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August 1989
7:315
研究〉
く
Comparative Studies on Various Ilerative
lmage Reconstruction Algorithms for
EHlission Tomography
Hideo MURAYAMA*1,Eiichi TANAKA*1*2 and Norimasa NOHARA*1
ALstract: Scvcral itcrativc rccOnstruction methods apprOpriate for clnission tomography arc
frcc and
comparcd in rcgard to thc ratc of convcrgencc and ilnagc quality, by using thc noisc‐
noisy prttcCtiOn scts gcncratcd frbm a mathcmatical phantom using a computcr.Two modincd
vcrsions Of thc silnultancous itcrativc rcconstrunction tcchniquc (SIRT) mcthOd arc intro‐
duccd as thc additivc SIRT(ASIRT)and the multiplicativc SIRT(ヽ
ISIRT).ThC MSIRT
mcthod is analogous to thc itcrativc spacc rcconstruction algorithln (ISRA)。
Silnulation
studics prOvcd that the ヽISIRT Incthod has ttLstcr cOnvcrgcncc and bcttcr imagc quality than
thc ISRA mcthod.It is suggestcd that,though thc expcctation maximization(E取
[)mcthOd
givcs good cstimates for thc salnc noisy prdcctiOn set alnong all the mcthods reportcd hcrc,
thc fast hcthOds such as thc iltcrcd itcrativc rcconstruction algorithm (FIRA)and thC
COttugatc gradicnt method in weightcd vcrsion(CONGRWヽ
り arC gOOd candidatcs for practical
image rcconstruction.
INTRODUCT10N
Recently, iterativc rcconstruction methods havc rcccivcd considcrablc attcntion for thcir use
in cmission cOmputed tomography(ECT),cSpecially in positron cmission tomography(PET)
which is rapidly emerging as a vcrSatilc diagnostic tool in nuclcar medicinc.
Comparcd to
1〕
analytical reconstruction methods such as the convolution backprttcCtiOn▼ (CONVO)methOd〔
and thc Fourier translorIII Incthod
〔
2〕
, iterativc
mcthods havc advantages in that they arc
tolcrant to incomplete sampling of prtteCtiOn data, and that it is casy to incorporatc a priori
infOrmation about thc clnission distribution, such as non―
pronlising methods for a stationary PET systcHェ
ncgativity, ctc.
ThCy are also
with ine spatial rcsolution and high Sensitivity
using a three dilnensiOnal detector arrangement, in which undesirable gaps among the
dctcctor banks are inevitable Or the nnite distance between the attacent detcctor elemcnts
3,4〕
. resolution 〔
restricts thc spatial resolution of the image over the intrinsic detcctor
It has becn alI■ ost 15 ycars sincc Gilbcrt introduced thc silnultaneous intcrative Reconロ
*l Division of Physics,National lnstitutc of Radiological Scicnccs[9-1,Anagawa-4-chome, Chiba‐
shi,
260」 APAN]
*2 Hamamatu Photonics K.K.
(Rcccived Scp.2, 1988 and AcccPtCd OCt,7, 1988)
inistry
在
This work was supportcd in part by a grant froln
theヽof Education of Japan.
Key words:Imagc rcconstruction,itcrativc mcthod,cmission computed tomottraphy,pOSitron cmissio
tomography
7:316
EDICAL IMACING TECHNOLOGY VoI.7 No.3(1989)
1近
struction tcchniquc(SIRT)methOd〔
5〕, and COitein proposed the gradicnt(GRADY)
method〔
6,7〕based On iterative least squares techniques.The cottugate gradient(CON
methOd 〔
7,8〕Was alSO dCveloped in order to accelerate the rate of convergence. In the last
maximization algorithm (EM)mcthOd〔9,10〕
ive ycars, thc expectationi―
, introduced by
/ardi〔
12-31〕
Shepp andヽ
11〕
, haS bCen studied extensively by a number of invcstigators〔
becausc it is bascd on thc rcalistic assumption that photon counts lollow a Poisson proccss.
The virtues of thc EAtt method arc not only that convcrgencc to thc maxirnum likclihood
ncgativity
cstiIIlatc can bc prOvcn theoretically, but also that automatic inclusion of non―
constraints and preservation Of tOtal image cOunts in every itcration are possible.
authors〔 18,19,21,23,24〕
Since its
在mcthod have bccn dcvcloped by scvcral
convergcnce ratc is iow, modined vcrsions Of the Eヽ
.
One silnplc mcthod is to rcducc the numbcr of vicws in thc initial itcration stcps.
Another modincation is to doublc or triple the sizc of the Ettl step while still procceding in
the EM direction. Lcwitt et al〔
24〕 SuggeSted accelerated algorithHls in
18〕 and KauFman〔
relaxation
which thc changes to thc ilnages arc multiplied at cach iteration by an over‐
parameter.
They obtained about three tilnes faster speed of convergence.
approach, Tanaka〔
From a difFcrent
34〕 propOSed the altered itcrative reconstruction algorithm (FIRA)
method which is based on a modined Eヽ
在 algorithm.
