[PDF] [PDF] Discrete Fourier Transform - WPI Computer Science (CS) Department

[PDF] Lecture 7 - The Discrete Fourier Transform

Figure 7 1: (a) Sequence of 8Д Е samples (b) implicit periodicity in DFT Since the operation treats the data as if it were periodic, we evaluate the DFT equation  

12 Discrete Fourier transform

17 nov 2006 · Fourier transform (CTFT) The formal definition of the DFT is presented in Section 12 2, including its matrix-vector representation Section 12 3 

[PDF] The Discrete Fourier Transform - Eecs Umich

This definition is the most important one since our primary use of the DFT is for Example Find the 8-point DFT of the signal x[n] = 6 cos2(π 4n) Expanding: 

[PDF] FFT Examples Lets calculate the Discrete Fourier Transform (DFT

(Remember, the FFT is just a fast algorithm for computing the DFT, it is not a transform, just a method for computing the DFT efficiently) 2 = −1 P(x)=1+2x + 3x2 + 4x3 + 5x4 + 6x5 + 7x6 + 8x7 =(1+3x2 + 5x4 + 7x6) + x(2 + 4x2 + 6x4 + 8x6) =(1+3y + 5y2 + 7y3) + x(2 + 4y + 6y2 + 8y3)

[PDF] Discrete Fourier Transform (DFT)

Sample the spectrum X(ω) in frequency so that X(k) =⇒ the DFT spectrum is periodic with period N (which is expected, Example: DFT of a rectangular pulse :

[PDF] Chapter 1 Discrete Fourier Transform - Physics

Figure 1 1: Example of a periodic function with the period of 10 A more mathematically solid definition of a function that can be transformed is: any periodic

[PDF] Discrete Fourier Transform (DFT)

2D Discrete Fourier Transform • Definition Assuming f(m n) m = 0 1 Direct computation of N-point DFT takes N2 Calculate Linear Convolution Using DFT

[PDF] Discrete Fourier Transform - WPI Computer Science (CS) Department

Definition of 1D DFT ○ Suppose is a sequence of length N ○ Interpretation: Basis functions completes x cycles over distance N ○ Similar to Fourier series 

[PDF] Discrete Fourier Series & Discrete Fourier Transform - CityU EE

cannot use a digital computer to calculate a continuum of sample points of are considered According to Example 7 2 and the relationship between DFT

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DigitalImageProcessing(CS/ECE545)

Lecture10:DiscreteFourierTransform

(DFT)

ProfEmmanuelAgu

ComputerScienceDept.

WorcesterPolytechnicInstitute(WPI)

FourierTransform

summationofsinesandcosines

Complex function

Sine function 1Sine function 2Sine function 3

Complex function expressed

as sum of sines

FourierTransform:Why?

effectoncomplexsignal

FourierTransform:SomeObservations

Observation 1: The

sines have different frequencies (not same)Observation 2: Frequencies of sines are multiples of each other (called harmonics)

Frequency = 1x

Frequency = 2x

Frequency = 4x

Observation 3: Different amounts of

different sines added together (e.g. 1/3, 1/5, etc)

FourierTransform:AnotherExampleSquare wave

Approximation

Using sines

Observation 4: The sine terms go to infinity.

The more sines we add, the closer the

approximation of the original.

WhoisFourier?

Frenchmathematicianand

physicist

Lived1768Ͳ1830

FourierSeriesExpansion

Iff(x)isperiodicfunctionofperiod2T

Fourierseriesexpansion

Where a n andb n calledFouriercoefficients

ComplexFormofFourierSeriesExpansion

Fourierseriesexpansionoff(x)

canbeexpressedincomplexformas: where

FourierSeriesofPeriodicFunctions

frequencyʘ 0 canbedescribedasasumofsinusoids

ThisinfinitesumiscalledaFourierSeries

frequency (harmonics) A k andB k calledFouriercoefficients

Fourieranalysis

Infinite

sum ofCosines Sines

FourierIntegral

(integrationofdenselypackedsines andcosines wherecoefficientscanbefoundas

FourierTransform

spectrumG(ʘ)

1DDiscreteFourierTransform

Imageisadiscrete2Dfunction!!

Fordiscretefunctionsweneed

onlyfinitenumberoffunctions

Forexample,considerthediscrete

sequence

1,1,1,1,Ͳ1,Ͳ1,Ͳ1,Ͳ1

Aboveisdiscreteapproximation

tosquarewave

CanuseFouriertransformto

expressassumof2sinefunctions

Definitionof1DDFT

Suppose

isasequenceoflengthN

SimilartoFourierseriesexpansion

Insteadofintegral,wenowhaveafinitesum

Compare with complex form of coefficients

Inverse1DDFT

Formulaforinverse

ComparedtoDFTequation,

Theinversehasnoscalingfactor1/N

DFT equation

FastFourierTransform(FFT)

ManywaystocomputeDFTquickly

way

OneFFTcomputationmethod

Dividesoriginalvectorinto2

CalculatesFFTofeachhalfrecursively

Mergesresults

FFTComputationTimeSavings

2DDFT sines andcosines expressedasasumofasines andcosinesalong2dimensions

2DFourierTransform

distanceofMunits distanceofNunits

2DFourierTransform

ForMxNmatrix,forwardandinversefourier transformscan bewritten where

2DCosinesfunctions

Orientation depend on m and n

2DFourierTransform:

CorrugationofFunctions

Previousimagejustsummedcosines

bysumming sinesandcosinesin2direction=corrugations Corrugations result when sines and cosines are summed in 2 directions

Propertiesof2DFourierTransform

properties scalefactor1/MNininversetransform 2.

