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Chapter1
Binarytomographicimage
reconstructionfromlimited-angle
projections
RafałZdunek,
WroclawUniversityofTechnology,
InstituteofTelecommunications,TeleinformaticsandAcoustics,
WybrzezeWyspianskiego27,50-370Wroclaw,Poland,
e-mail:rafal.zdunek@pwr.wroc.pl
Thischapterisconcernedwithimagereconstructionsmethodsthatarein-
tendedforreconstructingbinaryimagesfromlimited-angleprojections.Themeth-
odsaremotivatedbycross-boreholeimaging,however,ourconsiderationscanbe
alsousefulforotherapplicationsofbinarytomography.Amongseveralobjec-
tivefunctionsthatarewell-knowninbinarytomography,wefocussontheleast-
squaresobjectivefunctionpenalizedwiththesmoothnesstermderivedfromthe
Markovrandomfieldmodel.Wepresentafewstrategiesthatmaybeusedfor
minimizationoftheobjectivefunction.Oneofthemincorporatesanadditional
non-convextermintotheobjectivefunctiontoenforcebinarity,andthentheD.C.
algorithmisappliedtominimizethedistancebetweentwoconvexfunctions.In
thesecondone,themean-fieldreferencedistributionandmean-fieldannealing
areappliedtoestimatetheglobalmaximizeroftheGibbs-Boltzmanndistribution
associatedwiththeMRFpenalizedobjectivefunction.Finally,wediscussthe
algorithmthatappliesabinarysteeringtotheprojectedgradientalgorithm.Very
efficientimplementationsofthealgorithmsarealsogiven.Thenumericalresults
arepresentedfornoise-freeandnoisydatatakenfrombinaryboreholetomogra-
phy.