This enhances the convergcnce spced
dramatically and ilnproves thc frcquency rcsponse.
method,it is well known〔
1 1,14,15,28〕that reconstructed
In spite of thc virtues of thc I E脱
ilnages with the Eヽ
江 mcthod become noisy and havc large distortions ncar the edges as
iterations prOcccd beyond a ccrtain point.
oisc and edge
In order to suppress both the ■
artifacts, ilnproved vcrsiOns of thc EMI method have recently been proposcd to include
constraints with a pcnalty function 〔 23,31〕 Or tO usc a method of sicves〔
15,16,22〕 .
在1■
ethod
based
the Eヽ
VcrklerOv and Llaccr〔
30〕intrOduccd a stopping rulc of iterations lor
on statistical hypothcsis testing.
As an altcrnative to the EM algorithm,the image space reconstruction algorithm(ISRA)
. In terIIIs of
【
uchllchncr〔
Withcrspoon and 単
method has been proposed by Daube―
92,35〕
asymptotic theory, the ISRA mcthod provides an image which is not as good as thc maximum
littclihOOd in tcrlns of precision
〔 iS advantageous since it reduces the size of largc,
33〕
, but
sparsely populatcd arrays of prtteCtiOn dattt by backprttCCting them directly into a more
compact imagc matrix.
ising tool particularly for volume
Thc ISRA method is a proH■
ilnaging, namely, three― dilnensional image reconstruction.
The purposc of this papcr is to prescnt comparisons among several iterative reconstruction
methods for both their algorithms and propcrties Of convergence, froIII computcr silnulation
studics, and to prottidc useful infOrmation to choosc a mcthod for practical usc.
mcthod as a fast algorithm is invcstigatcd, comparing with thc CONGR mcthod.
modined versions of the SIRT method, which are called ASIRT and ヽ
The FIRA
Two
狂SIFごr, are als。
introduced in this paper.
METHOD
l. Reconstrucdon methods
We assumc that a sct of l measurcmcnts yi(1≦
ヽ
i≦ I)are availablc,wherc yi is thc nuttber
August 1989
7:317
to estimate the cEliSSion
prttectiOn. The basic problem iさ
Of events counted in thethi‐
density xs(1≦
of」
j≦J)frOm the metturements,where j is a source pixel in a sctdements_
A sct of the J e・
liSSiOn densities xj dennes an image vector x. Let us denote the norllrlalized
probability that an event4 CInitted from pixel j is assigned to prttcCtiOn i by aij. Then for
each j,
Σa i 5 = 1 ・
(1)
i = 1
1n this paper, it is assumed that the efFects of scattering and attenuation of photons are
I)are
negligible, and the aij are known ttOm the geometry. A set of l elements i≦
zi(1≦
the expected values of yi, where
Z i = Σa i j X 5 。
(2)
3=1
1)GRADY method〔
6,7〕
), where lhe index n specines the
This IIIcthod generates a sequcnce of sets of estilnates ・
xjく
……
・
iteratiOn,according to the following step.For each
n=0,1,・
jE≦
J),
, and for each j(1≦
■
こ
),
(3)
)〃
1)=xj(n)十
く
xj(■
β Xj(・
・
))名
Zxj(D=Σ
i )2,
Kyi一れく
ノΣ(名j/σ
i=l
i‐
1
th iteration, and zi(■ ) is tho expectcd
where Zxj(n)is an error image at pixcl xs at thc n‐
valuc of yi, namely,
る( D = Σa i j x 5 ( D
(4)
1
j単
In eq.(3),β
)iS a damping ttctOr which is determined so as to minimize the difFerence
(・
between the measured proJectiOns yi and the estirnated proJectiOtts zi ln a least squares sensc.
The function to be lninillnizcd is
2 ( x ()め
) 2 /iび
2,
χ
■( め
= ΣK y i 一
(5)
i = 1
wherc び i is_lhe unCertainty with which yi is mcasured, and x(m)represcnts the image vector
・
).In this method,we choseび
With」 COmpOnents,xj(・
i=l for all i・
2)CONGRコ methOd〔 7,8〕
｀
This method is silnilar to the GRADY method except that the convergencc is improved
by mttking an error image vcctor in each step Of iteration orthogonal to those itt the previous
stepso while the arst step is taken to btt the same asin the GRADY method given by
eq。
(3),
the succeeding steps are given by
),
)〃
■
X j・
く
x j ( lⅢ) = x j)(十
β( ・
(6)
1)ぅ
))ais/主
(D〃
x j(n二
xj(D=童
〃
i)2_γ
(ai3j/σ
Kyi一
ィ・
i‐ l
i=1
where r(・
'iS dCtermined so as to make all steps of iteratiOtt OrthOgonal in the sense that
I
J
J
Xj(m))/び
x5(め
辞=0,
Baij〃
B(透
Baij〃
)。
(透
め
r
1‐l j=l
j=1
て
) plays
the same rolc as the d4mping ractor itt the GRADY method.
(・
fOr ttl kttmj and
β
In thl CONGR method, we chose び
i==l fOr all i.
