4-Image Enhancement in the Frequency Domain

7/15/2019 4-Image Enhancement in the Frequency Domain

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School of Info-Physics and Geomatics Engineering, CSU1 2013/6/28

Chapter4 Image Enhancement in theFrequency Domain

Enhance: To make greater (as in value, desirability, orattractiveness.

Frequency: The number of times that a periodic

function repeats the same sequence of values during

a unit variation of the independent variable.Websters New Collegiate Dictionary


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Content

Background

Introduction to the Fourier Transform and the

Frequency Domain Smoothing Frequency-Domain Filters

Sharpening Frequency-Domain Filters

Homomorphic Filtering Implementation

Summary

This chapter is concerned primarily with helping the reader develop a

basic understanding of the Fourier transform and the frequency

domain, and how they apply to image enhancement.


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4.1 Background

Any function that periodically repeats itself can be

expressed as the sum of sines and/or cosines of differentfrequencies, each multiplied by a different coefficient

(Fourier series).

Even functions that are not periodic (but whose area

under the curve is finite) can be expressed as the integralof sines and/or cosines multiplied by a weighting function

(Fourier transform).

The advent of digital computation and the discovery of

fast Fourier Transform (FFT) algorithm in the late 1950srevolutionized the field of signal processing, and allowed

for the first time practical processing and meaningful

interpretation of a host of signals of exceptional human

and industrial importance.


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The frequency domain

refers to the plane ofthe two dimensional

discrete Fourier

transform of an image.

The purpose of theFourier transform is to

represent a signal as a

linear combination of

sinusoidal signals of

various frequencies.


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4.2 Introduction to the Fourier Transform

and the Frequency Domain The one-dimensional Fourier transform and its inverse

Fourier transform (continuous case)

Inverse Fourier transform:

The two-dimensional Fourier transform and its inverse

Fourier transform (continuous case)

Inverse Fourier transform:

dueuFxfuxj 2)()(

dydxeyxfvuF

vyuxj )(2),(),(

1where)()( 2

jdxexfuF uxj

dvduevuFyxf vyuxj )(2),(),(

sincos je j


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4.2.1The one-dimensional Fourier transform and

its inverse (discrete time case)

Fourier transform (DFT)

Inverse Fourier transform (IDFT)

The 1/Mmultiplier in front of the Fourier transform sometimes

is placed in the front of the inverse instead. Other times bothequations are multiplied by

Unlike continuous case, the discrete Fourier transform and its

inverse always exist, only iff(x) is finite duration.

1,...,2,1,0for)(1

)(1

0

/2

MuexfM

uFM

x

Muxj

1,...,2,1,0for)()(1

0

/2

MxeuFxfM

u

Muxj

1/ M


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Since and the fact

then discrete Fourier transform can be redefined

Frequency (time) domain: the domain (values ofu) over

which the values ofF(u) range; because u determines

the frequency of the components of the transform.

Frequency (time) component: each of the Mterms of

F(u).

1

0

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

for 0,1, 2,..., 1

M

x

F u f x ux M j ux MM

u M

sincos je j cos)cos(


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F(u) can be expressed in polar coordinates:

R(u): the real part ofF(u)

I(u): the imaginary part ofF(u)

Power spectrum:

)()()()( 222

uIuRuFuP

( )

1/22 2

1

( ) ( )

where ( ) ( ) ( ) (magnitude or spectrum)

( )( ) tan (phase angle or phase spectrum)( )

j uF u F u e

F u R u I u

I uu R u


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Some One-Dimensional Fourier Transform Examples

Please note the relationship between the value of K and the height of

the spectrum and the number of zeros in the frequency domain.


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The transform of a constant function is a DC value only.

The transform of a delta function is a constant.


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The transform of an infinite train of delta functions

spaced by T is an infinite train of delta functions spacedby 1/T.

The transform of a cosine function is a positive delta at

the appropriate positive and negative frequency.


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The transform of a sin function is a negative complex

delta function at the appropriate positive frequency and a

negative complex delta at the appropriate negative

frequency.

The transform of a square pulse is a sinc function.


