Chapter 4 Image Enhancement in the Frequency Domainjan/204584/04-image_enhancement_freq.pdf ·...
Transcript of Chapter 4 Image Enhancement in the Frequency Domainjan/204584/04-image_enhancement_freq.pdf ·...
Chapter 4Image Enhancement in the Frequency Domain
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 Domain
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 =
Chapter 4Image Enhancement in the Frequency Domain
Chapter 4Image Enhancement in the Frequency DomainImage Enhancement in the Frequency DomainImage Enhancement in the Frequency Domain
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)( +++=
Chapter 4Image Enhancement in the Frequency Domain
Chapter 4Image Enhancement in the Frequency DomainImage Enhancement in the Frequency DomainImage Enhancement in the Frequency Domain
)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)(
Chapter 4Image Enhancement in the Frequency Domain
Chapter 4Image Enhancement in the Frequency Domain
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.:∞−
Chapter 4Image Enhancement in the Frequency Domain
Chapter 4Image Enhancement in the Frequency Domain
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
Chapter 4Image Enhancement in the Frequency Domain
Chapter 4Image Enhancement in the Frequency Domain
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
Chapter 4Image Enhancement in the Frequency Domain
Chapter 4Image Enhancement in the Frequency Domain
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
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
Chapter 4Periodicity of 2-D DFT
Chapter 4Periodicity of 2-D DFT
∑∑− −
+−=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
(Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2nd Edition.
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
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.
Chapter 4Example of DFT
Chapter 4Example of DFT
• 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.
Chapter 4Example of DFT
Chapter 4Example of DFT
• Example 2Example 2
Chapter 4Example of DFT
Chapter 4Example of DFT
E l 3• Example 3
Chapter 4Fourier transforms of images
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));( ( ))
Chapter 4Fourier transforms of images
Chapter 4Fourier transforms of images
• Example 2>> a=zeros(256,256);>> a(78:178,78:178)=1;>> imshow(a)( )>> af=fftshift(fft2(a));>> figure,fftshow(af,'abs')g , ( , )
Chapter 4Fourier transforms of images
Chapter 4Fourier transforms of images
• 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));
Chapter 4Fourier transforms of images
Chapter 4Fourier transforms of images
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.
Chapter 4Example of 2-D DFT
Chapter 4Example of 2-D DFT
2D DFT
Example of 2 D DFTExample of 2 D DFT
2D DFT
Original image
2D FFT Shift
Chapter 4Example of 2-D DFT
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');
Chapter 4Example of 2-D DFT
Chapter 4Example of 2-D DFT
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
(Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2nd Edition.
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
(Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2nd Edition.
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!
(Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2nd Edition.
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.
(Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2nd Edition.
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
Linear Convolution by using Circular Convolution and Zero Padding Linear Convolution by using Circular Convolution and Zero Padding
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.
Linear Convolution by using Circular Convolution and Zero Padding Linear Convolution by using Circular Convolution and Zero Padding
Linear Convolution by using Circular Convolution and Zero Padding Linear Convolution by using Circular Convolution and Zero Padding
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
Chapter 4Chapter 4Image Enhancement in the
Frequency DomainImage Enhancement in the
Frequency Domain
Chapter 4Chapter 4Image Enhancement in the
Frequency DomainImage Enhancement in the
Frequency Domain