、
デ
ヽ在EDICAL IPItACING TECHNOLOGY V01.7 No,3(1989)
7:318
め CONGRW】 methOd〔 7,8〕
This lnethod is a special casc Of the weighted least squares lnethod and is run usittg the
salne algorithlltl, as given by eq。
i is difFerent froln the
(6), CXCept that the choice Ofび
CONGR method.In the qONGRW method,び
i is given by
,
=瞥苦携
ぃ
≧
柱
4)EM】 method〔 11〕
The iteratiOn step for this methOd is givcn by
l x j ( n + 1 ) = xΣ(
j(め
aijyi/4).
(7)
i = 1
Eqllation(7)Can be rewrittcn in thc f0110wing additive form:
,
x j ( n t l ) = x)j+(〃
■x j ()・
(8)
),
= Σ (yi一
ttjXj(・
〃xs(め
孔(め
)乳3Xj(D/Σ
i=l
j=1
where cqs。
(1)and(4)are uSed to transform eq。
(7)intO eq。
(8)。
5.FIRAコ method〔 21〕
This is a method modined fronl the EM algorithm. Each iteration step is given by
n的
= C芋
ポ
;皆
;岳
_1).β
章
壬
裾
特
)苦
戦
沖
:,
Ilts, and
(9)
where αand βare constを
C j = Σ( y i t t D /)4,( ・
ヽ
i = 1
h ) s ) /( (D 孔
(yi*h)s)],
U 5 = Σ[ ( y i a i(jD(*名
)*h)s)/(Zi(・
)(yi*h)s)].
As=(xjtt )Σ[(ais(Zi(・
ρ
ヽ
=1
二
Hcre βis a small positive constant,an4 h iSpass
nlter,namely
a low―
h≡h(S)=(S3+2s2+3) 1(-10≦S≦10),
,
(10)
where sis the bin number of the prtteCtiOns in the samc vicw angle ofthe prdectiOn i.Two
c6nvolutions,(zi(・)*h)s and (yi*h)s, are pCrformed bn the sampling points on the s axis.
The total dcnsity of thc reconstructcd imagc is nOrmalized to the total numbcr or cvcnts
at every iteration. For noisy data, the correction matrix Uj fOr the high frequency component
is smoothed by a nine‐
point wcighted nlter(1:2:l fOr the X and Y directions)berOrc
=2 and β =0.8 in the
inserting it intO (9)・
eq。 In thiS Study, we used thc paramと
ters α
arst itcration, but the
β Was halved in the f01lowing itcrations.
valuC
工〔
6)ISRAコ meth。
32〕
This method uses thc following itcratiOn step,
)(め
X3(n+1)=x3(・
Byャ“う/(め
B4(DttD。
i=l
(11)
i=1
Pro」ection data are back‐proJected into inage space, and the back― proJectさ
s compared
d ilnagゃェ
to an image produced by prttectiOns calculated frOm the corrcsponding itcrative reconstrucロ
tion ilnage in each itcration.
ヽ
Thc total dcnsity Of the ilnage is norlnalizcd tO the total number of cvcnts at eヤ
￨
ery
Augtst 1989
7:319
iteratiOn.
7)ASIRT method
Wc introduce a new algorithtt givcn by
｀
埼
(nt D=MaX:杓
(1)+量
(比
_名 (D)乳
ノbi,0]!
(12)
i=i
where
bi=Σaij.
1車
(13)
1
The total density oF the ilnage is normalized to the total number of events at every
iteration.This algorithm is analogOus to thc additive fOrm of the Cilbert,s SIRT〔 5〕,WhiCh
th ray, a4d where the number
is described using components whose lettgth is that of the i“
of pixels lying along thc particular i,lh ray arc substituted for aij and bi in eq.(12).
For
this reason, the new method is called the additive simultancous iterative reconstruction
techniquc(ASIRT).
Compared with cq。
(12)iS a similar expression
(8)in tho additivc form of EM, cq。
)).In ASIRT the ttctor iS proportional
Zi(■
except for the factor multipling the residual(yi一
在the ttctOT is aijXj and its
to ais and its nOrllflalizatiOn over all the」pittels, while in Eヽ
normalizationf The rOrmer is a silnplined algebraic expression of the latter.
8)MSIRT】
method
Another new algorithHl is introduced by
あ
1■) 圭
(/ Dめa i j / b i 〉
x j (■
x 5 () ・
B ( y i a i j / bBi()■
i‐ l
where bi is given by eq。
age is nOrlnalized to the total
(13)and tOtal density Of the il■
number oF events at every iterあ tion,
(14)
i=1
This algorithHl is analogous to the multiplicative form
Of Gllbert's SIRT〔
5〕
, and it iS thcrefore called the multiplicative simultaneous iterative
reconstruction technique(MSIRT).
Equation(14)iS a similar algebraic Cttpression to (11)in
eq。 lsRA.At the backprdectiOn
h order
to correct the total de,cctiOn prObability
RT uses a dividing factor, bi,
ュ
stage, MSI―
f o r e a c h i ‐t h p r t t e C t i O n , b u t I S R A n e g l e l t s t h i S f a c t o r .