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4.2.2 The two-dimensional Fourier transform and

its inverse (discrete time case)

Fourier transform (DFT)

Inverse Fourier transform (IDFT)

u, v: the transform or frequency variables

x, y: the spatial or image variables

1,...,2,1,0,1,...,2,1,0for

),(1

),(1

0

1

0

)//(2

NvMu

eyxfMN

vuFM

x

N

y

NvyMuxj

1,...,2,1,0,1,...,2,1,0for

),(),(1

0

1

0

)//(2

NyMx

evuFyxfM

u

N

v

NvyMuxj


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We define the Fourier spectrum, phase anble, and power

spectrum of the two-dimensional Fourier transform as

follows:

R(u,v): the real part ofF(u,v) I(u,v): the imaginary part ofF(u,v)

spectrum)(power),(),(),()(

angle)(phase),(

),(

tan),(

spectrum)(),(),(),(

222

1

2

122

vuIvuRvuFu,vP

vuR

vuI

vu

vuIvuRvuF


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Some properties of Fourier transform:

)(symmetric),(),(

symmetric)(conujgate),(*),(

(average)),(

1

)0,0(

(shift))2

,2

()1)(,(

1

0

1

0

vuFvuF

vuFvuF

yxfMNF

Nv

MuFyxf

M

x

N

y

yx


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(a) f(x,y) (b) F(u,y) (c) F(u,v)

The 2D DFT F(u,v) can be obtained by

1. taking the 1D DFT of every row of image f(x,y), F(u,y),

2. taking the 1D DFT of every column ofF(u,y)

Steps and some example of two-dimensional DFT

y or v

x or u

Convention of coordination:


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shift

Consider the relationship between the separation of zeros in u- or v- direction

and the width or height of white block of source image.


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Shape of three dimensional spectrum


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DFT

DFT

The Property of Two-Dimensional DFT Rotation


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DFT

DFT

DFT

A

B

0.25 * A+ 0.75 * B

The Property of Two-Dimensional DFT Linear Combination


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DFT

DFT

A

Expanding the original image by a factor of n (n=2), filling

the empty new values with zeros, results in the same DFT.

B

The Property of Two-Dimensional DFT Expansion


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Two-Dimensional DFT with Different Functions

Sine wave

Rectangle

Its DFT

Its DFT


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Two-Dimensional DFT with Different Functions

2D Gaussian

function

Impulses

Its DFT

Its DFT


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4.2.3 Filtering in the Frequency DomainFrequency is directly related to rate of change. The frequency of fast

varying components in an image is higher than slowly varying components.


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Basics of Filtering in the Frequency Domain

Including multiplication the input/output image by (-1)x+y.

What is zero-phase-shift filter?


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Some Basic Filters and Their Functions

Multiply all values ofF(u,v) by the filter function (notch filter):

All this filter would do is set F(0,0) to zero (force the average value of

an image to zero) and leave all other frequency components of the

Fourier transform untouched and make prominent edges stand out

otherwise.1

)2/,2/(),(if0),(

NMvuvuH


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Lowpass filter

Highpass filter

Circular symmetry


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Low frequency filters: eliminate the gray-level detail and keep the generalgray-level appearance. (blurring the image)

Low frequency filters: have less gray-level variations in smooth areas and

emphasized transitional (e.g., edge and noise) gray-level detail. (sharpening

images)


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4.2.4 Correspondence between Filtering in the

Spatial and Frequency Domain

Convolution theorem: The discrete convolution of two functions f(x,y) and h(x,y) of

size MNis defined as

The process of implementation:

1) Flipping one function about the origin;

2) Shifting that function with respect to the other by changing the

values of (x, y);

3) Computing a sum of products over all values ofm and n, for

each displacement.

1 1

0 01 1

0 0

1( , ) ( , ) ( , ) ( , )

1( , ) ( , )

M N

m nM N

m n

f x y h x y f m n h x m y nMN

h m n f x m y nMN


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),(),(),(),( vuHvuFyxhyxf

),(),(),(),( vuHvuFyxhyxf Eq. (4.2-32)

Let F(u,v) and H(u,v) denote the Fourier transforms off(x,y) and

h(x,y), then

Eq. (4.2-31)

an impulse function of strengthA, located at coordinates

(x0,y0): and is defined by

:

where : a unit impulse located at the origin

The Fourier transform ofa unit impulse at the origin (Eq4.2-35) :

1

0

1

0

0000 ),(),(),(M

x

N

y

yxAsyyxxAyxs

),( 00 yyxxA

1

0

1

0

)0,0(),(),(M

x

N

y

syxyxs

1

0

1

0

)//(2 1),(1

),(M

x

N

y

NvyMuxj

MNeyx

MNvuF

),( yx

The shifting property of impulse

function


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Let , then the convolution (Eq. (4.2-36))

Combine Eqs. (4.2-35) (4.2-36) with Eq. (4.2-31), weobtain:

),(1

),(),(1),(),(1

0

1

0

yxhMN

nymxhnmMN

yxhyxfM

m

N

n

),(),( yxyxf

),(),(

),(

1

),(

1

),(),(),(),(

),(),(),(),(

vuHyxh

vuHMNyxhMN

vuHyxyxhyx

vuHvuFyxhyxf

That is to say, the response of

impulse input is the transfer

function of filter.


