fft hift(F))+1)F3=log(abs(fftshift(F))+1);figure;imshow(F3,[-1 5],'notruesize');



2D DFT

Example of 2 D DFTExample of 2 D DFT

2D DFT

Original image

2D FFT Shift

Chapter 4Basic Concept of Filtering in the Frequency Domain

Chapter 4Basic Concept of Filtering in the Frequency Domain

From Fourier Transform Property:

Basic Concept of Filtering in the Frequency DomainBasic Concept of Filtering in the Frequency Domain

),(),(),(),(),(),( vuGvuHvuFyxhyxfyxg =⋅⇔∗=

We cam perform filtering process by usingWe cam perform filtering process by using

M ltiplication in the freq enc domainMultiplication in the frequency domainis easier than convolution in the spatialDomain.

Filtering in the Frequency Domain with FFT shiftFiltering in the Frequency Domain with FFT shift

F(u,v) H(u,v)(User defined) g(x,y)

FFT hifFFT shift 2D IFFTX

2D FFT FFT shift

f(x,y) G(u,v)f(x,y) G(u,v)

In this case, F(u,v) and H(u,v) must have the same size andhave the zero frequency at the center.

Multiplication in Freq. Domain = Circular ConvolutionMultiplication in Freq. Domain = Circular Convolutionf( ) DFT F( )f(x) DFT F(u)

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

g(x)IDFT

1

h(x) DFT H(u)

Multiplication of DFT f 2 i l f( )

0 20 0 60 80 100 1200

0.5DFTs of 2 signalsis equivalent toperform circular

f(x)

0 20 40 60 80 100 120

0 5

1

perform circular convolutionin the spatial domain.

h( )

0 20 40 60 80 100 1200

0.5 h(x)

“Wrap around” effect 0 20 40 60 80 100 120

40g(x)

p

0 20 40 60 80 100 1200

20g(x)

convolution_discrete_B.swf

Applications of the Fourier TransformApplications of the Fourier TransformApplications of the Fourier TransformApplications of the Fourier TransformLocating Image Features

Th F i f l b d f l i hi hThe Fourier transform can also be used to perform correlation, which is closely related to convolution. Correlation can be used to locate features within an image; in this context correlation is often calledfeatures within an image; in this context correlation is often called template matching.

bw = imread('text.png');a = bw(32:45,88:98); C l(iff 2(ff 2(b ) *C = real(ifft2(fft2(bw) .*

fft2(rot90(a,2),256,256)));figure imshow(C []);figure, imshow(C,[]); max(C(:))ans = 68.0000thresh = 60;figure, imshow(C > thresh)

Multiplication in Freq. Domain = Circular ConvolutionMultiplication in Freq. Domain = Circular Convolution

H(u,v)Gaussian

Original image

LowpassFilter with D0 = 5

Filtered image (obtained using circular convolution)circular convolution)

Incorrect areas at image rims

Fil i i h F D i E lFil i i h F D i E lFiltering in the Frequency Domain : ExampleFiltering in the Frequency Domain : Example

In this example, we set F(0,0) to zerowhich means that the zero frequency

t i dcomponent is removed.

N t Z fNote: Zero frequency = average intensity of an image

Filtering in the Frequency Domain : ExampleFiltering in the Frequency Domain : Example

Lowpass Filter

Highpass Filter


Filtering in the Frequency Domain : Example Filtering in the Frequency Domain : Example Filtering in the Frequency Domain : Example (cont.)

Filtering in the Frequency Domain : Example (cont.)

Result of Sharpening Filter

Filter Masks and Their Fourier TransformsFilter Masks and Their Fourier Transforms


Ideal Lowpass FilterIdeal Lowpass Filter

Ideal LPF Filter Transfer function

⎩⎨⎧

>≤

=0

0

),( 0),( 1

),(DvuDDvuD

vuH⎩

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

Examples of Ideal Lowpass FiltersExamples of Ideal Lowpass Filters

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

Results of Ideal Lowpass Filters Results of Ideal Lowpass Filters

Ringing effect can be obviously seen!