This algottthIIl is derivcd by
etween yi and zi as in〔
. The quantity
squares distance Hleasuresむ
33〕
miniIIlizing the least‐
to be minimized with respect to the(X3)iS
i乳
一
主w i ( yj=1生
jxj)2,
i=l
(15)
matrix
where the wi's are weighting ttctors.If we denote the vector(yi)by y,
thc lx」
], then this can be written as
(4i3)by [A], and the diagOnal matrix (wi)by Ⅳ
(y一 [A]x)TEW](yr[A]主
),
(16)
where T denotes transposc. i
Th1 least squares estimates,″長 of x are Obtained by solving
[A]TEW][A]食
=[A]TEW]y
(17)
in(17)iS
Written
The link with the MSIRT ctt bc made by notingithat the j_th
cquation
a : . X r ) =yΣ
i4isWi;
Σa :r=lメ
i(Σ
i‐
i=l
1
(18)
7:320
3(1989)
1MEDICAL IMACING TECHNOLOGY Vol.7 No。
I
I
J
l = Σ ( y i a i 5 w i () Σ
/ (aΣi r X r ) a i s W i ) .
i=l
(19)
i=l r=1
ヽ在ultiplicatiOn Of bOth sides by xj provides thc following equation
=
yiaijwi)/(Σ
埼(ntl)=X3(D(Σ
孔(D aijWi)・
i=l
(20)
i=1
The weighting factor(wi)ShOuld be chosen to be about thc same sizc fOr all thc cOmponcnts
in cxprcssiOn (15).
If all the components of x arc equal, thcn wc should choose
a対
Wi=1/Σ
。
(21)
j=1
This introduces thc ヽ
在SIRT algorithm. Thc abOve assumption is reasonable when wc ch00sc
a unilbrHlly distributcd source for the initial gucss of x. Thc ISRA algorithFn iS alsO derived
from the samc equation,(20),for the case that wi=l for all i.
2. Simulated data
The mathe甲
住atical silnulation phantom as shown in Fig。
Fig。 l B shows thc clnission dcnsity distribution.
l A was gencratcd by computcr.
■is digitized in
The mathematical phanto■
a64、 64 matrix and thc clnission dcnsity xs is unifrom within a square pixcl j.Wc rcstr
the reconstructiOn region to a circle inscribcd in a 64×
64 Square, and chose cquispaced
prtteCtiOns in which rays are prarallcl lincs with uniforIII spacing of thc salne side of a pixel
frec imagc was preparcd for the silnulation study. Corrcsponding to thc
A `ttruc''noisc―
image vectOr T, a set of nOischfrcc prttectiOn data yl's was calculated analytically for 240
i≦I),
from the following cquation.FOr cach i(1≦
cqually spaced anglcs over 180°
Y i = Σt t j T j ,
(22)
j=1
w h c r e T j i s t hleh jC―
O m p o n e n t o f t h e t r u c i l n a g e v c c t o r T , J = 3 2 2 8 2a4n0d. AI =s6e8t×
numbers froIIl a
Of noisy prttcctiOn data was also obtaincd by generating randon■
pscudo―
Poisson distribution with a mean cqual to the noisc_frec valuc of Yi.
In the following silnulation study, wc uscd the threc sets of prtteCtiOn data based on the
same phantoHュ : the nrst was the noisc―frce data with a tOtal count of l x 106, the Second was
thc noisy data with a total cOunt of l×
count of 2×105.
106, and thc third was the nOisy data with a total
Logan convolution method 1〕
〔, thC reconstructed
By uSing the Shepp―
ilnages shown in Figs. l C, lD and E wcrc obtaincd form thc arst, sccond and third
prtteCtiOn data scts, respectively.
In this study, ali thc iterative reconstruction procedures were started from a unilbrln disk,
namely thc initial set of estimates xj(0)is givcn by
I
杓( 0 ) = Σy i / J , f O r a l l j .
(23)
i‐ 1
No constraints Or a priori information were used in any Of thc rcconstruction mcthOds in
order to clcarly show dinお
rences bctween the algorithms.
Three functiOns were dcancd as measurcs of thc goodness of thc cstimatcd imagc.
Thc
nrst was the mean absolute error, m(X), Of thC image doaned by
m(X)=童
j=1
I X j 一島 1 / 主 T j .
3=1
(24)
Thc sccond was the squarc of residual nOrIIl, N2(x), Of thC prttcctiOn data deflncd by
｀￨
August 1989
7: 321
守( 5 )
C
２
３
讐( 0 )
３
一
A
D
B
Fig。 l ThC Inathclnatical phantom uscd in thc simulation(A), and itS imagc(B)・
The valucs in front of thC parcnthcscs arc diamctcrs Or distanccs in units of
pixel width, and thc valucs in parcnthescs are rclativc cnlission dcnsitics.
IInagcs C, D and E arc rcconstructed with thc convolutiOn backprttcctiOn
oF prdectiOn data;thc noisc‐frcc data,
(CONVO)mcthOd for thc three scぃ
106 and 2 x 105,
and thc two noisy data scts with thc total counts of l×
rcspectivcly.
Thcse imagcs arc normalizcd by their total coun体
.
N2(x)=Σ
Kyi一
つ2.
