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2.Numericalproblemanalysis
2.4.Objectivefunction
Thealgorithmofimageconstructioninelectricalimpedancetomographyis
basedontheoptimizationmethod,inwhichthewaytodeterminethesolutionisto
minimizethefunctionofthetarget.Theobjectivefunctionisdefinedasthemean
squareerrorvalueofthereconstructionreferredtoastheinter-electrodevoltage
vector:
(2.38)
whereFjisthemeansquareerrorofreconstructionforthej-thmeasurementpair
Thetaskofthereconstructionalgorithmistodeterminesuchaγvectorforwhich
theobjectivefunctionreachesatleastthelocalminimum.Itshouldbenotedthat
thevalueoftheobjectivefunction,withdeterminedmeasurementdata,dependson
thematrixU,whichisnotonlyafunctionofthevectorγ,butalsoafunctionofthe
positionofthestimulussources,andthusisafunctionoftheprojectionangle.The
matrixUwasdeterminedintheanalysisofaforwardproblem.
2.5.Sensitivityanalysis
Optimizationalgorithmsforinverseproblems,inwhichaproperlydefined
objectivefunctionisminimized,requiretheuseofsensitivityanalysis[154,304].
Ingeneral,thetaskofsensitivityanalysisistodeterminethegradientoftheobjec-
tivefunction.Determiningthecorrectionsfordecisionvariablesinthesemethods
ispossibleafterdeterminingthegradientvectoroftheobjectivefunction,relative
tothedecisionvariables.Todeterminethisvector,theobjectivefunctionisoften
directlydifferentiatedwithrespectdecisionvariables.
Theadjointvariablemethodisconsideredtobethemosteffectiveandre-
quiresasmallnumberofnumericaloperations.Theefficiencyandspeedofthe
methodimplementedinthesensitivityanalysisisimportantintheconstructionof
theimage.Thisisduetothesizeofthenon-lineartaskandmultiplerepetitionof
calculationsintheiterativeprocess.Imagereconstructionrequirestheuseofsen-
sitivityanalysisforeachprojectionangleateachstepoftheiterativeprocess.The
objectivefunctionforthej-pairofelectrodes:
(2.39)
where:ndeisthenumberofindependentmeasurementsofinter-electrodevoltages
attheedgeofthearea.
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