Chapter 4 Image Enhancement in the Frequency DomainCSIE/ycliaw/DIP/04_Enhancement_Frequency.pdf · Enhancement methods Spatial Domain (in chapter 3) based on direct manipulation of

1

Digital Image Processing, 2nd ed.Digital Image Processing, 2nd ed.www.imageprocessingbook.com

© 2002 R. C. Gonzalez & R. E. Woods

The principal objective of enhancementThe principal objective of enhancementto process an image so that the result is more suitablethan the original image for a specific application.

Enhancement methodsEnhancement methodsSpatial Domain (in chapter 3)

based on direct manipulation of pixels in an image

Frequency Domain (in this chapter)based on modifying the Fourier transform of an image

The viewer is the ultimate judge of how well ofThe viewer is the ultimate judge of how well ofa particular method works.a particular method works.

Chapter 4 Chapter 4 Image Enhancement in theImage Enhancement in theFrequency DomainFrequency Domain



Chapter 4 Chapter 4 Image Enhancement in theImage Enhancement in theFrequency DomainFrequency Domain

4.1 Background4.2 Introduction to the Fourier Transform and the

Frequency Domain4.3 Smoothing Frequency-Domain Filters4.4 Sharpening Frequency-Domain filters4.5 Homomorphic Filtering4.6 Implementation

2



4.1 Background4.1 Background

Source:http://www-groups.dcs.st-and.ac.uk/~history/PictDisplay/Fourier.html

Jean Jean BaptisteBaptiste Joseph Fourier(1768~1830) Joseph Fourier(1768~1830)French mathematicianFrench mathematician




3




Fourier SeriesFourier SeriesAny Any periodically repeated functionperiodically repeated function can be expressed can be expressedof the sum of of the sum of sinessines/cosines/cosines of of different frequenciesdifferent frequencies,,each multiplied by a different coefficienteach multiplied by a different coefficient

Fourier TransformFourier TransformFinite curvesFinite curves can be expressed as the integral of can be expressed as the integral ofsinessines/cosines multiplied by a weighing function/cosines multiplied by a weighing functionwildly used in wildly used in signal processingsignal processing field field

Fourier Series/TransformFourier Series/Transform can be reconstructed can be reconstructedcompletely via completely via an inverse processan inverse process



4.2 Introduction to the Fourier Transform and4.2 Introduction to the Fourier Transform andthe Frequency Domainthe Frequency Domain

This Section will introduceThis Section will introduceOne dimensionOne dimension and and Two dimensionTwo dimension Fourier Fouriertransformtransformmostly on a mostly on a discrete formulationdiscrete formulation of the continuous of the continuoustransform and some of its propertiestransform and some of its properties

4




One dimensional Fourier transform and itsOne dimensional Fourier transform and itsinverse (Fourier transform pair)inverse (Fourier transform pair)

Fourier transform

Inverse Fourier transform

其中, f(x)是單變數連續函式，F(u)是f(x)的FourierTransform結果，j= 1−

( ) ( ) 1)-(4.2 2 dxexfuF uxj π−∞

∞−∫=

( ) ( ) 2)-(4.2 2 dueuFxf uxj π∫∞

∞−=

像素強度

頻率強度

頻率強度

像素強度




Two dimensional Fourier transform and itsTwo dimensional Fourier transform and itsinverse (Fourier transform pair)inverse (Fourier transform pair)

Fourier transform

Inverse Fourier transform

( ) ( ) ( ) 3)-(4.2 ,, 2∫ ∫∞

∞−

∞

∞−

+−= dydxeyxfvuF vyuxj π

( ) ( ) ( ) 4)-(4.2 ,, 2∫ ∫∞

∞−

∞

∞−

+= dvduevuFyxf vyuxj π

5




Discrete Fourier Transform (DFT)Discrete Fourier Transform (DFT)

Discrete Fourier Transform

Inverse Discrete Fourier Transform

( ) ( ) 5)-(4.2 1 -210for 1 1

0

2 ., ..., M, , uexfM

uFM

x

Muxj == ∑−

=

− π

( ) ( ) 6)-(4.2 1 -210for 1

0

2 ., ..., M, , xeuFxfM

x

Muxj == ∑−

=

π




Frequency DomainFrequency Domain

Euler’s formula

7)-(4.2 sincos θθθ je j +=

substituting this expression into Eq. (4.2-5)