(25)
i=1
The third was the log likelihood function l(x), dCancd by
1(X)=Σ (yi 10g名一名)・
(26)
Thc function m(x)providcs a measure of``source convergencc",N2(x)prOVidCs a mcasure
of``prttcctiOn convergence"in the sensc of lncan square error and l(x)indiCates how closely
thc rcconstructcd ilnage approaches thc maxilnum likclihood cstimatce
Silnulation studies
allow thc cvaluation of m(X)by the direct comparison of reconstructed imagё
sourcc distribution, but in practicc, thc other two functions only arc ｀
true imagc is usually unknown.
with the truc
availablc bccausc thc
3(1989)
MEDICAL IMACING TECHNOLOGY Vol.7 No。
7:322
︵Ｃ︶ＸＹＥ ” 任０任Ｅロロ，⊃ヨ０の０＜Ｚ＜凹〓
︵
0
5
1o
15
20
40
60
oo
loo
NUMBER OF ITERAT10NS i n
F i g . 2 P 1 0 t S
o f
t h c
m c a n )a )b as So l au t fc u nc cr tr io or n
itcration numbcr n,for thc noisc‐frcc prttectiOn sct. Thc point
markcd on thc ordinate shows thc valuc of m(x)for thO
m o( Fx ( t・h c
CONVO imagc.
RESULTS AND DISCUSS10N
free data
l. Convergence for ncise口
First of all,reconstruction was perforIIIcd using noisc― free prttectiOn set to delnOnstrate the
dcpcndcncc of ratc of convergencc on the reconstruction method.
・2 ShOWS plots of thc
F七
mcan absolute crror m(x(n))aS a function of the numbcr of interations n,for thc nrst 100
iterations with the cight difrcrent methods.The valuc of m(x)for thC CONVO image
(Fig。 lC)iS l■ arkcd by ``x"on the ordinate of Fig.2.
PIots of the squarc of rcsidual norェ ■N2(x(■))and the 10g likelihood l(x(n))are ShOWn in
Figs. 3 and 4, respectivcly.
m(X(n)) and N2(x(■
It can be sccn that, as the numbcr of iteration increascs,
))decrease monotonically, and l(x(・
))inCrcases monofoniCally.
shows thc rcconstructed ilnages at 20 itCratiOns with all mcthods.
Fig. 5
As comparcd with thc
CONVO image,FIRA,CONGRW and CONGR yicld superior images,while GRADY and
Eヽ江provide allnOst the samc ilnage quality, and ASIRT, MSIFピ
iIIlages,
r and lsRA provide inferiOr
We found that by the loo、 th iteration thc images of all methods are close enough
to the truc image shown in F七
・1■ ・
In ordcr tb evaluate the ratc of convergence quantitativcly, we introduced thrcc CONVO
cquivalent iteration numbcrs n(m), m(N2)and n(1), WhiCh give the closest reconstruction
ilnages to the CONVO ilnage by lneasurcs of the]■
norm and the log likelihOOd, respectively.
ean absolute error, the square of residual
Table l lists the CC)NVO equivalent iteration
numbers fOr all the methods reported here. The fastest convergcnce was obtained with FIRA
and the sccond ttstest with CONGRW. The Slowest is ISRA.The Eヽ
than ASIRT, whilc ISRA is slower than MSIRT.
｀
在 methOd is faster
t
Compared with GRADY, EM is infcrior in terms of both n(m)and n(N2), but Supcrior
7:323
小uguSt 1989
将︶ＮＺ車︵〓αＯＺコ＜ＤＯ︻
︵︵ｃ︼
の回任︶
0
5
10
15
20
40 60
NUMBER OF ITERAT10NS: n
orm N2(x(■
Fig.8 P10tS of the square 6f residual■
80
100
))as a funcdOn of
the iteration number n for the noiscttfrce prdeCtiOn set.The
っ fOr the
point marked on the ordinate shows the vttuc of N2ぐ
CONVO image.
3■5
―free
t x 10V counts(nois●
３・
ヱ３。
ヨ叫︼︻コＯＯョ
︵ｏＯ一Ｘ︶︵︵ＥＸ︶一いＯＯＯ工︻
︶
CONVO
FIRA
CONGRW
CONeR
3,25
0
5
10
15
20
40
60
80
100
NUMBER OF iTERAT10NS i n
))aS a function of thc iteraton
Hge 4 P10tS Of the log likellh00d X(・
lぐ
d On
numbcr n fOr the noise"frcc pttj,cdOn set.The point ttarkゃ
the ordinatc shows thc value ofo l(】
fOr the CONVO imange.
in terms of n(1)s the value of■
(1)is abOut a half of n(N2)fOr EM, but n(1) iS nearly
d to
equal to n(N2)for GRADY. This Fenccts the fact that the EM algorithm is designも
pursuc■ he車
牢 lmutt likeliho側
Ⅲo車lmぞtel‐Thatぉ ,,EM hcorporates larger,wl七
理 atchilg^rOf・ pr弱ゃctlon,data withす
N2(→
1。
w coutttS as_comparゃ
is afFected by the difFerencP tttween mPaSuredあ
weight as given by cq。
htS・On the
d with GRADY.The value of
nd estⅢ ated pttCttiOIIS With equal
(25), Whil,the matching of th1 10w count data Seriously agLcts the
MEDICAL IMAGING TECHNOLOGY Vol.7 No.3(1989)
7:324
江 (E), ASIRT
(C), GRADY(D), Eヽ
ｈ
Rcconstructcd ilnagcs at 20 itcrations
ｔヽ
九
．
ｉ
ＷＣ
Fig. 5
FIRA(A), CONGRW (B), CONGR
MSIRT(G)and ISRA(H), fOr thc
nOisc―
frcc prttcCtiOn sct.
