Digital Image Processing3.docx
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DigitalImageProcessing3
DigitalImageProcessing
ComputerProjectReportIII
FourierTransformandFrequencyDomainFiltering
StudentID:
20091613310032
Name:
XiaopengJi
Date:
April18,2012
A.Objectives
i.FamiliarwiththeconceptsandprinciplesofFouriertransform;
ii.ProcessingdigitalimagewithMATLABinFrequencydomain;
iii. Familiarwiththefilterdesignfunctions:
fsamp2,fwind1,andfwind2.
B.Methods
Forthisprojectusetheimagelenna.gif.
1） Computetheforward2DFFTofthe lenna imageusingtheMATLABIMAGEPROCESSINGTOOLBOXfunctionFFT2.
2） LowpassFilterDesign
3） HighpassFilterDesign
4） TwoDimensionalFilterDesign
C.Results
1） Computetheforward2DFFTofthe lenna imageusingtheMATLABIMAGEPROCESSINGTOOLBOXfunctionFFT2.
Thecommandis:
imglenna=imread（'lenna.gif'）;
imgFFT=fft2（double（imglenna）./255）;
imgFFT=fftshift（imgFFT）;
ThefunctionfftshiftisusefulforvisualizingtheFouriertransformwiththezero-frequencycomponentinthemiddleofthespectrum.
a） Computethelogmagnitudeandphase（i.e.,MATLABIMAGEPROCESSINGTOOLBOXfunction ANGLE.
Thecommandis:
imgLogMag=log（abs（imgFFT）+1）;
imgPhase=angle（imgFFT）;
b） Computetheinverse2DFFTofthe lenna imageusingtheMATLABIMAGEPROCESSINGTOOLBOXfunction IFFT2.
Thecommandis:
imgIFFT=abs（ifft2（imgFFT））
c） Plottheoriginal lenna image,logmagnitude,phase,andinversetransformedimages.
Thecommandis:
figure;
subplot（221）;
imshow（imglenna）;
title（'OriginalImage'）;
subplot（222）;
imshow（imgLogMag,[]）;
title（'LogMaganitudeofFFT'）;
subplot（223）;
imshow（imgPhase,[]）;
title（'PhaseofFFT'）;
subplot（224）;
imshow（imgIFFT,[]）;
title（'InverseFFT'）;
Theresultis:
Figure1:
Original,logmagnitude,phase,andinversetransformedimages
2） LowpassFilterDesign
a）UsetheMATLABIMAGEPROCESSINGTOOLBOXfunctionFSPECIALtodesignan11x11Gaussianlowpassfilterwithavalueofsequalto1.3.
Thecommandis:
LowpassFilter=fspecial（'gaussian',[1111],1.3）;
b）Computetheforward2DFFTofthefilterkernelusingthesamesizeFFTasthatofthelennaimage.UtilizetheSIZEfunctionfromtheexampleonthewebsite.
Thecommandis:
imgSize=size（imglenna）;
imgRows=imgSize
（1）;
imgCols=imgSize
（2）;
LowpassFFT=fftshift（fft2（LowpassFilter,imgRows,imgCols））;
Wecangettherowandcolumnoftheimageandusethefunctionfft2toComputetheforward2DFFTofthefilterkernel
c）Fromtheresultsinb）computeandplotthelogmagnitudeandphaseoftheGaussianLowpassFilterkernel.
Thecommandis:
figure;
subplot（121）;
imshow（log（abs（LowpassFFT）+1）,[]）;
title（'LogMagnitude'）;
subplot（122）;
imshow（angle（LowpassFFT）,[]）;
title（'Phase'）;
Figure2:
logmagnitudeandphaseoftheGaussianLowpassFilterkernel
Fromtheresult,wecanfindthecenteroflogmagnitudeisbrightandthephaseisalternatinglightanddarkstripes.
d）Utilizingtheresultsin1）and2b）performthefilteringfunctionG（u,v）=H（u,v）*F（u,v）,whereH（u,v）=2DFFToftheGaussianFilterKernel,andF（u,v）=2DFFTofthelennaimage.Plotthelogmagnitudeandphaseofthelowpassfilteredimage.