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The distinction and links between spatial and frequency filtering

If the size of spatial and frequency filters is same, then thecomputation burden in spatial domain is larger than in frequency

domain;

However, whenever possible, it makes more sense to filter in the

spatial domain using small filter masks.

Filtering in frequency is more intuitive. We can specify filters in the

frequency, take their inverse transform, and the use the resulting

filter in spatial domain as a guide for constructing smaller spatial

filter masks.

Fourier transform and its inverse are linear process, so thefollowing discussion is limited to linear processes.


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Let H(u) denote a frequency domain, Gaussian filter

function given the equation

where : the standard deviation of the Gaussian curve.

The corresponding filter in the spatial domain is

222

22)( xAexh

22 2/)( uAeuH

There is two reasons that filters based on Gaussian functions are of

particular importance: 1) their shapes are easily specified; 2) both

the forward and inverse Fourier transforms of a Gaussian are real

Gaussian function.


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2 2 2 21 2/ 2 /2

1 2( ) , ( , )u uH u Ae Be A B

The corresponding filter in the spatial domain is

2 2 2 2 2 21 22 2

1 2( ) 2 2x xh x Ae Be

We can note that the value of this types of filter has both negative and

positive values. Once the values turn negative, they never turn positive again.

Filtering in frequency domain is usually used for the guides to design the

filter masks in the spatial domain.


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One very useful property of the Gaussian function is that both it and

its Fourier transform are real valued; there are no complex valuesassociated with them.

In addition, the values are always positive. So, if we convolve animage with a Gaussian function, there will never be any negativeoutput values to deal with.

There is also an important relationship between the widths of aGaussian function and its Fourier transform. If we make the widthof the function smaller, the width of the Fourier transform gets larger.This is controlled by the variance parameter2 in the equations.

These properties make the Gaussian filter very useful forlowpassfiltering an image. The amount of blur is controlled by 2. It can be

implemented in either the spatial or frequency domain. Other filters besides lowpass can also be implemented by using two

different sized Gaussian functions.

Some important properties of Gaussian filters funtions


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4.2 Smoothing Frequency-Domain Filters

The basic model for filtering in the frequency domain

where F(u,v): the Fourier transform of the image to besmoothed

H(u,v): a filter transfer function

Smoothing is fundamentally a lowpass operation in thefrequency domain.

There are several standard forms of lowpass filters (LPF).

Ideal lowpass filter Butterworth lowpass filter

Gaussian lowpass filter

),(),(),( vuFvuHvuG


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Ideal Lowpass Filters (ILPFs)

The simplest lowpass filter is a filter that cuts off all

high-frequency components of the Fourier transform that

are at a distance greater than a specified distance D0

from the origin of the transform.

The transfer function of an ideal lowpass filter

where D(u,v) : the distance from point (u,v) to the center

of ther frequency rectangle (M/2, N/2)

21

22 )2/()2/(),( NvMuvuD

),(if0

),(if1),(

0

0

DvuD

DvuDvuH


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cutoff frequency

ILPF is a type of nonphysical filters and cant be realized with electronic

components and is not very practical.


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The blurring and ringing

phenomena can be seen, inwhich ringing behavior is

characteristic of ideal filters.

A th l f


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Another example ofILPF

Figure 4.13 (a) A frequency-domain

ILPF of radius 5. (b) Corresponding

spatial filter. (c) Five impulses in thespatial domain, simulating the values

of five pixels. (d) Convolution of (b)

and (c) in the spatial domain.frequency

spatial

spatial

spatial

),(),(),(),( vuHvuFyxhyxf

Notation: the radius of center

component and the number of

circles per unit distance from

the origin are inversely

proportional to the value of thecutoff frequency.

diagonal scan line of (d)


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nDvuDvuH

2

0/),(1

1),(

Butterworth Lowpass Filters (BLPFs) with ordern

Note the relationshipbetween ordern and

smoothing

The BLPF may be viewed as a transition between ILPF AND GLPF, BLPF of

order 2 is a good compromise between effective lowpass filtering and

acceptable ringing characteristics.


