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38
Binarytomographicimagereconstruction000
ThustheMLestimateisasfollowed:
2N
δ;
ˆ
∑
j11
N
n∈Nj
∑
wjn
aδ
a
V(xj−xn,δ)|
δ1ˆ
δ1−1.
(1.53)
FortheimageshowninFig.1.3,weroughlyestimatedˆ
δ10.5forV
j
(HL)
,and
δ10.01forV
ˆ
j
(GR)
.Thediscussedapproachcanbealsoconsideredasin-line
estimationforrealdata.
Theestimationof;canbeefficientlydonewithmarginalization,i.e.maxi-
mizationofTypeIIlikelihood[21].Anotheroptionistoapplythetoolssuchas
GeneralizedCross-Validation(GCV),L-curve[6](especiallyforAlgorithm4),or
eventheMLestimatepresentedin[52].
TheresultspresentedinFigs.1.4–1.7(Algorithm3)aresurprisinglygood.
Fornoise-freedatanearlyallthecliqueenergyfunctionsgivesatisfactoryresults
(seeFig.1.4).Fornoisydatathebestreconstructioncanbeobtainedwiththe
Greenfunctionwhichisconvex(seeFig.1.5).FortheGGMRFmodel,there-
sultsstronglydependonp.Ifthisparameterisclosetoitslowerbound(p11),
thereconstructedimageisspeckledwithnoise.ThisisbecausetheBesagfunc-
tionenforcessparseobjects,andthiscaseshouldbeavoidedinverynoisyDEG
data,evenifsomesparseobjectsareexpectedtobereconstructed.Forp12,we
observetheover-smoothing,andsuchacaseisalsounacceptable.
ThegoodresultsarealsoobtainedwithAlgorithm4–seeFigs.1.8–1.10.
InspiteofapplyingonlytheGaussianpriorthatisanequivalenttotheGGMRF
model,theover-smoothingisnotvisibleintheimagesreconstructedfromnoisy
data(seeFig.1.9).WedecidedtousetheGaussianprioronlyduetothecomputa-
tionalcost.FortheregularizationtermwiththeGreenfunction,theresultswould
bepresumablymuchbetter.
Inourexperiments,wehaveusedthenoisydatawithSNR120dB,andinnon-
discreteelectromagnetictomography,suchdatawouldberegardedasverynoisy
data.TheresultsreconstructedwiththeART-likealgorithmsfromlessnoisydata
(SNR125dB)canbefoundin[47].
105Conclusions
Inthechapter,differentbinaryimagereconstructionalgorithmsarediscussedin
thecontextoftheirapplicationtoadiscreteversionofelectromagneticgeotomog-
raphywhichwecallDiscreteElectromagneticGeotomography(DEG).Allthe
presentedalgorithmsminimizetheregularizedleast-squaresobjectivefunctionin
whichtheregularizationtermisbasedontheMRFmodel.Sinceourgoalisto
findabinarysolution(oringeneraladiscreteone),theminimizationtechnique