Thecommandis:
imgFiltered=LowpassFFT.*imgFFT;
figure;
subplot（121）;
imshow（log（abs（imgFiltered）+1）,[]）;
title（'LogMagnitude'）;
subplot（122）;
imshow（angle（imgFiltered）,[]）;
title（'Phase'）;
Figure3:
logmagnitudeandphaseofthelowpassfilteredimage
e）Computeandplottheinverse2DFFTofthelowpassfilteredimage.
Thecommandis:
figure;
imgLowpassFiltered=abs（ifft2（imgFiltered））;
imgLowpassFiltered=circshift（imgLowpassFiltered,[-1.*floor（length（LowpassFilter）/2）-1.*floor（length（LowpassFilter）/2）]）;
imshow（imgLowpassFiltered,[]）;
title（'InverseFFTofLowpassFilteredImage'）;
Figure4:
theinverse2DFFTofthelowpassfilteredimage
Fromtheresult,wecanfindtheimageafterlowpassfiliterisfuzzyandthehigh-frequencycomponentsoftheimagearelost.
3） HighpassFilterDesign
a）UsetheMATLABIMAGEPROCESSINGTOOLBOXfunctionFSPECIALtodesignalaplacianhighpassfilter.
Thecommandis:
HighpassFilter=fspecial（'laplacian'）;
b）Computetheforward2DFFTofthefilterkernelusingthesamesizeFFTasthatofthelennaimage.UtilizetheSIZEfunctionfromtheexampleonthewebsite.
Thecommandis:
HighpassFFT=fftshift（fft2（HighpassFilter,imgRows,imgCols））;
c）Fromtheresultsinb）computeandplotthelogmagnitudeandphaseoftheLaplacianhighpassFilterkernel.
Thecommandis:
figure;
subplot（121）;
imshow（log（abs（HighpassFFT）+1）,[]）;
title（'LogMagnitude'）;
subplot（122）;
imshow（angle（HighpassFFT）,[]）;
title（'Phase'）;
Figure5:
logmagnitudeandphaseoftheLaplacianhighpassFilterkernel
Fromtheresult,wecanfindthecenterofthelogmagnitudeisdarkandthereisacleardividinglineinphase.
d）Utilizingtheresultsin1）and3b）performthefilteringfunctionG（u,v）=H（u,v）*F（u,v）,whereH（u,v）=2DFFToftheGaussianFilterKernel,andF（u,v）=2DFFTofthelennaimage.Plotthelogmagnitudeandphaseofthelowpassfilteredimage.Thecommandis:
imgFiltered=HighpassFFT.*imgFFT;
figure;
subplot（121）;
imshow（log（abs（imgFiltered）+1）,[]）;
title（'LogMagnitude'）;
subplot（122）;
imshow（angle（imgFiltered）,[]）;
title（'Phase'）;
Figure6:
logmagnitudeandphaseofthelowpassfilteredimage
e）Computeandplottheinverse2DFFTofthehighpassfilteredimageusingtheIFFT2function.
Thecommandis:
figure;
imgHighpassFiltered=abs（ifft2（imgFiltered））;
imgHighpassFiltered=circshift（imgHighpassFiltered,[-1.*floor（length（HighpassFilter）/2）-1.*floor（length（HighpassFilter）/2）]）;
imshow（imgHighpassFiltered,[]）;
title（'InverseFFTofHighpassFilteredImage'）;
Figure7:
InverseFFTofHighpassFilteredImage
Fromtheresult,wecanfindtheimageafterHighpassFilteronlycontainsthecontourwithfrequencycomponents.
4） TwoDimensionalFilterDesign
a） Theobjectiveofthisexerciseidtoutilizethefilterdesignfunctions:
fsamp2,fwind1,andfwind2.
1. Use[f1,f2]=freqspace（21,'meshgrid'）;commandtodesignthesamplinggridforthefilter.
2. Once1）iscompletedcomputetheradiusvectorsforthefollowingfilterdesignsforthefilterdesignfunctions:
fsamp2,fwind1,andfwind2.
a） Theradiusvectorsarethefollowing:
Bandpass:
（r<0.1）|（r>0.6）
Lowpass:
r>0.6
Highpass:
r<0.6
Thecommandis:
[f1,f2]=freqspace（21,'meshgrid'）;
r=sqrt（f1.^2+f2.^2）;
Hd=ones（size（f1））;
Bandpass=Hd;
Lowpass=Hd;
Highpass=Hd;
Bandpass（（r<0.1）|（r>0.6））=0;
Lowpass（r>0.6）=0;
Highpass（r<0.6）=0;
b） Foreachofthefilteringalgorithmsdothefollowing:
1） Designabandpass,lowpass,andhighpassfilter
2） Computetheforward2DFFTofthefilterkernelsusingthesamesizeFFTasthatofthelennaimage. Utilizethe SIZE functionfromtheexampleonthewebsite.