( ) ( )[ ]

8)-(4.2 1-M ..., 2, 1,for

/2sin/2cos1 1

0

=

−= ∑−

=

u

MuxjMuxxfM

uFM

x

ππ

F(u)是f(x)的頻率成份，Fourier Transform就像菱鏡的功能一樣可將f(x)的頻率成份分離出來分離出來的值為F(u)

6




To express F(x) in polar coordinates(To express F(x) in polar coordinates(極座標極座標))( ) ( ) ( ) 9)-(4.2 ujeuFuF φ−=

( ) ( ) ( )[ ] 10)-(4.2 21

22 uIuRuF +=

|F(u)| is called the magtitute(強度) or spectrum(頻譜)

( ) ( )( ) 11)-(4.2 tan 1

= −

uRuIuφ

φ is called the phase angle (相位角) or phase spectrum

R(u) 與 I(u)分別是F(u)的實部與虛部

( ) ( ) ( ) ( ) 12)-(4.2 222 uIuRuFuP +==

p(u) is called power spectrum of spectrum density



( ) ( )

( )

MAKMA

eMAF

eMA

AeM

exfM

uF

K

x

K

x

Mxj

K

x

Muxj

K

x

Muxj

M

x

Muxj

=

=

=

=

=

=

∑

∑

∑

∑

∑

−

=

−

=

−

−

=

−

−

=

−

−

=

−

1

0

1

0

02

1

0

2

1

0

2

1

0

2

1

0

1

1

π

π

π

π


ExampleExample

7







8




22D DFT and its inverseD DFT and its inverse




Spectrum, Phase angle, and Spectrum densitySpectrum, Phase angle, and Spectrum density

9




Shifts the origin of F(u, v) to (M/2, N/2)Shifts the origin of F(u, v) to (M/2, N/2)

dc (direct current) valuedc (direct current) valuezero frequencyzero frequencymean value of an imagemean value of an image




if f(x,y) is real, its Fourier transformation isif f(x,y) is real, its Fourier transformation isconjugate conjugate ((共軛共軛)) symmetricsymmetric

10




ExampleExample




11




Filtering in the frequency domainFiltering in the frequency domain影像灰階內容與頻率的關係



1. Multiply the input image by (-1)x+y to center the transform,as indicated in Eq. (4.2-21).

2. Compute F(u, v), the DFT of the image from (1).3. Multiply F(u, v) by a filter function H(u,v).

4. Compute the inverse DFT of the result in (3).

5. Obtain the real part of the result in (4).6. Multiply the result in (5) by (-1)x+y


Steps of Filtering in the frequency domainSteps of Filtering in the frequency domain

12







Some basic filters and their propertiesSome basic filters and their propertiesNotch filter - remove the average value of an imageNotch filter - remove the average value of an image

13




Low frequenciesLow frequenciesSmooth areasSmooth areas

High frequenciesHigh frequenciesEdge, Texture, noiseEdge, Texture, noise

LowpassLowpass filter filter濾掉高頻部份，保留低頻部份濾掉高頻部份，保留低頻部份

highpass highpass filterfilter濾掉低頻部份，保留高頻部份濾掉低頻部份，保留高頻部份




14







Correspondence between filtering in the Correspondence between filtering in the spatialspatialand and frequencyfrequency domains domains

Convolution theorem(Convolution theorem(捲積定理捲積定理))

( ) ( ) ( ) ( ) ( )31-4.2 ,,,*, vuHvuFyxhyxf ⇔

( ) ( ) ( ) ( ) ( )32-4.2 ,*,,, vuHvuFyxhyxf ⇔

捲積捲積

頻域相乘等於空間域做捲積，空間域相乘等於頻域做捲積

15




Impulse Function(Impulse Function(脈衝函式脈衝函式))