Table l
ComparisOn of thc CC)Nヽ
/O cquivalcnt itcratiOn
numbcrs n(m),n(N2)and n(1),WhCrc thc rccO‐
nstructed ilnagc is closest to thc CONヽ
!
/O ilnagc
in Fig。 lC, by measure of thc mcan absolutc
crror, the squarc of residual norm and the log
止
kelihood.
CC)Nヽア
O equivalcnt itcration number
ヽIcthod
n(m)
n(N2)
n ( 1 )
FIRA
2
3
2
CONGRW
6
6
4
CONGR
8
6
6
GRADY
25
21
20
Eヽ1
31
35
19
ASIRT
46
48
35
取ISIRT
62
57
>100
ISRA
82
86
>100
valuc of n(1)duc tO the terms of
lbg zi in cq.(26).
Fbrithe same r9ason, bOth MSI]RT
and ISRA prOvide ttuch larger
valucs Of n(1)than
thc valucs of n(m)and n(N2)when
compared to Eヽ在.
7=325
August 1989
。
l x 106と
unts(nOisy)
ISRA
ASiRT
MSlRT
EM
CRADY
と_宅ダ毛茅
‐
: 二
10
15
20
才
40
60
６
０
ぱ︲
一
二伍ＯＺコ＜Ｄａ中の口 α ︶
︵ＣＸ︶Ｚい
田︵
側
︵︶
６。
４・
２・
０００
Ｘ︶Ｅ ”ＥＯαα回Ｗ卜Ｄコ０”口くＺ＜口 Σ
︻︵Ｅ︶
l x106 counts(noに
y)
ISRA
ASIRT
MSIRT
EM
GRADV
80
5
NUMBER OF ITERAT10NS i n
Ｘ︶一
︵。０一Ｘ︶︵︵
“
ａＯＯＩ︻
ヨＯＯヨ
ヨロ支中
Ｃ︶
孝
80
ー
ザ
戸
無恵
0
l
10
15
20
40
60
NUMBER OF iTERAT10NS i n
5
10
15
、
20
40
60
80
100
NUMBER OF ITERATiONS:n
‐c
Fig.6 P10tS of(A)thc mean absolute frror m(xく
N2(x(■)), and(C)thC 10g likelihood l(x(・
n)),(B)the Square of rcsidual norm
2),rcspectively,as a functio■ oF itera‐
ムtion numbcr n for the noisy prttcctiOn sct with the total count of l x 106.
2, COnvergente of slow methOds fOr■
oisy data
ln this section, we shall be concerned with the nve slow methods of GRADY, EM,
ASIRT,MSIRT and ISRA only,in order to avOid complexity in the comparativゃ
The others,FIRA,CONGRW ttld CONGR,will be discussed in thと
discttssion.
ncxt section.Inl order
of convergence, the reconstruction w4s
to cxalline the efFects of statistical nbisc on the ratё
perforIIled fOr the nOisy prdeCtiOn set with a total count of l x 106. Ftts・
￨IA,■
,and C
))
)), the square Of thl res車
立響 ■Ortt N?(x(・
show plots of the mean absolute cttor h(X(・
and the log likelihood l(X(・
)), iespectively, forithe arst 100 iterations. irhe valu●
s oF In(ま ),
MEDICAL Iヽ
7:‐326
在
ACING TECHNOLOGY Vol.7 Noi 3(1989)
．
．
，
０００
oisy)
Σ
2 2
コ＜ＤＯ中
の口伍︶
¨
Ｘと
︵︵Ｅ︶
に伍〇区伍回ロト⊃コ０の口＜Ｚ＜口Σ
Ｘ︶Ｚ
︵ゆ〇一Ｘ︶︵︵
Ｅ︶
Ｎ
2x105counts(い
5
10
15
20
40
60
NUMBER OF iTERAT10NS i n
80
5
100
10
15
20
40
60
NUMBER OF lTERAT10NS i n
80
100
2x105counts(noisy)
Ｘ︶一 ”
︵Ｏ一Ｘ︶
００Ｉ︻
コロ支中
コＯＯＨ
ｏ
ｒＣ︶
℃
/CONY0
″有
5
10
15
20
40
60
80
orm
Fig.7 P10tS of(A)the mean absolutl crrOr五
(x(・)),(B)thC Squarc of residual■
N2(x(■)), and (C)th1 16g likclihood l(x(n)), r6Spectively, as a:unctiott of iteraⅢ
,tiOn number n for the noisy prttectiOn set with thも
total count of 2 x 105.
N2(x)and l(X)fOr the CoNVO image(Fig。
lD)are marked by`牧
''on the ordittateslof
the corresponding p16tS.
Although theヤ alucs or m(x(・))各 pproach zerb For noise‐ free data, Figt 6 A demonstrates
altes to inlrcase after a
that noisc in the proJlctiOn data prevents this attd causes thcず
certain humber Of iteralions regardlも轟 tO th● recontructiott method.