Chapter 4Chapter 4Image Enhancement in the
Frequency DomainImage Enhancement in the
Frequency Domain
Chapter 4Chapter 4Image Enhancement in the
Frequency DomainImage Enhancement in the
Frequency Domain
Chapter 4Chapter 4Image Enhancement in the
Frequency DomainImage Enhancement in the
Frequency Domain
Chapter 4Chapter 4Image Enhancement in the
Frequency DomainImage Enhancement in the
Frequency Domain
Chapter 4Image Enhancement in the
F D i
Chapter 4Image Enhancement in the
F D iFrequency DomainFrequency Domain
Chapter 4Image Enhancement in the
F D i
Chapter 4Image Enhancement in the
F D iFrequency DomainFrequency Domain
Chapter 4Image Enhancement in the
Chapter 4Image Enhancement in theImage Enhancement in the
Frequency DomainImage Enhancement in the
Frequency Domain
Chapter 4Image Enhancement in the
F D i
Chapter 4Image Enhancement in the
F D iFrequency DomainFrequency Domain
Chapter 4Image Enhancement in the
Chapter 4Image Enhancement in theImage Enhancement in the
Frequency DomainImage Enhancement in the
Frequency Domain
Chapter 4Chapter 4Image Enhancement in the
Frequency DomainImage Enhancement in the
Frequency Domain
Chapter 4Chapter 4Image Enhancement in the
Frequency DomainImage Enhancement in the
Frequency Domain
Chapter 4Chapter 4Image Enhancement in the
Frequency DomainImage Enhancement in the
Frequency Domain
Chapter 4Chapter 4Image Enhancement in the
Frequency DomainImage Enhancement in the
Frequency Domain
Chapter 4Chapter 4Image Enhancement in the
Frequency DomainImage Enhancement in the
Frequency Domain
Chapter 4Chapter 4Image Enhancement in the
Frequency DomainImage Enhancement in the
Frequency Domain
Chapter 4Chapter 4Image Enhancement in the
Frequency DomainImage Enhancement in the
Frequency Domain
Chapter 4Chapter 4Image Enhancement in the
Frequency DomainImage Enhancement in the
Frequency Domain
Chapter 4Chapter 4Image Enhancement in the
Frequency DomainImage Enhancement in the
Frequency Domain
Chapter 4Chapter 4Image Enhancement in the
Frequency DomainImage Enhancement in the
Frequency Domain
Chapter 4Chapter 4Image Enhancement in the
Frequency DomainImage Enhancement in the
Frequency Domain
Chapter 4Chapter 4Image Enhancement in the
Frequency DomainImage Enhancement in the
Frequency Domain
Chapter 4Chapter 4Image Enhancement in the
Frequency DomainImage Enhancement in the
Frequency Domain
Chapter 4Chapter 4Image Enhancement in the
Frequency DomainImage Enhancement in the
Frequency Domain
Chapter 4Chapter 4Image Enhancement in the
Frequency DomainImage Enhancement in the
Frequency Domain
Chapter 4Chapter 4Image Enhancement in the
Frequency DomainImage Enhancement in the
Frequency Domain
Chapter 4Chapter 4Image Enhancement in the
Frequency DomainImage Enhancement in the
Frequency Domain
Chapter 4Chapter 4Image Enhancement in the
Frequency DomainImage Enhancement in the
Frequency Domain
Chapter 4Chapter 4Image Enhancement in the
Frequency DomainImage Enhancement in the
Frequency Domain
Chapter 4Chapter 4Image Enhancement in the
Frequency DomainImage Enhancement in the
Frequency Domain
Chapter 4Chapter 4Image Enhancement in the
Frequency DomainImage Enhancement in the
Frequency Domain
Chapter 4Chapter 4Image Enhancement in the
Frequency DomainImage Enhancement in the
Frequency Domain
Chapter 4Chapter 4Image Enhancement in the
Frequency DomainImage Enhancement in the
Frequency Domain
Chapter 4Chapter 4Image Enhancement in the
Frequency DomainImage Enhancement in the
Frequency Domain
Chapter 4Chapter 4Image Enhancement in the
Frequency DomainImage Enhancement in the
Frequency Domain
Chapter 4Chapter 4Image Enhancement in the
Frequency DomainImage Enhancement in the
Frequency Domain
Chapter 4Chapter 4Image Enhancement in the
Frequency DomainImage Enhancement in the
Frequency Domain
Chapter 4Chapter 4Image Enhancement in the
Frequency DomainImage Enhancement in the
Frequency Domain
Chapter 4Chapter 4Image Enhancement in the
Frequency DomainImage Enhancement in the
Frequency Domain
Chapter 4Chapter 4Image Enhancement in the
Frequency DomainImage Enhancement in the
Frequency Domain
Chapter 4Chapter 4Image Enhancement in the
Frequency DomainImage Enhancement in the
Frequency Domain