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16
Binarytomographicimagereconstruction000
tionsindata,andusuallytheycanbeimplementedveryefficiently.
Thelayoutofthetheremainderofthischapterisasfollows:Thenextsection
shortlydescribesthemodelofimagereconstructioninDEG.Inthethirdsection,
somealgorithmsfrombinarytomography,whichmaybeusefulinourapplication,
aresurveyed.Wedrawaparticularattenuationtopenalizedreconstructionwith
theMarkovRandomField(MRF).Thissectionalsocontainsourpropositions
foradaptingwell-knownalgorithmstoDEGaswellasthetipsontheirefficient
implementation.Thenumericalresultsarepresentedintheforthsection.Finally,
thefifthsectioncontainssomeconclusions.
102Model
Weassumeadiscreteapproximationoftheforwardprojectionmodel,i.e.:
N
∑
aij˜
˜
xj+ni1˜
bi,
i11,...,M,
j11
(1.1)
whereMisanumberofray-paths,Nisanumberofpixelsintheimage,and
usuallyM≥N,˜
xjisanattenuationofthej-pixel,˜
aijisacontributionofthei-th
ray-pathtothej-thpixel,niisnoisyperturbationofattenuation˜
bimeasuredalong
thei-thray-path.Inamatrixnotation,wehave:
A˜
˜
x+n1˜
b,
where
A∈IRM×N,˜
˜
x∈IRN,n∈IRM,˜
b∈IRM.
(1.2)
10201Rank-deficiencyinsystemmatrices
Toshowthemotivationfortheusageofdiscretetomographyinourapplication,
firstwebrieflydiscusstheoriginofthelimited-angleimagereconstructionprob-
lem.Letusconsiderthesimplestcase,i.e.thesurveyedareaisdividedintofour
regularpixelsasshowninFig.1.2.
Figure102:Ray-coveragemodeloftheareadividedinto2by2pixels