How ringing effect happens How ringing effect happens

⎩⎨⎧

>≤

=0

0

),( 0),( 1

),(DvuDDvuD

vuH

Surface Plot

⎩

Ideal Lowpass FilterIdeal Lowpass Filterwith D0 = 5

Abrupt change in the amplitudeAbrupt change in the amplitude

How ringing effect happens (cont ) How ringing effect happens (cont ) How ringing effect happens (cont.) How ringing effect happens (cont.)

x 10-3

Surface Plot

15

5

10

Spatial Response of Ideal Lowpass Filter with D = 5

2020

0Lowpass Filter with D0 = 5

-200

20

-20

0Ripples that cause ringing effect

How ringing effect happens How ringing effect happens g g pp(cont.)

g g pp(cont.)

Butterworth Lowpass Filter Butterworth Lowpass Filter

1

Transfer function

[ ] NDvuDvuH 2

0/),(11),(

+=

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

Results of Butterworth Lowpass Filters Results of Butterworth Lowpass Filters

There is less ringing effect compared toeffect compared to those of ideal lowpassfilters!filters!

Spatial Masks of the Butterworth Lowpass Filters Spatial Masks of the Butterworth Lowpass Filters

Some ripples can be seen.

Gaussian Lowpass Filter Gaussian Lowpass Filter

20

2 2/),()( DvuDevuH −=

Transfer function

),( evuH =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.) Gaussian Lowpass Filter (cont.)

1

20

2 2/),(),( DvuDevuH −=

0 4

0.6

0.8 Gaussian lowpass filter with D0 = 5

0.2

0.4

-200

20

-20

0

20

0.03

0 01

0.02

0.03

Spatial respones of the Gaussian lowpass filter

2020

0

0.01Gaussian lowpass filterwith D0 = 5

-200

20

-20

0

Gaussian shape

Results of Gaussian Results of Gaussian Results of Gaussian Lowpass Filters

Results of Gaussian Lowpass Filters

No ringing effect!

Application of Gaussian Lowpass FiltersApplication of Gaussian Lowpass Filters

Better LookingOriginal image

Th GLPF b d t j d dThe GLPF can be used to remove jagged edges and “repair” broken characters.

Application of Gaussian Lowpass Filters (cont.) Application of Gaussian Lowpass Filters (cont.) R i klRemove wrinkles

Original image

Softer-Looking

Application of Gaussian Lowpass Filters (cont.) Application of Gaussian Lowpass Filters (cont.)

Filtered imageOriginal image : The gulf of Mexico and Filtered imageOriginal image : The gulf of Mexico andFlorida from NOAA satellite. (Images from Rafael C. Gonzalez and Richard E.

Wood, Digital Image Processing, 2nd Edition.

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

Hi h Filt Hi h Filt Highpass Filters Highpass Filters

Hhp = 1 - Hlp

Ideal Highpass Filters Ideal Highpass Filters Ideal Highpass Filters Ideal Highpass Filters

⎨⎧ ≤

= 0),( 0)(

DvuDvuH

Ideal HPF Filter Transfer function

⎩⎨ >

=0),( 1

),(DvuD

vuH

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

Butterworth Highpass Filters Butterworth Highpass Filters Butterworth Highpass Filters Butterworth Highpass Filters

T f f ti

[ ]vuH 1)( =

Transfer function

[ ] NvuDDvuH 2

0 ),(/1),(

+

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

Gaussian Highpass Filters Gaussian Highpass Filters Gaussian Highpass Filters Gaussian Highpass Filters

Transfer function2

02 2/),(1),( DvuDevuH −−=

Transfer function

Where D0 = spread factor.