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Butterworth LowpassFilters (BLPFs)

n=2

D0=5,15,30,80,and 230


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Butterworth Lowpass Filters (BLPFs)Spatial Representation

n=1 n=2 n=5 n=20

l ( )


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Gaussian Lowpass Filters (FLPFs)

20

2 2/),(),( DvuDevuH


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Gaussian LowpassFilters (FLPFs)

D0=5,15,30,80,and 230


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Additional Examples of Lowpass Filtering

Character recognition in machine perception: join the broken character

segments with a Gaussian lowpass filter with D0=80.


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Application in cosmetic processing and produce a smoother,

softer-looking result from a sharp original.


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Gaussian lowpass filter for reducing the horizontal sensor scan lines and

simplifying the detection of features like the interface boundaries.

4 4 Sh i F D i Fil


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( , ) 1 ( , )hp lpH u v H u v

Ideal highpass filter

Butterworth highpass filter

Gaussian highpass filter

),(if1

),(if0),(

0

0

DvuD

DvuDvuH

nvuDDvuH

2

0 ),(/1

1),(

20

2 2/),(1),( DvuDevuH

4.4 Sharpening Frequency Domain Filter


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Highpass Filters Spatial Representations

Id l Hi h Filt (IHPF )


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Ideal Highpass Filters (IHPFs)

),(if1

),(if0),(

0

0

DvuD

DvuDvuH

non-physically realizable with electronic component and have the

same ringing properties as ILPFs.

Butterworth Highpass Filters


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Butterworth Highpass Filters

nvuDDvuH 2

0 ),(/11),(

The result is smoother than that of IHPFs and sharper than that of GHPFs

h l


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Gaussian Highpass Filters

2

0

2

2/),(1),( DvuDevuH

The result is the smoothest in three types of high-pass filters

The Laplacian in the Frequency Domain


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The Laplacian in the Frequency Domain

)(),( 22 vuvuH

2 2

( , ) ( / 2) ( / 2)H u v u M v N

The FT of n-order differential of a function f(x) is

( ) / ( ) ( )n n nd f x dx ju F u

2 22 2 2 2 2

2 2

( , ) ( , )[ ( , )] [ ] ( ) ( , ) ( ) ( , ) ( ) ( , )

f x y f x yf x y ju F u v jv F u v u v F u v

x y

For a two-dimensional function f(x,y), it can be shown that

So, Laplacian can be implemented in the frequency domain by using the filter

Shift the center to (M/2, N/2) and obtain

We have the following Fourier transform pairs

2 2 2( , ) ( / 2) ( / 2) ( , )f x y u M v N F u v

The plot of Laplacian in frequency and spatial domain


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Frequency

domain

Spatial domain

The plot of Laplacian in frequency and spatial domain

2 1 2 2( ) ( ) ( ) 1 (( ) ( ) ) ( )M N

g x y f x y f x y u v F u v


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( , ) ( , ) ( , ) 1 (( ) ( ) ) ( , )2 2

g x y f x y f x y u v F u v

For display

purposes only

A integrated operation

in frequency domain


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Unsharp masking, high-boost filtering, and high-frequency emphasis

filtering (refers to page187-191)

4 5 Homomorphic filtering


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4.5 Homomorphic filteringProblems:

When the illumination radiating to an object is non-

uniform, the detail of the dark part in the image ismore discernable.

aims:Simultaneously compress the gray-level range and

enhance contrast, eliminate the effect of non-uniform

illumination, and emphasis the details.Principal:

Generally, the illumination component of an image is

characterized by slow spatial variations, while the

reflectance components tends to vary abruptly,

particularly at the junctions of dissimilar objects.These characteristics lead to associating the low

frequencies of the Fourier transform of the logarithm

of an image with illumination and the high

frequencies with reflectance.


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),(),(),( yxryxiyxf

The illumination-reflectance model of an image

),( yxi ),( yxrIllumination coefficient: reflectance coefficient:

Steps: ),(ln),(ln),(ln),( yxryxiyxfyxz

)],([ln)],([ln)],([ yxryxiyxz FFF ),(),(),( vuRvuIvuZ

),(),(),(),(),( vuRvuHvuIvuHvuS

2)

1)

3) Determine the H(u, v), which must compress the dynamic

range ofi(x,y), and enhance the contrast ofr(x,y) component.