By 16ntrast, the ミミ ueS
))in Fig1 6 C idO not clangさ
))i■ 副ヒ・6■ an卓 =(X(・
(・
theiF血6■otonicあ
bf N2(土
!behaⅣ iOr,
))4aVe almost thettame shape as
r t6 the plots or Fig.3 attd 4.The plむ
ts of l(x(・
立m工今
the plots of F七
・4, ThiS implieS that the relat市
e valdes of l(主
))are insensitive tO
(五
ｋｌｌ、ど、，，，ｉ，１︰，・
ヽｒトーｒヽ１ユＩＡｌｌｆｌムーー、４ヽ
﹂可１，１ヽ１１１１ヽ
NUMBER OF ITERAT10NS i n
―
c
7:327
August 1989
ASIRT
E M
CRADY
n=8
n=20
C2
n=100
C3
Fig.8 Rcconstructcd imagcs at 8, 20 and 100 iterations with GRADY(Al,A2,A3),
Eヽ江 (Bl,B2,B3),and ASIRT(Cl,C2,C3),for thC nOisy prttcctiOn sct with
the total count of 2×105. Thc top, Iliddlc and bottoln imagcs wcre obtaincd at
itcration numbcrs of n=8, 20 and 100, rcspcctivcly.
cxistcncc of noisc in thc prttCCtiOn data.
To demonstratc a noisiCr situation, wc ran the methods for 100 iterations, using another
)),N2(x(■ ))
105,Thc rcsulting plots of m(x(・
nOisy prtteCtiOn sct with the total count of 2×
and l(X(・ ))are shown in Figs.7A,B and C, rcspcctively. For all the IIlethods, the
))in Fig.7 A arc obtained at a smallcr numbcr of iterations than
minimum valucs of m(x(・
in Fig.6A. The monotonical bchavior of the functions N2(x(n))and l(X(n))dOCS nOt
change in Figs.7B and C.
Fig. 8 illustratcs thc reconstructcd ilnagcs obtaincd froHl the noisy pro」cction set with 2×
105 cOunts at 8, 20 and 100 itCrations with the GRADY,Ettl and ASIRT methods.All the
eye view ilnagcs at 100 iteratiOns demonstratc ``noisc artifacts", and the reconstructed
bird's―
ilnages difFcr frOm thc true irnage at high iteration numbers.
that both the functions l(x)and 卜
From this, it is apparcnt
r2(x)are not good mcasurcs to Obtain desirablc rccons‐
truction with noisy data.
Outside the boundary of thc phantom in Fig. 8, GRADY intrOduccs noisc enhanccment
cvcn at low iteratiOns, duc to thc lack of non― negativity constraints, and ASIRT produccs
dot-like artifacts duc to the non―
lincar operation carricd out in Order to remOVe n♀
ilnage density. E脱I produces much lcss noise duc to thl automatic non‐
Onc remarkablc result from Figs. 6A and 7A is that Eヽ
gatiVe
negativity constraints.
狂 provides thc slnallcst valuc of
m(X)among all thc mcthods for ,he same noisy prttcctiOn scte ThC minimum valuc of
m(X(・
))fOr EM is O。
129 at n=28 fOr l×
106 cOunts,and O。
188 at n=16 fOr 2×
105 cOunts.
7:328
NEIEDICAL
IMAGING TECHNOLOGY V01,7 No。
l x 106 countS
(n=23)
3(1989)
2x105 countS
( n = 1 6 )
E M
A
B
Fig。9 Reconstructcd imagcs at 28 itcratiOns with Eヽ在for the noisy prttcctiOn set with
thc total count of l x lo6(A), and at 16 itcratiOns with 2× 105(B).
ISRA(n=20)
MSIRT(n=20)
x106
counts
2x105
counts
Fig。lo Reconstructcd imagcs at 20 itcrations with ISRA(Al,A2),and MSIRT(Bl,
B2)。 Thc tOp and bottOm imagcs wcrc obtaincd by using thc ■
Oisy prdectiOn
scts with total cOunts of l x 106 and 2 x 105, rCSpcctively.
The corresponding reconstructed images arc shown in Figs。
9A a■ dB.
F七・10 illustratcs the rccOnstructcd images at 20 iterations with ISRA and MSIRT for
the noisy proJcctiOn scts with tOtal cOunts Of l× 106 and 2× 105.
The ttISIFご r method has
allnost the same recovery rate of spatial resOlution and the same noise propcrties as ISRA.
Thc only difrerence between the two‐
methOds is that, Outside the ottcct,MSIRT has faster
recovery than ISRA.
This diSLrence gives an insight intO the reason why the ■
linimum
valuc of m(x)fOr MISIRT is smallcr tlan that for ISRA,as shottn in both Figs.6A and
7A.
Thus, we can expect that ヽ
ISIFごr provides bettcr images as compared with ISRA for
7:329
August 1989
4
6
8
10
12
14
16
//
２・
４・
い００
Ｘ〓岳 ”
にＯ任伍ロロトコョＯの口＜Ｚ＜ｕ Σ
︵︵Ｃ︶
６・
０ “ 硬
一
︵︵Ｅ︺Ｘ︶Ｅ ” 任〇伍任ロロトＤヨ〇切的くＺ＜口夏
V0
0
0
′
c′
o′u n t s ( n o i s y )
す
/′
2 x 1 0 5
counts(noisy)
2
4
6
8
12
14
16
B
A
)) as a Ftlnctlon of itcratlon numbcr n for thc
Plots of thc mcan absolutc crror m(x(・
and 2 x 105(B).