Gaussian Highpass Filters (cont.) Gaussian Highpass Filters (cont.) g p ( )g p ( )

20

2 2/),(1),( DvuDevuH −−=1

0.6

0.8

Gaussian highpass filter with D = 5

0

0.2

0.4 filter with D0 = 5

1020

3040

5060

20

40

60

2000

3000

0

1000Spatial respones of the Gaussian highpass filter

ith D 5

1020

3040

5060

20

40

60with D0 = 5

Spatial Responses of Highpass Filters Spatial Responses of Highpass Filters

Ripplespp

R lt f Id l Hi h Filt R lt f Id l Hi h Filt Results of Ideal Highpass Filters Results of Ideal Highpass Filters

Ringing effect can be b i l !obviously seen!

R lt f B tt th Hi h Filt R lt f B tt th Hi h Filt Results of Butterworth Highpass Filters Results of Butterworth Highpass Filters

Results of Gaussian Highpass Filters Results of Gaussian Highpass Filters g pg p

Laplacian Filter in the Frequency DomainLaplacian Filter in the Frequency Domain

( ) )()( uFjuxfd nn

⇔

From Fourier Tr. Property:

( ) )(uFjudxn ⇔

Then for Laplacian operator22 ff ∂∂ ( ) ),(22

2

2

2

22 vuFvu

yf

xff +−⇔

∂∂

+∂∂

=∇

We get( )222 vu +−⇔∇

We get

Image ofg–(u2+v2)

Surface plot

Laplacian Filter in the Frequency Domain (cont.)Laplacian Filter in the Frequency Domain (cont.)

Spatial response of –(u2+v2) Cross section

Laplacian mask in Chapter 3p p

Sharpening Filtering in the Frequency DomainSharpening Filtering in the Frequency DomainSharpening Filtering in the Frequency DomainSharpening Filtering in the Frequency Domain

Spatial Domain

),(),(),( yxfyxfyxf lphp −=

),(),(),( yxfyxAfyxf lphb −=

)()()()1()( yxfyxfyxfAyxf + ),(),(),()1(),( yxfyxfyxfAyxf lphb −+−=

),(),()1(),( yxfyxfAyxf hphb +−=

Frequency Domain Filter

),(),()(),( yfyfyf hphb

q y

),(1),( vuHvuH lphp −=

),()1(),( vuHAvuH hphb +−=

Sharpening Filtering in the Frequency Domain (cont.)Sharpening Filtering in the Frequency Domain (cont.)

p P2∇

P2∇ PP 2∇P2∇ PP 2∇−

Sharpening Filtering in the Frequency Domain (cont.)Sharpening Filtering in the Frequency Domain (cont.)

),(),()1(),( yxfyxfAyxf hphb +−= ),(),()(),( yfyfyf hphb

Pfhp2∇=f

A = 2 A = 2 7A = 2.7

High Frequency Emphasis FilteringHigh Frequency Emphasis Filtering

),(),( vubHavuH hphfe += ),(),( hphfe

Original Butterworthhighpass g passfilteredimage

High freq. emphasis Afterg q pfiltered image Hist

Eq.

a = 0.5, b = 2

Homomorphic FilteringHomomorphic Filtering

An image can be expressed asg p),(),(),( yxryxiyxf =

i(x y) = illumination componenti(x,y) illumination componentr(x,y) = reflectance component

We need to suppress effect of illumination that cause image Intensity changed slowly.

Homomorphic FilteringHomomorphic Filtering

H hi Filt iH hi Filt iHomomorphic FilteringHomomorphic Filtering

More details in the room can be seen!More details in the room can be seen!