Th f ll i f ti t th b i


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rL1

2 20( ( , ) / )( , ) ( )[1 ]c D u v DH L LH u v e

The following function meet the above requires

The curve shape shown in above figure can be approximated using basic formof the ideal highpass filters, for example, using a slightly modified form of the

Gaussian highpass filter and can obtain

Steps: 4)


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)],(),([),( 1' vuIvuHyxi F

)],(),([),( 1' vuRvuHyxr F

)],(exp[),(

'

0 yxiyxi

)],(exp[),( '0 yxryxr

),(),(),( 00 yxryxiyxg

Steps: 4)

5)


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ln FFT H(u,v)

FFT-1 exp

f(x,y)

g(x,y)

The flow-chart ofHomomorphic filtering

T l


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Two examples


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4.6 Implementation


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4.6 Implementation4.6.1Some Additional Properties of the 2D Fourier Transform

, distributivity, scaling, and :

0 0

0 0

2 ( / / )0 0

2 ( / / )

0 0

( , ) ( , )

( , ) ( , )

j ux M vy N

j xu M yu N

f x x y y F u v e

f x y e F u u v v

1 2 1 2[ ( , ) ( , )] [ ( , )] [ ( , )]1

( , ) ( / , / )

f x y f x y f x y f x y

f ax by F u a v bab

0 0

( , ) ( , )f r F w

Translation

Distributivity andscaling

rotation

Periodicity, conjugate symmetry, and back-to-back properties


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y j g y y

shift

( , ) ( , ) ( , ) ( , )

( , ) ( , ) ( , ) ( , )

( , ) ( , )

F u v F u M v F u v N F u M v N

f u v f u M v f u v N f u M v N

F u v F u v

Separability


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Separability

1 1 12 / 2 / 2 /

0 0 0

1 1 1( , ) ( , ) ( , )

M N Mj ux M j vy N j ux M

x y x

F u v e f x y e F x v eM N M

A similar process can be applied to computing the 2-D inverse Fourier

transform.

4 6 2 computing the inverse FF using a forward


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4.6.2 computing the inverse FF using a forward

transform algorithmRepeat the one-dimensional inverse FF:

1 2 /

0

( ) ( )M j ux M

u

f x F u e

Take the complex conjugate of two side and multiply M

12 /

0

1 1( ) ( )

Mj ux M

u

f x F u eM M

Which is the form of forward FF. Take the complex conjugate of the result of

the above equation and will get the inverse FF by forward transform.

For two-dimensional case, similarly have

1 12 ( / / )

0 0

1 1( , ) ( , )M N

j ux M vy N

u v

f x y F u v eMN MN

Here, we can treat F(u,v) as a simple function presenting on the forward

transform equation

4.6.3 More on Periodicity: the need for padding


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y p g

1

0

( ) ( )

1( ) ( )

M

m

f x h x

f m h x mM

Convolution process

Aliasing or wraparound error


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extend

extend

The methods of solving the

aliasing problem is to

extend and pad.

( ) 0 1( )

0 1

( ) 0 1

( ) 0 1

where 1

e

e

f x x Af x

A x P

h x x B

h x B x P

P A+ B

For two-dimensional case


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An example


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*

=

4 6 4 convolution and correlation theorems


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4.6.4 convolution and correlation theorems

1 1

0 0

1( , ) ( , ) ( , ) ( , )

M N

m n

f x y h x y f m n h x m y n

MN

1 1

0 0

1( , ) ( , ) ( , ) ( , )

M N

m n

f x y h x y f m n h x m y nMN

Except for the complex offand h not mirrored about the origin, everythingelse in the implementation of correlation is identical to convolution,

including the need for padding.

convolution

correlation

( , ) ( , ) ( , ) ( , )

( , ) ( , ) ( , ) ( , )

f x y h x y F u v H u v

f x y h x y F u v H u v

Correlation theorem

Correlation includes across- and auto-correlation, and its main use is for

matching and sure the location where h (template) finds a correspondence

in f.


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4.6.5 Summary of Some Important Properties


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of the 2-D Fourier Transform


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Summary


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y

The basic concept about Fourier Transform (FT)

The algorithm of FT and the process of frequency filtering

The physical meaning related to FT

The relationship of resolution between spatial and frequency

domain

Correspondence between filtering in the spatial and frequencydomains

The general type of smoothing and sharpening filters and their

main features

Homomorphic filtering Some important properties of FT

Convolution and correlation theorems

Spatial and frequency filtering are both highly subjective processes

Documents

4-Image Enhancement in the Frequency Domain