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101Introduction
101Introduction
13
Intomographicimaging,theinteriorstructureoftheobjectofinterestisrecon-
structedfromasetofprojectiondata.Reconstructionfromcompleteorsuffi-
cientlyabundantdataisbynowwellunderstood,andmanyimagereconstruction
algorithmscanprovideanexactreconstructionfromconsistentdata.However,
whentheobserveddataarelimitedinthesenseofalimitednumberofprojections
orlimitedangularviewofprojections,theproblemofobtainingartifact-freeim-
agefromsuchprojectiondataisgenerallymuchmoredifficulttosolveandstilla
subjectofextensivestudy[1,4,5,8,25,27,32,35,36,51,59,63,66,68].
Limited-datascanningtakesplaceinmanyimagingmodalitiesduetointrinsic
limitationwherethecompleteprojectiondataisimpossibletobeacquiredorby
intendedactivity,e.g.tominimizetheradiationdoseabsorbedbyapatientin
medicalX-rayimaging.
Theexampleoftheformercaseisboreholeimagingthatmostlyreferstogeo-
physicalimagingmodalities[42]suchasseismictomography[4,8,17,23,45]or
electromagneticgeotomography[16,38,49].
Boreholetomographymapsthegeologicalstructureofacross-boreholesec-
tionwithwell-to-wellmeasurementsofsomeenergycarrierthatispropagated
throughthemediumofinterest.Thereceivedcharacteroftheenergycarrier(e.g.,
amplitude,travel-time)isthenusedtoinferthedistributionofthemediumproper-
ties(e.g.,velocity,density,permittivity,attenuationcoefficient).Anenergycarrier
canbeaseismic,ultrasonicorelectromagneticwave.Thedataisgatheredalong
multipleraypathsthatlinkthepositionsofthewavesource(transmittingprobe)
andthemeasurementpoint(receivingprobe).Thegeometryofinterestiscross-
boreholescanningthatisschematicallydepictedinFig.1.1a.Inthiscase,the
angularrangeofprojectionsdependsonthesizeofthesurveyedarea,anditcan
beestimatedas[−arctan(h/d),arctan(h/d)],wherehistheheightofthesurveyed
area,anddisthedistancebetweenboreholes.Forthesquaredarea,theangular
rangeis[−450,450].
Theexampleoftheintendeddatalimitationisthelimited-viewX-raytomog-
raphy,wherethelimitationofanumberofprojectionsconsiderablyreducesthe
X-rayradiationdoseabsorbedbytheexaminedpatient.Asamplescanningge-
ometryforlimited-viewparallel-beamX-raytomographyisshowninFig.1.1b.A
challengeistolimitanumberofprojectionstoasfewaspossible,keepingthe
imageresolutionatahighlevel.
Bothcasesoflimited-datatomographysharethesamepropertythatthefor-
wardprojectionmodelisunderdetermined,andhencetheinversionisseverelyill-
posed.Sincetheobserveddataisincomplete,itisobviousthattheunknowntrue
(artifact-free)imagecannotbereconstructedwithoutadditional(prior)knowledge
orconstraints.
