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30
Binarytomographicimagereconstruction000
c(GR)≃S[(5N2)m/d+(5N2)a/s+(MN)m/d+(MN)a/s+
+(43N)m/d+(27N)a/s+(16N)f].
ForthefunctionV(HL)(xn,δ)andV(BS)(xn,p),thecostsdecreasebyS(8N)f
andS((8N)m/d+(8N)a/s),respectively.AssumingM1N,V(HL)(xn,δ),and
skippingthetypeofoperations,weneedabout12N2+78Noperationstoperform
onestepoftheouterloopintheAlgorithm3,whereastheAlgorithm2withtypical
K110requiresnearly20N2+560Noperationstodothesamestep.Thus,the
Algorithm3hasalowercomputationalcost.
TheAlgorithm3givesabinarysolution,however,aslightmodificationcanbe
donetomakethisalgorithmrecoveramulti-discretesolution.LetΘ1{θ1,...,θL}
beasetofLdiscretevalues,andlet∀j∈{1,...,N}:xj∈Θ.
Thesemodificationscanbeeasilyintroducedtotheequation(1.37).Thus:
xj(T)1
∑x
∑x
j∈Θxjexp{−1
j∈Θexp{−1
TF(xj)}
TF(xj)}
.
(1.41)
Theorem1canbealsoappliedto(1.41),whichgiveslimT→0⟨xj⟩|P
F(xj)→x∗
j.
TheproofforthisissimilarasforTheorem1.
AssumingthediscretevalueswrittenonP+1bits,andΘ1{θ010,...,θl1
l/2P,...,θL11},whereL12P,wehave:
xj(T)1
2P∑L
∑
L
l11lexp{−1
l10exp{−1
TF(xj1l)}
TF(xj1l)}
.
(1.42)
Unfortunately,theimplementationof(1.42)isnotsosimpleasinabinary
case(Algorithm3).
10303BinarySteeringofProjectedGradientAlgorithm
Theprojectedgradientmethod[2]findsastationarypointx∗∈X⊆IRNforthe
problem:
x∗1argmin
x
F(x),
s.t.x∈X,
withthefollowingiterativeformula:
x(k+1)1P[x(k)−I(k)∇xF(x)|
x1x(k)],
fork10,1,...
,
(1.43)
(1.44)