3） Usetheresultsin2）computeandplotthelogmagnitudeandphaseofeachrespectivefilterkernel.
4） Utilizingtheresultsin2）performthefilteringfunctionG（u,v）=H（u,v）*F（u,v）,whereH（u,v）=2DFFToftherespectivefilterkernel,andF（u,v）=2DFFTofthelennaimage. Plotthelogmagnitudeandphaseofthefilteredimage.
5） Computeandplottheinverse2DFFTofeachfilteredimage.
Withthefilterdesignfunctionsfsamp2,thecommandis:
BandpassFilter=fsamp2（Bandpass）;
LowpassFilter=fsamp2（Lowpass）;
HighpassFilter=fsamp2（Highpass）;
BandpassFilterFFT=fftshift（fft2（BandpassFilter,imgRows,imgCols））;
LowpassFilterFFT=fftshift（fft2（LowpassFilter,imgRows,imgCols））;
HighpassFilterFFT=fftshift（fft2（HighpassFilter,imgRows,imgCols））;
Theresultsare:
Figure8:
LogMagnitudeandPhaseofbandpass,lowpass,andhighpassfilter
Figure9:
logmagnitude,phaseandinverse2DFFTofeachfilteredimage
Fromtheresults,wecanfindthattheimageafterLowpassFilterisfuzzy.TheimageafterHighpassandBandpassisdistortionbutretaindifferentfrequencycomponentsoftheimage.
Withthefilterdesignfunctionsfwind1,thecommandis:
BandpassFilter=fwind1（Bandpass,hamming（21））;
HighpassFilter=fwind1（Highpass,hamming（21））;
LowpassFilter=fwind1（Lowpass,hamming（21））;
BandpassFilterFFT=fftshift（fft2（BandpassFilter,imgRows,imgCols））;
LowpassFilterFFT=fftshift（fft2（LowpassFilter,imgRows,imgCols））;
HighpassFilterFFT=fftshift（fft2（HighpassFilter,imgRows,imgCols））;
Theresultsare:
Figure10:
LogMagnitudeandPhaseofbandpass,lowpass,andhighpassfilter
Figure11:
logmagnitude,phaseandinverse2DFFTofeachfilteredimage
Fromtheresults,wecanfindthatimageafterLowpassFilterisbetter.TheimageafterHighpassFilterretainsminimalinformation.TheimageafterBandpassFilterretainpartofinformation.
Withthefilterdesignfunctionsfwind2,thecommandis:
window=fspecial（'gaussian',21,2）;
window=window./max（max（window））;
BandpassFilter=fwind2（Bandpass,window）;
HighpassFilter=fwind2（Highpass,window）;
LowpassFilter=fwind2（Lowpass,window）;
BandpassFilterFFT=fftshift（fft2（BandpassFilter,imgRows,imgCols））;
LowpassFilterFFT=fftshift（fft2（LowpassFilter,imgRows,imgCols））;
HighpassFilterFFT=fftshift（fft2（HighpassFilter,imgRows,imgCols））;
Theresultsare:
Figure12:
LogMagnitudeandPhaseofbandpass,lowpass,andhighpassfilter
Figure13:
logmagnitude,phaseandinverse2DFFTofeachfilteredimage
Fromtheresults,wecanfindthattheimageafterHighpassFilterretainhighfrequencycomponentsoftheimagecontour.TheimageafterLowpassFilterlostlessinformationcomparedwiththeoriginalimage.AndtheimageafterBandpassFilterisdarkcomparedwiththeoriginalimage.
D.Conclusions
TheeffectofLowpassFilterinFrequencydomainistoremovethehighfrequencynoiseoftheimage,andtheabilitydependsontheformandcutofffrequencyofthefilter.LowpassFilterwillalsocauseimagefuzzybyremovingthenoise.
HighpassFil
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