Negativesigninexponentofforwardtransform

Propertiesof2DFourierTransform

independentoffandF)

Separability

expressedasproducts 1DDFT

2D DFT1D DFT (row)1D DFT (column)

Properties:Separabiltyof2DDFT

rowsthencolumnsof2DFourierTransform

Implementationof2DDFT

DFTsonrowsandcolumns

Propertiesof2DDFT

theindividualDFT's expressedasasum(e.g.noise)

Wecanfindfourier transformas:

k is a scalar

ConvolutionusingDFT

withspatialfilterS1.

PadStomakeitsamesizeasM,yieldingS'

2.

FormDFTsofbothMandS'

3.

MultiplyMandS'elementbyelement

1.

Takeinversetransformofresult

Essentially

OrequivalentlytheconvolutionM*S

ConvolutionusingDFT

LargespeedupsifSislarge

Example:M=512x512,S=32x32

Directcomputation:

32
2 =1024multiplicationsforeachpixel multiplications

UsingDFT:

Eachrowrequires4608multiplications

NeedsameforDFToffilterandforinverseDFT.

DCComponent

Recallthat:

Ifweputu=v=0,then

CenteringDFTSpectrum

F(0,0)attopleftcorner

JustswapfourquadrantsofFouriertransform

Swap 4 quadrants to

center DC component

DFT spectrum after

centering

CenteringDFTSpectrum

Original ImageNon-centered

spectrumCentered spectrum

ConjugateSymmetry

DFTshowsconjugatesymmetry

Otherhalfisredundant

ConjugateSymmetry

Then foranyintegerspandq

DisplayingTransforms

directly ofthetransform

ExamplesofDFTs

DFTofImage

ConsiderDFTofimagewithsingleedge

Fordisplay,DCcomponentshiftedtocenter

Image DFT

DFTExample:ABox

Box DFT

DFTExample:RotatedBox

Rotated BoxDFT

DFTExample:ACircle

Note:Ringingcausedbysharpcutoffofcircle

CircleDFT

DFTComputation:RepeatedImage

vertically middleofspectrumaftershifting

Windowing

transitions

SomeProposed

Windowing

Functions

SomeProposed

Windowing

Functions

SomeProposed

Windowing

Functions

Applicationof

Windowing

Functions

FilteringinFrequencyDomain

duetoconvolutiontheorem matrix"

IdealLowPassFiltering

Lowpassfilteringcausesblurring

aretowardscenter

Mayspecifyfrequencycutoffascirclec

IdealLowPassFiltering

IdealLowPassFiltering

productofFandm

ImageDFT

IdealLowPassFiltering

Applying

low pass filter to DFT

Cutoff D = 15Image after

inversion low pass filter

Cutoff D = 5low pass filter

Cutoff D = 30

Note: Sharp filter

Cutoff causes

ringing

IdealHighPassFiltering

frequencyvalues),keepingothers

Highpassfilteringcausesimagesharpening

Largecutoff=Moreinformationremoved

DFT of Image after

high pass FilteringResulting image after inverse DFT

IdealHighPassFiltering:EffectofCutoffs

Low cutoff

frequency removes

Only few lowest

frequenciesHigh pass filtering of DFT with filter

Cutoff D = 5

High pass filtering

of DFT with filter

Cutoff D = 30High cutoff

frequency removes many frequencies, leaving only edges

HighpassFilteringExample

Original image

Highpass filtering result

High frequency

emphasis result

After histogram

equalisation Images taken from Gonzalez & Woods, Digital Image Processing (2002)

ButterworthFiltering

Sharpcutoffleadstoringing

Ideallowpassfilter

Butterworth

lowpassfilter n= 2n= 4

ButterworthHighPassFiltering

Idealhighpassfilter

Butterworthhigh

passfilter n= 2n= 4

LowPassButterworthFiltering

Gentlercutoffeliminatesringingartifact

DFT of Image after low

pass Butterworth filteringResulting image after inverse DFT

HighPassButterworthFiltering

DFT of Image after high

pass Butterworth filteringResulting image after inverse DFT

GaussianFiltering

Samesteps

Creategaussianfilter

Multiply(DFTofimage)by(gaussianfilter)

Invertresult

transform)

Frequency

DomainLowPass

GaussianFiltering

brokentext Images taken from Gonzalez & Woods, Digital Image Processing (2002) blemishesinaphotograph Images taken from Gonzalez & Woods, Digital Image Processing (2002)

FrequencyDomainHighPassGaussian

Filtering

FrequencyDomain

RemovalofPeriodicNoise

Image with periodic Noise DFT of Image

FrequencyDomain

RemovalofPeriodicNoise

NotchFilter

Bandrejectfilter

Removesmuchoftheperiodicnoise

Notch FilterResult after notch filter

applied then inverted

FrequencyDomainRemovalofPeriodic

Noise:BandRejectFilter

ApplyfiltertoDFT

Band Reject FilterResult after band reject filter

applied then inverted

2DFourierTransformExamples:Scaling

Stretchingimage=>Spectrumcontracts

Andviceversa

2DFourierTransformExamples:Periodic

Patterns

correspondingpositionsinspectrum

Enlarging image ( c) causes

Spectrum to contract (f)

2DFourierTransformExamples:Rotation

2DFourierTransformExamples:Oriented,

elongatedStructures dominantinspectrum

2DFourierTransformExamples:Natural

Images

madeones,lessobviousinspectra

2DFourierTransformExamples:Natural

Images

orientation=>donotstandoutinspectra

2DFourierTransformExamples:Printed

Patterns

visible/removableinfrequencyspectrum.

References

ProcessingusingMATLAB

WilhelmBurgerandMarkJ.Burge,DigitalImage

Processing,Springer,2008

Spring2012

andMultimedia,Fall2012 rd edition),PrenticeHallquotesdbs_dbs17.pdfusesText_23