Optimized knot placement for B-splines in deformable image registration

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1、OptimizedknotplacementforB-splinesindeformableimageregistrationa)TravisJ.JacobsonandMartinJ.MurphyDepartmentofRadiationOncology,VirginiaCommonwealthUniversity,Richmond,Virginia23298(Received15February2011;revised13June2011;acceptedforpublication15June2011;published22July2011

2、)Purpose:TodevelopanautomaticknotplacementalgorithmtoenabletheuseofNonUniformRationalB-Splines(NURBS)indeformableimageregistration.Methods:Theauthorsdevelopedatwo-stepapproachto?taknowndisplacementvector?eld(DVF).Aninitial?twasmadewithuniformknotspacing.Theerrorgeneratedbyth

3、is?twasthenassignedasanattractiveforcepullingontheknots,actingagainstaresistivespringforceinaniter-ativeequilibrationscheme.Todemonstratetheaccuracygainofknotoptimizationoveruniformknotplacement,wecomparedthesumofthesquarederrorsandthefrequencyoflargeerrors.Results:Fitswerem

4、adetoaone-dimensionalDVFusing1–20freeknots.Giventhesamenumberoffreeknots,theoptimized,nonuniformB-spline?tproducedasmallererrorthantheuniformB-spline?t.Theaccuracywasimprovedbyameanfactorof4.02.TheoptimizedB-splinewasfoundtogreatlyreducethenumberoferrorsmorethan1standarddevi

5、ationfromthemeanerroroftheuni-form?t.TheuniformB-splinehad15sucherrors,whiletheoptimizedB-splinehadonlytwo.Thealgorithmwasextendedto?tatwo-dimensionalDVFusingcontrolpointgridsizesrangingfrom88to1515.Comparedwithuniform?ts,theoptimizedB-spline?tswereagainfoundtoreducethesum

6、ofsquarederrors(meanratio?2.61)andnumberoflargeerrors(meanratio?4.50).Conclusions:NonuniformB-splinesofferanattractivealternativetouniformB-splinesinmodelingtheDVF.TheycarryforwardthemathematicalcompactnessofB-splineswhilesimultaneouslyintro-ducingnewdegreesoffreedom.Theincr

7、easedadaptabilityofknotplacementgainedfromthegeneral-izationtoNURBSoffersincreasedlocalcontrolaswellastheabilitytoexplicitlyrepresenttopologicaldiscontinuities.VC2011AmericanAssociationofPhysicistsinMedicine.[DOI:10.1118/1.3609416]Keywords:Deformableimageregistration,B-splin

8、es,NURBSI.PURPOSEvatedresearchintoB-splinecon?gurationsthatcanbeDeformablei