Semi-Blind Spectral Deconvolution with Adaptive Tikhonov Regularization

Semi-Blind Spectral Deconvolution with Adaptive Tikhonov Regularization

spectroscopictechniques

Semi-BlindSpectralDeconvolutionwithAdaptiveTikhonovRegularization

LuxinYan,HaiLiu,ShengZhong,*HouzhangFang

NationalKeyLaboratoryofScienceandTechnologyonMultispectralInformationProcessing,InstituteforPatternRecognitionandArti cialIntelligence,HuazhongUniversityofScienceandTechnology,Wuhan,Hubei,430074,China

Deconvolutionhasbecomeoneofthemostusedmethodsforimprovingspectralresolution.Deconvolutionisanill-posedproblem,especiallywhenthepointspreadfunction(PSF)isunknown.Non-blinddeconvolutionmethodsuseaprede nedPSF,butinpracticethePSFisnotknownexactly.BlinddeconvolutionmethodsestimatethePSFandspectrumsimultaneouslyfromtheobservedspectra,whichbecomeevenmoredif cultinthepresenceofstrongnoise.Inthispaper,wepresentasemi-blinddeconvolutionmethodtoimprovethespectralresolutionthatdoesnotassumeaknownPSFbutmodelsitasaparametricfunctionincombinationwiththeaprioriknowledgeaboutthecharacteristicsoftheinstrumentalresponse.First,weconstructtheenergyfunctional,includingTikhonovregularizationtermsforboththespectrumandtheparametricPSF.Moreover,anadaptiveweightingtermisdevisedintermsofthemagnitudeofthe http://www.wendangwang.comparativeresultswithotherdeconvolutionmethodsonsimulateddegradedspectra,aswellasonexperimentalinfraredspectra,arepresented.

IndexHeadings:Infraredspectroscopy;Spectraldeconvolution;Semi-blinddeconvolution;Tikhonovregularization.

INTRODUCTION

Deconvolutionisamathematicaltechniquethatremovesthebroadeningeffectoftheinstrumentalresponsefunction(orpointspreadfunction,PSF)toimprovethespectralresolution.Manyspectraldeconvolutionmethodshavebeendeveloped.Therearetwomajortypesofmethods:thenon-blinddeconvolution(NBD)method,inwhichthePSFisassumedtobeknowninadvance,andtheblinddeconvolution(BD)method,inwhichthePSFisunknown.

Non-blinddeconvolutionmethodshavebeenwidelyre-searchedforspectraldeconvolution.Wiener lteringisthemostpopular.1Fourierself-deconvolution(FSD),developedbyKauppinenetal.,isthemostcommonmethodusedininfraredspectroscopy.2TheconstraineddeconvolutionalgorithmwitharelaxationweightingfunctionproposedbyJanssonhasbeen

Received7February2011;accepted26July2012.

*Authortowhomcorrespondenceshouldbesent.E-mail:zs2971@http://www.wendangwang.com.

DOI:10.1366/11-06256

provedtobepracticallyeffective.3Othermethods,suchasthemaximumlikelihood,maximumentropy,andalternatingprojectionmethods,havealsoachievedacertainamountofsuccess.4,5ThemaximumBurgentropy(MaxEntD)methodsuggestedbyLo´renz-Fonfr ´ahasobtainedimpressivenarrow-ingandnoisesuppression.5Inpractice,however,theinstrumentalresponsefunctionisnotknownexactly.Inadditiontothephysicalmechanismandthedesignscheme,theadjustmentandagingoftheinstrumentcanalsoaffectitsresponsefunction.Also,NBDmethodsareverysensitivetothemismatchbetweentheusedandthetruePSF,andapoorknowledgeofthePSFnormallyleadstoapoorrecoveryresult.Incontrast,withoutassumingaknownPSF,blinddeconvolutionmethodsestimatethePSFandthespectrumsimultaneouslyfromthemeasureddata.Sengaetal.developedahomomorphic lteringmethodandappliedittotheinfraredmolecularabsorptionspectrasuccessfully.6Zouetal.proposedaniterativemethod,startingfromaninitial ttingresponsefunction,toiterativelyestimatethetruespectrumandupgradetheresponsefunction.7Yuanetal.proposedahigh-orderstatisticalGauss–Newtonalgorithmtoblindlydeconvolvethemeasuredspectroscopicdata.8Blinddeconvolutionbecomesevenmoredif cultbecausethePSFisnotknowninadvance,andthepresenceofnoisewillfurtherdeterioratetheproblem.7,8

BecausethePSFisveryimportantfordeconvolution,weshouldadequatelyutilizetheaprioriknowledgeabouttheinstrumentalPSFtoimprovedeconvolutionperformance.Inthecaseofnon-blinddeconvolutionmethods,assumingaknownPSFisirrational,butforblinddeconvolutionmethods,notutilizinganyinformationaboutthePSFisalsoirrational.Infact,manystudieshaveshownthattheoverallinstrumentalresponsefunctionapproachesaGaussianform,whichisespeciallytrueifthefactorsofbroadeningaremultifold,suchasslitfunction,gratingresponse,circuitresponse,etc.3,7Therefore,thePSFcanbemodeledbyaparametricfunctionandappliedinthedeconvolutionprocess,forexample,aGaussianfunction,whichiscompletelyde nedbyasingleparameter(theGaussian’svariance).Borrowingfromasimilarconceptinimagerestoration,werefertothisasthesemi-blindspectraldeconvolution(SBD)methodinthispaper.ItisexpectedthattheSBDmethodwillproducebetterresultsbecausethenumberofunknownsinthePSFisreducedfrom

1334Volume66,Number11,2012

0003-7028/12/6611-1334$2.00/0

Ó2012SocietyforAppliedSpectroscopy

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