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101Introduction
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theimageintoafinitesetofhomogeneousareas,andtorestricttherealspace
oftheimagetothespaceofdiscretevalues.ThisstrategyisproposedbySingh
etal[58]forlimited-viewCTreconstruction.Thelimitationoftheimagespace
canbeevenmuchmorerestrictive.Applyingtomographicimagingfordetection
orlocalizationofinsideanomalies,theimagespacecanberestrictedonlytotwo
binaryvalues,andhence,abinarymapissearched.
Theproblemofreconstructingabinaryimagefromprojectionshasbeendis-
cussedbymanyresearchers[11,24,28,39,53,54,64,65,67,70],mostlyinthecon-
textofmedicalapplicationssuchasangiography,bloodvesselsimaging,limited-
dataX-raytomography,mammography,intraoraldentalimaging,etc.
Inthechapter,ourconsiderationsarefocussedongeophysicalapplicationsof
binarytomography,whichwerefertoasDiscreteElectromagneticGeotomogra-
phy(DEG)[67,70],however,theycanbeeasilyextendedtootherapplications,
especiallyformedicalimagingmodalities.
TheaimofDEGistovisualizesubsurfaceair-filledvoidsthatmayoccurdue
tocollapseofinactivecoalmines.Air-filledanomalieshaveconsiderablydifferent
electricalpropertiesthanthebackground(sedimentaryrocks),andhence,theycan
beeasilydetectedwithelectromagneticprobingsbetweenboreholes.
Theelectricalpropertiesofthesurveyedmediumalongeachray-pathaffect
thetransmissionpropertiesoftheelectromagneticwavespropagatedalongthat
ray-path.Therefore,themeasureddatacarriestheinformationabouttheunknown
geologicalstructure.
Ourstudyisbasedonanabsorptionmodelthatdescribestheattenuationof
electromagneticwavesalongthestraight-lineray-path,andtherefore,thismodel
canbeexpressedintermsoftheRadontransform.The”straight-ray”approxima-
tionneglectstheeffectsofreflection,refractionanddiffraction,anditisjustified
providedthattheelectromagneticwavesofhighfrequenciesareused[16,37,49].
Aninversionofthemodelyieldsanimageofspatialdistributionofelectromag-
neticwaveattenuationcoefficientinthesurveyedarea.
Assumingtheconstantattenuationcoefficientsfortheanomaliesandback-
ground,theimagereconstructionprobleminDEGboilsdowntotheproblemof
findingabinarysolution.Moreover,usingsomelineartransformationswecan
assignzero-valuestoanomalies,andonestorocks.Thusthespaceoffeasible
solutionscanbenarroweddowntothespaceofdiscretepoints,oreventobinary
ones.
Suchharshconstraintscanbeinterpretedintermsofregularizationwithstrong
priorinformation.Thebinaryapproachcanconsiderablyrelaxtheambiguitiesof
theleast-squareinversion,whichimpliesthereductionofthe”ghost”effects.For
thisreason,theusageofimagereconstructionalgorithmsfromdiscretetomog-
raphy[28]maybeveryhelpfulinsolvingtheimagereconstructionproblemin
DEG.Furthermore,suchalgorithmsaremuchmoreresistanttonoisyperturba-
