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105Conclusions
39
indiscretetomographyconsiderablydiffersfromthewell-knowntechniquesthat
areusedforleast-squaresproblems.Wefirstrevisesomewell-knownalgorithms,
whichcanbepotentiallyusefulforDEG.Notethat,unliketypicalapplicationsof
discretetomographywhereanumberofprojectionsislimited,DEGhasalimita-
tioninanangularrangeofprojections.Thismakessomealgorithmsfromdiscrete
tomographyuselessinDEG.Butthereisstillavarietyofalgorithmsthatcanbe
appliedforourapplication.Theselectedalgorithmshavebeenapplieddirectly
orinmodifiedversionstoourobjectivefunctions.WeusetheD.C.algorithm,
MeanFieldAnnealing(MFA)algorithm,andProjectedGradient(PG)algorithm
withbinarysteering.Allthealgorithmsiterativelysteertheupdatestowardsa
binarysolution.Thementionedmodifications(trickyimplementations)arebasi-
callyrelatedwithreductioninacomputationalcomplexity.Wealsotestmany
potentialfunctionsinvolvedintheMRF,butfinallywepresenttheresultsonly
fortheGreenfunctionandfortheGGMRF,whicharewell-knowintheliterature.
Wealsoapplysomesimplifiedmethodforestimatingthescalingparameterthat
isassociatedwiththeMRF.
Thenumericaltestsperformedfornoise-freeandverynoisy(20dB)data
demonstratethatthediscussedalgorithms(especiallytheMFAalgorithm)areable
togiveverygoodreconstructionfromDEGdata.Butonemustbeawareofthe
factthattheresultsareparameter-dependant.Also,theMRFstronglyaffectsthe
performanceofthealgorithms.WeareunabletotunetheparametersinGGMRF
modelinthiswaytoachievethesameresultsasfortheGreenfunctions.Un-
fortunately,therobustpotentialfunctionsconsiderablyincreaseanoverallcom-
putationalcost.Thisisbecausetheevaluationofsuchfunctionsismuchmore
expensive,andusuallythefunctionshavemorefreeparameterstobeestimated.
Weshortlydiscusssometoolsthatcouldbeappliedforestimatingthefreeparam-
eters,however,thisproblemisstillopeninDEG.
Fornoisydata,thereconstructionalsodependsonthedampingfactorthat
steersthesolutiontowardsabinaryone.Thedescentrateshouldnotbetoofast
inordertoavoidgettingstuckinlocalminimabuttooslowdescentslowsdown
thereconstruction.ThePGalgorithmwithbinarysteeringistheleastsensitivefor
thisparameter,buttherobustpotentialfunctionsaredifficulttobeimplemented
inthisalgorithm.
AnotheropenissuesinDEGarerelatedwithappropriatemodelingofartifacts
inDEG.Inthisapproach,weassumedaverysimplifiedimagereconstruction
modelinwhichpossibleparasiteeffects,likediffractionorrefractionofelectro-
magneticwaves,areneglected.Inamoregeneralcase,themodelshouldalso
assumeageneraldiscretecase,i.e.afewintermediatevaluestomodelbound-
aryeffects(aroundanomalies).Thus,theextensiontoageneralcaseofdiscrete
tomography(notonlybinary)isalsoanopenissueinDEG.