Fourier transform of Impulse FunctionFourier transform of Impulse Function

( ) ( ) ( ) 33)-(4.2 ,,,1

0

1

00000∑∑

−

=

−

=

=−−M

x

N

y

yxAsyyxxAyxs δ

( ) ( ) ( ) 34)-(4.2 0,0,,1

0

1

0∑∑−

=

−

=

=M

x

N

y

syxyxs δ

( ) ( ) ( ) 35)-(4.2 1,1,1

0

1

0

//2

MNeyx

MNvuF

M

x

N

y

NvyMuxj == ∑∑−

=

−

=

+− πδ

函式函式與與脈衝函式脈衝函式做做捲積捲積就是將函式就是將函式複製複製到脈衝函式所在位置到脈衝函式所在位置



( ) ( ) ( ) ( )

( ) 36)-(4.2 1

,,1,*,1

0

1

0

x,yhMN

nymxhnmMN

yxhyxfM

x

N

y

=

−−= ∑∑−

=

−

=

δ


Convolution between h(x,y) and Impulse FunctionConvolution between h(x,y) and Impulse Function

16




Correspondence between filtering in the Correspondence between filtering in the spatialspatialand and frequencyfrequency domains domains

Spatial Domain Frequency Domain




GaussianGaussian filter ( filter (高斯濾波器高斯濾波器))

( ) 38)-(4.2 22/2 σuAeuH −=

( ) 39)-(4.2 22222 xAexh σπσπ −=

( ) 40)-(4.2 -22

221

2 2/2/ σσ uu BeAeuH −−=

( ) 41)-(4.2 2 -222

2222

12 2

22

1xx BeAexh σπσπ σπσπ −−=

17






4.3 Smoothing Frequency-Domain Filters4.3 Smoothing Frequency-Domain Filters

( ) ( ) ( ) 1)-(4.3 ,,, vuFvuHvuG =

4.3.1 4.3.1 Ideal Ideal LowpassLowpass Filters Filters

( ) ( )( ) 2)-(4.3

if 0if 1

,0

0

>≤

=Du,v DDu,v D

vuH

D(u,v)表距中心點距離, MxN的影像，中心點為(M/2, N/2)

( ) ( ) ( )[ ] 3)-(4.3 2/2/, 2122 NvMuvuD −+−=

D0 is called cutoff frequency

18







Choosing the cutoff frequency according to theChoosing the cutoff frequency according to thepower spectrum of an imagepower spectrum of an image

( ) 4)-(4.3 ,1

0

1

0∑∑−

=

−

=

=M

u

N

vT vuPP

( ) 5)-(4.3 /,100

= ∑∑

u vTPvuPα

19







a b cd e f

20




blurring and ringing effectsblurring and ringing effects

( ) ( ) ( )( ) ( ) ( )yxfyxhyxgDomainSpatial

vuFvuHvuGDomainFrequency,*,,:

,,,: =

=

Blurring : Low frequencies are removedBlurring : Low frequencies are removed

Ringing : Cutoff is too sharpRinging : Cutoff is too sharp




21




4.3.2 4.3.2 ButterworthButterworth LowpassLowpass Filters Filters

( )( )[ ]

6)-(4.3 ,11, 2

0nDvuD

vuH+

=



4.3 Smoothing Frequency-Domain Filters4.3 Smoothing Frequency-Domain Filtersno ringing

a b cd e f

FIGURE 4.15 (a) Original image. (b)-(f) Results of filtering with BLPFs of order 2,with cutoff frequencies at radii of 5, 15, 30, 80, 230, as shown in Fig. 4.11 (b).Compare with Fig. 4.12.

