DIP Chapter 2 Part2 Dft Dct Conv

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Chapter 2 :Enhancement

Part 2 : Fourier Transform & Filters

Dr. Hojeij youssef

Digital image processing

1

Chapter 2 : Part 2

Fourierandimage:

Fourier transformFT

AnalogFT :

AnalogFT 1Dcase

AnalogFT 2Dcase

Digital FT:

Digital FT 1Dcase

Digital FT 2Dcase

Propertiesof DFT

DFT applicationsDiscretecosinetransformation(DCT)

Maskandconvolution

Dr. Hojeij youssef2


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Fourier and image

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What is a frequency in an image ? Low frequencies : homogeneous, regions, blurHigh frequency : Edges, sudden change of intensity, noise

High Frequency

Low frequency

Rem: The largest energy of the any picture is located in the low frequencies

Fourier

T, T-1 exist

Good properties in transformdomain : using the transformdomain is a

better way to solve problemsFind bases : exp, sin, cos, wavelets (choose your family of functions)

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Spatial domain Transformdomain


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Fourier

Function function

FT is an easy tool. It was sometimes difficult to compute, calculate the coefficients.Now it s easy with computers.

Gives properties of spectrum: frequencies.

Ex : If large number of low frequencies means smoothed signal, low varying signal.

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Used to filterSysteminterpretation :spatial domainconvolution

frequency domainmultiplication

FT : analog1D case

Let f(x)be a continuous function of a real variable x. The Fouriertransformisdefinedby:

GivenF(u), f(x)can be obtained using the inverse Fourier transform:

Conditions :

- f(x)continuous & integrable

- F(u)integrable

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FT analog1D case

If we are concerned with real valued functionsf(x)

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Fourier or frequency domain

complex

Direct domain

real

-| F(u) |Fourier spectrum (magnitude function)-| F(u)| Power spectrum(spectral density)

-(u)phase angle

-ufrequency

FT : example

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FT analog 2D case

TheFT caneasilyextendedtoafunction f(x,y)of 2variables:

GivenF(u,v), f(x,y)canbeobtainedusingtheinverseFourier transform:

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Conditions :-f(x,y)continuous & integrable

-F(u,v)integrable

FT analog 2D Case

If we are concerned with real valued functionsf(x,y)

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-| F(u,v) |Fourier spectrum (magnitude function)-| F(u,v)| Power spectrum(spectral density)

-(u,v)phase angle

-u,vfrequency variable

Direct domain

real

Fourier or frequency domain

complex


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FT 2D Example


FT 2D Example



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FT 2D Example


DiscreteFourier transform (DFT)

Let f(x)isdiscretizedintoasequencebytakingN samplesxunitsapart :

If we set f(x)=f(x0+ xx)where x now assumes discrete values :0,1,2,3,,N-1

- This sequence denotes any uniform spaced samples froma correspondingcontinuous

- RememberShannonstheorembeforesamplingasignal. fe>2fmaxDr. Hojeij youssef

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DFT : 1D case

The DFT is defined by :

Rem:

- the values of u=1,2,,N -1in the DFT correspond to samples of the

continuoustransformsatvalues0, u, 2u,, (N-1)u

- F(u)representsF(uu)


DFT : 2D case

TheDFT 2D isdefinedby:


Rem:-The sampling is on a 2D grid

- Sampling increments in spatial and frequency domain


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DFT : 2D case

Normalizationcoefficient:


Properties of DFT

Most 2D DFT proprietiesarestraightforwardextend fromthe1 D DFTproperties

1.Separability:

The2D Fourier transformcanbeperformedasseriesof 1D DFT(complexexponential isseparable)

Rem:performthe1D DFTonyvariable(axis) first F(x,v), andthenperformthe1DDFTonxvariable F(u,v)



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Properties of DFT

2.Scaling:

Foranyconstantsaandb:

3.Shifting:

Shiftingthefunction f(x,y)resultsinaphaseshift intheFT :

4.Modulation:

The complement of the previous property-multiplying by a complex exponential. iemodulationresultsinashiftof theFT

5.Convolution:

Convolutionof twofunctioncorrespondsto amultiplicationof their FTs


)bv,au(Fab

1

)by,ax(f

)v,u(Fe)yy,xx(f)vyux(2j

0000

)vv,uu(F)y,x(fe 00)yvxu(2j 00

)v,u(H)v,u(H)v,u(G

dydx)y,x(f)yy,xx(h)y,x(h)y,x(f)y,x(g ''''''

Properties of DFT

6. Multiplication:

MultiplyingtwofunctioncorrespondstoconvolvingtheirsFts:

7.Rotation:

Usingthepolarcoordinates:

Rotatinganimagef(r,)byanangle rotatestheFT F(u,v)bythesameangle

8. Averagevalue:

Tofindaveragenumberscaleof anMNimage f(x,y)


)v,u(H)v,u(F

)y,x(h)y,x(f)y,x(g

sinwv&coswu

sinry&cosrx

),w(F),r(f 00 0

)0,0(FMN

1)y,x(f

MN

1)y,x(f

1M

1M

1N

1N


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Properties of DFT


Properties of DFT

9.Periodicity:

Moving the zero-frequency component to the center of the array. It is useful forvisualizingaFourier transformwith thezero-frequencycomponent in themiddleof thespectrum.


)n,mM(f)n,m(f

)vN,u(F)v,u(F

)nN,m(f)n,m(f

)v,uM(F)v,u(F

)v,u(F)vbN,uaM(F )n,m(f)nbN,maM(f


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Examples of DFT


Examples of DFT



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DFT applications


Filtering:

Spacefilteringisdonebyconvolution. In thefield spectral (frequency), it isdonebymultiplication(hidingtheimage).

original image Image transformed

Filtered imageImage transformed

Filtered

Spectral filtering

(multiplication)

Filtering Spatial

(convolution)

DFT applications


spectral

filtering


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DFT applications


Information percentage of the image included in the circles (smallest to largest) :90% 95% 98% 99% 99.5% 99.9%

DFT applications


High-passfilter : Low-passfilter :

Removes the high frequencies by putting thepixelsawayfromthecentertozero

Removes thebass frequencies by putting thepixels in thecenter to zero


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DFT applications


De-noisingimage:

Noisy image

Fourier spectrum Filtered image

DFT applications


Contrastmodification(Enhancement):

Original image

High-pass filtering

Imageenhanced

Filtered image(High Pass)Original image

Imageenhanced Enhanced +Equalization Histogram


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DCT isdefinedby:

DCT(Discrete Cosine transformation)


Example DCT



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Example DCT


Example DCT



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Numerical convolution


The discrete convolution is a tool to use filters or linear filter travel

invariants .The general equation of convolution, notedg(x), original function f(x)with a functionh(x)is :

f(x)is the original function and g(x) is the convoluted function (resultof convolution).

In our case, an image is seen as a mathematical function

h(x)is calledconvolution mask, convolution kernel, filter, kernel, ...

k )k(f)kx(h)x(h)x(f)x(g



In practice, the convolution of a digital image made by a summonsmultiplication

A convolution filter is usually a matrix (image), his size is not always oddand symmetrical

3x3, 5x5, 7x7, ...

Convolution of an image through a filter (kernel) 2D :

u v

'

'

)v,u(filter)vj,ui(I)j,i(I

)j,i(filtre)j,i(I)j,i(I


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Kernel

Image

Result of

convolution, I by

the kernel K



Image

Solution : No miraclesolution1. edges(0)2. Mirror effect : f (-x, y) = f (x, y)

Problem: What to do with theedges of theimage?

DIP Chapter 2 Part2 Dft Dct Conv

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