Retinex Filter

jckI

The Relative reflectance of pixel i to j:

∑ +=ckc IjiR 1log)( δk

ckI 1+

∑=k

ckI

jiR log),( δ

{ Threshold log if 1 1 >+c

ck

II

i =δ { ckI

Threshold log if 0 1 <+ck

ck

II

kI

j=1 j=2 j=4Average Relative Reflectance at i:

jiRN c∑ )(

i

N

jiRiR jc ∑ == 1

),()(

ij=3 j=5

Chapter 4How to Perform 2-D DFT by Using 1-D DFT

Chapter 4How to Perform 2-D DFT by Using 1-D DFTHow to Perform 2-D DFT by Using 1-D DFTHow to Perform 2-D DFT by Using 1-D DFT

f(x y)

1-D DFT

by row F(u,y)f(x,y) by row (u,y)1-D DFT

by columnby column

F(u,v)

Chapter 4How to Perform 2 D DFT by Using 1 D DFT(cont )

Chapter 4How to Perform 2 D DFT by Using 1 D DFT(cont )

Alternative method

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

Alternative method

f(x,y)f( ,y)1-D DFT

by columnby column

1-D DFT

by rowF(x,v) F(u,v)

Linear Convolution by using Circular Convolution and Zero Padding Linear Convolution by using Circular Convolution and Zero Padding

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

h(x) DFT H(u)Zero padding

Zero padding

h(x) DFT H(u)IDFT

Zero padding

1

Concatenation0

0.5

g(x)0 50 100 150 200 250

1

0 50 100 150 200 2500

0.5

Padding zerosBefore DFT0 50 100 150 200 250

40

Before DFT

Keep only this part

0 50 100 150 200 2500

20p y p


The discrete Fourier transform assumes a The discrete Fourier transform assumes a transform assumes a digital image exists on a closed surface, a torus.

transform assumes a digital image exists on a closed surface, a torus.



Filtered image

Zero padding area in the spatialDomain of the mask image(th id l l filt )

Only this area is kept.(the ideal lowpass filter)

Correlation Application: Object DetectionCorrelation Application: Object Detection

Chapter 42 D DFT P ti

Chapter 42 D DFT P ti2-D DFT Properties2-D DFT Properties

Chapter 42 D DFT P ti ( t )

Chapter 42 D DFT P ti ( t )2-D DFT Properties (cont.)2-D DFT Properties (cont.)

Chapter 4Chapter 42-D DFT Properties (cont.)2-D DFT Properties (cont.)

Chapter 42 D DFT P ti ( t )

Chapter 42 D DFT P ti ( t )2-D DFT Properties (cont.)2-D DFT Properties (cont.)

Chapter 4C t ti l Ad t f FFT C d t DFT

Chapter 4C t ti l Ad t f FFT C d t DFTComputational Advantage of FFT Compared to DFTComputational Advantage of FFT Compared to DFT

Chapter 4Chapter 4Image Enhancement in the

Frequency DomainImage Enhancement in the

Frequency Domain



Frequency Domain



Frequency Domain



Frequency Domain



Frequency Domain



Frequency Domain



Frequency Domain

Chapter 4Image Enhancement in the

F D i


F D iFrequency DomainFrequency Domain


F D i




Chapter 4Image Enhancement in theImage Enhancement in the


Frequency Domain


F D i




Chapter 4Image Enhancement in theImage Enhancement in the


Frequency Domain



Frequency Domain



Frequency Domain



Frequency Domain



Frequency Domain



Frequency Domain



Frequency Domain



Frequency Domain



Frequency Domain



Frequency Domain



Frequency Domain



Frequency Domain



Frequency Domain



Frequency Domain



Frequency Domain



Frequency Domain



Frequency Domain



Frequency Domain



Frequency Domain



Frequency Domain



Frequency Domain



Frequency Domain



Frequency Domain



Frequency Domain



Frequency Domain



Frequency Domain



Frequency Domain



Frequency Domain



Frequency Domain



Frequency Domain



Frequency Domain



Frequency Domain

Documents

Chapter 4 Image Enhancement in the Frequency Domainjan/204584/04-image_enhancement_freq.pdf · Chapter 4 Image Enhancement in the Frequency Domain An periodic signals can be Fourier