22







4.3.3 4.3.3 Gaussian LowpassGaussian Lowpass Filters Filters

( ) ( ) 7)-(4.3 ,22 2, σvuDevuH −=

( ) ( ) 8)-(4.3 ,20

2 2, DvuDevuH −=

23




a b cd e f

FIGURE 4.18 (a) Original image. (b)-(f) Results of filtering with Gaussian lowpassfilters with cutoff frequencies set at radii of 5, 15, 30, 80, 230, as shown in Fig. 4.11(b). Compare with Figs. 4.12 and 4.15.

no ringing




24




4.3.4 4.3.4 Additional examples of Additional examples of LowpassLowpass Filtering Filtering




25



4.4 Sharping Frequency-Domain Filters4.4 Sharping Frequency-Domain Filters

( ) ( ) 1)-(4.4 ,1, vuHvuH lphp −=




26




4.4.1 4.4.1 Ideal Ideal HighpassHighpass Filters Filters

( ) ( )( ) 2)-(4.4

if 1if 0

,0

0

>≤

=Du,v DDu,v D

vuH




4.4.2 4.4.2 ButterworthButterworth HighpassHighpass Filters Filters( )

( )[ ] 3)-(4.4 ,1

1, 20

nvuDDvuH

+=

27



4.4.3 4.4.3 Gaussian HighpassGaussian Highpass Filters Filters


( ) ( ) 4)-(4.4 1,20

2 2, DvuDevuH −−=




4.4.4 4.4.4 The The Laplacian Laplacian in the Frequency Domainin the Frequency Domain

28




4.4.4 4.4.4 The The Laplacian Laplacian in the Frequency Domainin the Frequency Domain

An image can be enhanced by subtracting the Laplacianfrom the original image.

( ) ( ) ( ) 12)-(4.4 ,,, 2 yxfyxfyxg ∇−=

( ) ( ) ( )[ ] ( ){ } 13)-(4.4 ,221 , 221 vuFNvMuyxg −+−−ℑ= −

( ) ( ) ( )[ ] ( ){ } 10)-(4.4 ,2/2/, 2212 vuFNvMuyxf −+−−ℑ=∇ −

( ) ( ) ( )[ ] ( ) 11)-(4.4 ,2/2/, 222 vuFNvMuyxf −+−−⇔∇




29







4.4.5 4.4.5 UnsharpUnsharp Masking, High-Boost Filtering, Masking, High-Boost Filtering,and High-and High-GrequencyGrequency Emphasis Filtering Emphasis Filtering

Unsharp Unsharp filteringfiltering

High-Boost filteringHigh-Boost filtering

( ) ( ) ( ) 14)-(4.4 ,,, yxfyxfyxf lphp −=

( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) 17)-(4.4 ,,1-A

16)-(4.4 ,,,1-A

15)-(4.4 ,,,

yxfyxf

yxfyxfyxf

yxfyxAfyxf

hp

lp

lphb

+=

−+=

−=

30



UnsharpUnsharp filtering - frequency-domain filter filtering - frequency-domain filter

High Boost filtering - frequency-domain filterHigh Boost filtering - frequency-domain filter


( ) ( ) 18)-(4.4 ,1, vuHvuH lphp −=

( ) ( ) ( ) 19)-(4.4 ,1, vuHAvuH hphb −−=




31




High frequency emphasisHigh frequency emphasis

( ) ( ) 20)-(4.4 ,, vubHavuH hphfe +=

typically, 0.25≤ a ≤ 0.5, 1.5 ≤ b ≤ 2.0




32



4.5 Homomorphic Filtering4.5 Homomorphic Filtering

illumination-reflectance modelillumination-reflectance modelsimultaneous simultaneous gray-level range compressiongray-level range compression and andcontract enhancementcontract enhancement

( ) ( ) ( ) 1)-(4.5 .,,, yxryxiyxf =

( ) ( )( ) ( ) 3)-(4.5 .,ln,ln

,ln, yxryxi

yxfyxzLet+=

=

( )( ) ( )( ) ( )( ) 2)-(4.5 .,,, yxryxiyxf ℑℑ≠ℑ

( ) ( ) ( ) 4)-(4.5 ,,,Then vuFvuFvuZ ri +=




( ) ( ) ( )( ) ( ) ( ) ( ) 5)-(4.5

,,, u,vFu,vHu,vFu,vH

vuZvuHvuS

ri +==

( ) ( ){ }( ) ( ){ } ( ) ( ){ } 6)-(4.5

,,11

1

u,vFu,vHu,vFu,vHvuSyxs

ri−−

−

ℑ+ℑ=

ℑ=

( ) ( ) ( ){ }( ) ( ) ( ){ } 8)-(4.5

7)-(4.5 ,' 1

1

u,vFu,vHx,yr'

u,vFu,vHyxiLet

r

i−

−

ℑ=

ℑ=

( ) ( ) ( ) 9)-(4.5 .,',', yxryxiyxs =

33



( ) ( )

