壓縮感知入門matlab代碼

1、30 eginnersCodeforCompressiveSensingAlejandroWeinsteinSeptember2009SparseSignalsintheTimeDomain1.1UsingaRandomSensingMatrixInthisfirstexamplewewillmeasureasignalthatissparseinthetimedomain.Wewillusearandomsensingmatrix,andwewillsolvetherecoveryproblemusingthel1-Magictoolbox.Weusethefollowingfunction

2、stogeneratethesignalsandthesensingmatrix:Listing1:Sparsesignalandrandommeasurementmatrix.functionf=getsparsefun(n,s)tmp=randperm(n);f=zeros(n,1);f(tmp(1:s)=randn(s,1);functionA=getArandom(n,m)A=sqrt(1/m)*randn(m,n);Thefollowingscriptusethesefunctionstogeneratethesignal,takethemeasurementsanddotherec

3、overy.Figure1showstheresult.Listing2:Example1.1234567891011121314151617181920212223242526272829%CSexample1Sensingmatrixphiisrandom.RepresentationbasisPsiistheRecoveringusingl1magic.2;s15*525=nsm%SignallengthSparsitylevelNumberofmeasurementsf=getsparsefun(n,s);A=getArandom(n,m);y=A*f;%Takethemeasurem

4、ents%Solveusingl1magic.path(path,./Optimization);x0=pinv(A)*y;%initialguess=ticxp=l1eqpd(x0,A,y,1e-3);tocnorm(f-xp)/norm(f)plot(f)holdonplot(xp,r.)legend(Original,Recovered)canonicalbasis.minenergy30 #TherecoverycanbemadebyusingCVXinsteadofl1-Magic.Justreplacelines19to24by3 -3.1111ii1I111OriginRecov

5、alered-111111I111|210-1-250100150200250300350400450500Figure1:Script1results.Listing3:UsingCVXfortherecovery.%SolveusingCVX.cvxbeginvariablexp(n);minimize(norm(xp,1);subjecttoA*xp=y;cvxend1.2UsingaFourierSensingMatrixNowwearegoingtorepeatthesameexperiment,butusingasamplingmatrixbasedontheFourierbasi

6、s.Wegeneratethemeasurementmatrixwiththefollowingfunction:Listing4:Fourierbasedmeasurementmatrix.functionA=getAfourier(n,m)tmp=randperm(n);phi=inv(fft(eye(n);A=phi(tmp(1:m/2),:);A=real(A);imag(A);Inordertorecoverthesignalusingl1-magic,nowweneedtousethefunctionl1qc_logbarrierinsteadofl1eq_pd.Ontheothe

7、rhand,thereisnoneedtochangeanythingwhensolvingtheproblemwithCVX.SinceingeneralissimplerandclearertouseCVX,weonlyusethisapproachinthefollowingexamples.SparseSignalintheFrequencyDomainLetstrynowwithasignalsparseinthefrequencydomain.Wegeneratethesignalas:Listing5:Sparsesignalinthefrequencydomain.0:n-1;

8、cos(2*pi/256*t)+cos(2*pi/128*t); 1234567891011Figure2showstheresult.Asexpected,therecoveryisexact.Letsmodifyoursignalslightly,byreplacingoneofthecosinebyasine:LetssolvewitharandomsensingmatrixandCVX.Noticethatnowweneedtospecifytherepresentationbasis(seeline5):Listing6:RandommeasurementsandCVXrecover

9、y.A=getArandom(n,m);y=A*f;%SolveusingCVX.Psi=inv(fft(eye(n);cvxbeginvariablexp(n);minimize(norm(xp,1);subjecttoA*Psi*xp=y;cvxendFigure2:Recoveryofafrequencydomainsparsesignal.Listing7:Sparsesignalinthefrequencydomain.t=0:n-1;f=cos(2*pi/256*t)+sin(2*pi/128*t);Figure3showstheresult.Evidentlythereissom

10、ethingwrong.TheproblemisthatnowtheFouriercoefficientshaveanimaginarycomponent,butCVXissearchingforarealx.Thesolutioniseasy,wejustneedtotellCVXtoconsideracomplexx:Listing8:Sparsesignalinthefrequencydomain.%SolveusingCVX.Psi=inv(fft(eye(n);cvxbeginvariablexp(n)complex;%WeneedtotellCVXthatxpiscomplex!m

11、inimize(norm(xp,1);subjecttoA*Psi*xp=y;cvxendFigure4showstheresult.Nowtherecoveryisexact.Wecanalsotrywithahighersparsitylevel.Thefollowingscriptcreateasignalbyaddingsixsinusoidswithrandomperiod,amplitudeandphases.Noticethattheperiodsarechosenfromthevector163264128256512.Figure5showstheresult.Onceaga

12、in,therecoveryisexact.Listing9:Signalmadeof6randomsinuoids.s=6;amp=rand(s,1);%amplitudesperiods=163264128256512;4tmp=randperm(length(periods);freq=(2*pi./round(periods(tmp(1:s);%frequenciesphases=2*pi*rnd(s,1);f=zeros(n,1);t=0:n-1;fork=1:s,f=f+amp(k)*cos(freq(k)*t+phases(k);endWenowreplacetheperiods

13、weareusingby183264128256512.Noticethattheonlydifferenceisthatnowthesmallestperiodis18insteadof16.Figure6showstheresult.Nowtheresultisnotexact.Thereasonforthisisthatnowthesignalisnotreallysparse,sinceoneoftheperiodsisnotanintegermultipleofthesignallength.AcknowledgmentsThankstoDr.MichaelWakinandBorhanSanandajiforhelpingmetosolvesomeoftheissuesIhadwiththecode.1Ifthesignalissparseinthetimedomain,=Identitymatrix,thatswhywedidntspecifyinsection1.Figure4:SuccessfullyrecoveryaftertellingCVXtouseacomple