[PDF] Lecture 15 Applications of the DFT: Image Reconstruction

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Lecture15

ApplicationsoftheDFT:Image

Reconstruction

15.1Introductio n

Perhapsthemostdirec texample applicationo ftheFourierTransf orminimagingisMRI imagere construction.InMRI,dataaresampledin Fourierspace (termedk-spacein MRI lingo).In otherwords,during anMRIacquisition wetravelthroughk- space ,colle cting samplesatvariousk-spa celoc ationsaswego.Whenweha ve'enough'samples,we typicallyperformaniFFT(in2Do r3D,dependingonthespe cifica cquisi tion)inor der toobtainour recons tructed image.

15.2MRIReconst ruction

ReconstructionofMR imagesfromMR rawdata, whichis acquiredin theFourier space (termed"k-space"inMR Ilingo),canbeassimpleasaDFT in2D(f or2DMRim aging, seeillust rationinFigure15.1)oraDFTin3D(for3DMR imagi ng).

Figure15.1:Basi cimagereconstru ctioninMRI.

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62LECTURE15.APPLICATI ONSOFTHEDFT :IMAGERECONSTRUCTION

Additionally,reconstructionofMRspec tra(ie:inMRspectroscopyinsteadofMR imaging)fromrawMRdatais typicallybas edona1DDFT .Inot herwords,the DFT (andtherefor etheFFT)ispervasiveinthereco nstructi onofim agesan dspectrafrom MRdata .Inthenextsectio ns,we willcover severalexam plesofdatacorruptionand artifacts,whichcanbeundersto odbasedonourstu dyofFouri erTra nsformsandDFTs.

15.3Corrupted samples

15.3.1SpikeNoise

Occasionallywhenacquiring MRIdata,a singlek-spacesample atsomearbitrary k-space location(k x ,k y f measured (k x ,k y f(k x ,k y )+a(k x k x,0 ,k y k y,0 )(15.1) Whatwillbe thecorrespon dinger rorinther econstructedimage?

15.3.2Motion

Motionartifactsare acommonchallengeinMRIandot herimagi ngmod alities.InMRI, theseartifacts canbeeasilyreplicatedandund erstoodvi aFouri eranalysis.Motion arti- factsinMRIarety picall ymild/ne gligiblea longonedimension(wheretheco rresponding Fourierdimensionissam pledveryrapidly),whereasthe ycanb eseverealongtheother dimension(s)(wherethecorresp ondingFourierdimension(s)are sample dslowly).This e ectisill ustrat edinFigure15.2andcanbereplicatedw ithsom esimpleMatlabc ode, asshown below. im0=phantom(256); [sx,sy]=size(im0); t=1:sx; mymotion=2*sin(2*pi*0.1*t); %imageandmotiondefinedabove meas_imk=zeros(sx,sy); forcur_t=1:length(t) cur_im0=circshift(im0,[round(mymotion(c ur_t)),round(mymotion(cur_t))]); cur_im0k=fftshift(fftn(ifftshift(cur_im 0))); meas_imk(cur_t,:)=cur_im0k(cur_t,:); end

15.4.MISSINGSAMPLE S63

im=fftshift(ifftn(ifftshift(meas_imk))); imagesc(abs(cat(2,im0,im)));axisequaltightoff;colormap gray Figure15.2:Phys iologicalmotio ninMRIleadstomotionartifactsalongthose directions thataresample dslowly.T hise ectisduet oanin consisten cyinth elines ofFourierspace acquiredat di erentstagesofmotion. Inotherw ords,di erentsampledlinesinF ourier spacecorrespond toimagesthatareshiftedwithresp ectto eachother.Whenweper form theinvers eDFT(in2Dinthiscase)t orecon structour image ,these inconsistenc ieslead todele teriousmotionartifactsobservable inimagespace.

15.4Missingsa mples

15.4.1Artifactsfro mMissingSamples

Duetoacqu isitio nspeedlimitationsandotherchalleng es,inMRIwesome timesarenot abletocollec tsu cientsamplesinF ourierspacetoobtainan adequa teimage.The resultingartifactscanbeund erstoodviaFourieranalysis, asshownin Figure1 5.3.Below issome Matlabcode toreproduce thisartifact: %Generatesometrueimage

64LECTURE15.APPLICATI ONSOFTHEDFT :IMAGERECONSTRUCTION

im0= phantom(256); %Do2DDFT im0k=fftshift(fftn(ifftshift(im0))); %ZeroouteveryotherlineinDFTspace im1k=im0k; im1k(2:2:end,:)=0; %Do2DIDFT im1=fftshift(ifftn(ifftshift(im1k))); imagesc(abs(cat(2,im0,im1)));axisequaltight off;colormapgray Figure15.3:Missi ngdatainFourie rspaceleadstoartifac tsintherecon struct edimages. Inth ecasewherewe skip everyotherlinein Fouriersp ace,thecorrespon dingartifacts canbe understo odintermsofsamplingandoverlap( rememberou rsamplinglectur es!), andresultin 2-foldov erlapinour images.

15.4.MISSINGSAMPLE S65

15.4.2AdvancedImageR econstruction

Althoughnotthefocusof thiscours e,therea reavarietyofadv ancedi magereconst ruction techniquesthatareoftenabletoo verc omethepre senceofmissing samples inimaging 1 Thesetechn iquesarebasedonincludingadditi onalinformation intoth erecon struction. Thisaddition alinformationcanoriginatefromvari oussources,includingthepresence ofmult iplereceiversacquirings amplessimultaneously,orprior informationaboutimage properties(eg:sparsity). 1 Thefoundat ionsforanumberofthesetechn iquesw ill beexploredinMedical Phy sics/BME574.

66LECTURE15.APPLICATI ONSOFTHEDFT :IMAGERECONSTRUCTION

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