( ) ( )

( ) ( ) 10)-(4.5

,

00

,

x,yrx,yiee

eyxgx,yr'x,yi'

yxs

=⋅=

=

( ) ( ) 11)-(4.5 ,'0

yxiex,yi =

( ) ( ) 12)-(4.5 ,'0

yxrex,yr =





34






4.6 Implementation4.6 Implementation

Some Additional Properties of the 2D FourierSome Additional Properties of the 2D FourierTransform TranslationTransform Translation

ShiftingShifting

Shifts center to (M/2, N/2)Shifts center to (M/2, N/2)

35




DistributivityDistributivity((分配率分配率))

ScalingScaling

( ) ( )[ ] ( )[ ] ( )[ ] 5)-(4.6 ,,,, 2121 yxfyxfyxfyxf ℑ+ℑ=+ℑ

( ) ( )[ ] ( )[ ] ( )[ ] 6)-(4.6 ,,,, 2121 yxfyxfyxfyxf ℑ⋅ℑ≠⋅ℑ

( ) ( ) 7)-(4.6 ,, vuaFyxaf ⇔

( ) 8)-(4.6 ,1,

⇔

bv

auF

abbyaxf




RotationRotation

( ) ( ) 9)-(4.6 ,, 00 θϕωθθ +⇔+ Frf

ϕωϕωθθ sin ,cos sin ,cos ==== vuryrxLet

( ) ( ) ( ) ( )ϕω,F r,θfu,vFx,yfThen and become and

36




PeriodicityPeriodicity

Conjugate symmetryConjugate symmetry

( ) ( ) ( ) ( )( ) ( ) ( ) ( ) 11)-(4.6

10)-(4.6 ,,,,NM,yxfNx,yfM,yxfx,yf

NvMuFNvuFvMuFvuF++=+=+=++=+=+=

( ) ( ) 12)-(4.6 ,*, vuFvuF −−=

( ) ( ) 13)-(4.6 ,, vuFvuF −−=




37




SeparabilitySeparability

( ) ( )

( ) 14)-(4.6 ,1

,11,

1

0

/2

1

0

/21

0

/2

∑

∑∑−

=

−

−

=

−−

=

−

=

=

M

x

Muxj

N

y

NvyjM

x

Muxj

evxFM

eyxfN

eM

vuF

π

ππ

( ) ( ) 15)-(4.6 ,1,1

0

/2∑−

=

−=N

y

NvyjeyxfN

vxF π




Computing the inverse Fourier TransformComputing the inverse Fourier Transformusing a Forward Transform Algorithmusing a Forward Transform Algorithm

38




More on PeriodicityMore on Periodicity




39




避免產生避免產生wraparound errorwraparound error在函式後方補零在函式後方補零

( ) ( )21)-(4.6

PxA 01-Ax0

≤≤≤≤

=xf

xfe

( ) ( )22)-(4.6

PxB 01-Bx0

≤≤≤≤

=xg

xge

且P≥A+B-1




22DD

40







41






4.6 Implementation4.6 ImplementationThe convolution and Correlation TheoremsThe convolution and Correlation Theorems

Convolution TheoremsConvolution Theorems

42




Correlation TheoremsCorrelation Theorems




43



4.6 Implementation4.6 ImplementationSummary of Properties of the 2D Fourier TransformSummary of Properties of the 2D Fourier Transform




44







45



4.6 Implementation4.6 ImplementationThe Fast Fourier TransformationThe Fast Fourier Transformation

Complexity of Fourier TransformationComplexity of Fourier TransformationO(NO(N22))




( ) ( ) 35)-(4.6 1 1

0∑−

=

=M

x

uxMWxf

MuF

36)-(4.6 /2 MjM eW π−=

38)-(4.6 237)-(4.6 2Let

K MM n

==

( ) ( )

( ) ( ) ( ) ( ) 39)-(4.6 12K12

K1

2121

1

0

1

0

122

22

1

0

++=

=

∑ ∑

∑−

=

−

=

+

−

=

k

x

k

x

xuK

xuK

M

x

uxM

WxfWxf

WxfK

uF

46



( ) ( )

( ) ( ) ( ) ( ) 39)-(4.6 12K12

K1

2121

1

0

1

0

122

22

1

0

++=

=

∑ ∑

∑−

=

−

=

+

−

=

k

x

k

x

xuK

xuK

M

x

uxM

WxfWxf

WxfK

uF


( ) ( )∑−

=

1

0

222

K1 k

x

xuKWxf ( ) ( )∑

−

=

++1

0

12212

K1 k

x

xuKWxf




( ) ( )

( ) ( ) ( ) 40)-(4.6 12K12

K1

21 1

0

1

02

/22/2222

++=∴

===

∑ ∑−

=

−

=

−−

k

x

k

x

uK

uxK

uxK

uxK

KuxjKuxjuxk

WWxfWxfuF

WeeW ππQ

( ) ( )

( ) ( ) ( ) ( ) 39)-(4.6 12K12

K1

2121

1

0

1

0

122

22

1

0

++=

=

∑ ∑

∑−

=

−

=

+

−

=

k

x

k

x

xuK

xuK

M

x

uxM

WxfWxf

WxfK

uF

47




( ) ( ) 41)-(4.6 2K1

1

0∑−

=

=k

x

uxKeven WxfuFDefining

( ) ( ) 42)-(4.6 12K1

1

0∑−

=

+=k

x

uxKodd WxfuFDefining

( ) ( ) ( )[ ] 43)-(4.6 21

2uKoddeven WuFuFuF +=




( ) ( ) ( )[ ]

( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( )

( ) ( )[ ] 44)-(4.6 21

12K12

K1

21

12K12

K1

21

12K12

K1

21

21

and

2

1

02

1

0

1

02

1

0

1

02

1

0

2

22

uKoddeven

K

x

uK

uxK

K

x

uxK

K

x

uK

uxK

K

x

uxK

K

x

kuk

xKuK

K

x

xKuK

kukoddeven

uM

MuM

uM

MuM

WuFuF

WWxfWxf

WWxfWxf

WWxfWxf

WKuFKuFKuF

WWWW

−=

+−=

−++=

++=

+++=+∴

−==

∑∑

∑∑

∑∑

−

=

−

=

−

=

−

=

−

=

++−

=

+

+

++Q 週期性

48




( ) ( ) ( )[ ]

( ) ( ) ( )[ ]

( ) ( ) ( )[ ]

( ) ( ) ( )[ ]

( ) ( ) ( )[ ]

( ) ( ) ( )[ ]12

12

02

12

12

02

11211

11211

00210

11211

11211

00210

−

−

−−−=+−

−=+

−=+

−+−=−

+=

+=

KKoddeven

Koddeven

Koddeven

KKoddeven

koddeven

koddeven

WKFKFKKF

WFFKF

WFFKF

WKFKFKF

WFFF

WFFF

L




( ) ( )( ) ( ) 46)-(4.6 1 212na

45)-(4.6 1 212 1

≥+−=

≥+−= −

nnannmnm

n

n

FFT所需要的乘法與加法計算次數

( )

( ) 48)-(4.6 logna

47)-(4.6 log21

2

2

MM

MMnm

=

=

n2M =其中

49




( )

( ) 50)-(4.6 22M

49)-(4.6 log

log

n

2

2

2

nnC

MM

MMMMC

n

=⇒=

=

=

Advantage of the FFT over Fourier Transform




50






• 頻域可完全掌控filter特性，非常適合在實驗階段，Filter在頻域設計完成後，通常會轉成對等的空間域filter，然後使用韌體或硬體的方式處理影像．

4.6 Implementation4.6 ImplementationSome comments on filter designSome comments on filter design

Documents

Chapter 4 Image Enhancement in the Frequency DomainCSIE/ycliaw/DIP/04_Enhancement_Frequency.pdf · Enhancement methods Spatial Domain (in chapter 3) based on direct manipulation of