×1 0 6 ( A )of
noisy prttcCtiOn scts with tOtal lcounts
FIRA
CONGRW
CONGR
n=2
n=4
n=6
A3
Fig. 12
18
NUMBER OF ITERAT10NS i n
NUMBER OF ITERATIONS i n
Fig.11
10
18
C3
B3
Reconstructcd images at 2, 4 and 6 iterations
CONGRW (Bl,B2,B3)and FIRA(Cl,C2,
sct with thc total count of l×
106. Thc top,
obtained at itcration numbcrs of n=2, 4 and 6,
with CONGR(Al,A2,A3),
C3), for thC nOisy iprdCCtiOn
lniddlc and botton■ imagcs wcrc
rcspcctivcly.
the samc noisy prttcctiOn set.
3. COnvergence of the fast methods for ncisy data
are comparcd.
d
l n t h i s s c c t i o n , t h e t h sr te e l nf cあt h o d S O f F I R A , C O N G R t t V a n CONGR
7:330
ヽ在EDICAL I脱
CONeR
IAGING TECHNOLOGY Vol.7 No.3(1989)
FIRA
CONGRW
n=2
n=4
A2
C2
n〓 6
F 二g . 1 3 R e c o n s t r u c t c d i m a g e s a t 2 , 4 a n d 6 i t e r a t i O n s w i t h C O N G R ( A l , A 2 , A 3 ) ,
CONGRW (Bl,B2,B3)and FIRA(Cl,C2,C3),using the noisy prtteCtiOn
sct with a total count Of 2 x 105.
The tOp, Iniddlc and bottOm ilnagcs wcrc
obtaincd at itcratiOn numbcrs of n=2, 4 and 6, rcspectivcly.
))fOr the nrst 18 iterations
Figs.1l A and B shOw p10ts Of thc mean absolute crrOr m(x(・
using the noisy prttectiOn scts with thc tOtal cOunts of l×
106 and 2× 105, reSpectivcly.
These plots demonstratc the &LSt COnvergencc of FIRA for thc noisy data as wcll as for the
noise_frce data.
This is bccause the frequcncy responsc of thc iterative correctiOn is
signincantly ilnprovcd by incOrporating adequatc high frcquency cnhancPmCnt in the FIRA
rcconstruction.A suitable number Of itcratiOns lnay be froln 2 to 4. In addition, FIRA
prbved to be tt10re stable,based On the valucs of ttL(X(・
valuc had becn rcached, comparcd with CONGR and CONGRW.
)),WhCn itCrating aftcr the IIlinimum
This is becausc FIRA
uses a selective nlter functiOn such as eq。(10), With which the cOnvergencc of each frequency
component is cOntrollcd.
Figs. 12 and 13 11lustrate the recOnstructed ilnages at 2, 4 and 6 iteratiOns with the threc
mcthods for the twO noisy prtteCtiOn scts.It can bc sccn that FIRA prOduccs better images
at a few iterations.
Thc imagc Of C l with FIRA at n==2 provides a silnilar recovery of
res01ution as the image OF B 3 with CONGRW at n==6, While thc fOrmer ilnagc produccs
less noisc than the lattcr. This is becausc thc autOmatic non_negativity constraints involved in
FIRA efFectively suppress the nOisc enhancement in areas Of 10w image density, in a silnilar
way to Eヽ在.
From thc abOve cOmparison, we conclude that FIRA is a good candidate for
the image reconstruction of ECT data, in practice古
From Figs.1lA and B,it is scen that CONGRW has a smallcr minimum valuc Of m
))than CONGR, while thc mhimum valucs for both methods are obtained at almost
(X(・
August 1989
7:331
the same iteration number.
CONGRW in Figs。
A cOmparison of the rcconstructed ilnages with CONGR and
12 and 13 ShOWS that CONGRW is superior tO CONGR due to the
suppression Of the noisc in arcas outside thc ottect.ThiSis becausc CONGRW achieves better
matching of prtteCtiOn data with low counts than that with high lounts by virtuc Of thc
weighted lcast squares, with which thc low counts arc assigned largc weights to thc minilni‐
zing functiOn given by cq。
(5).
The image quality for the fast methods is not pariticularly worse than that for the slow
))With CONGRW is o.130 fOr
methods.From Fig.1l A,thc minimum valuc of m(X(・
129)with Eヽ
l× 106 cOunts, which is very c10se to thc corresponding valuc(o。
宝 from Fig.
6A.The CONGRW method secms to be practical because the minimum valuc of m(x(・
つ
is achicved at n==5 iteratiOns, which is about 6 times faster compared with n==28 iteratiOns
with EM.For 2×
105 cOunts, however, the minimum valuc of m(x(・
))With CONGRW
質shown in Fig。
becomes iarger than Eヽ在.
Compared with thc imagc of E取
of CONGRW in Fig.13■
2 iS inferior especially in areas of low image density.This is duc
9B, the imagc
tO thc cxccsS enhancement of thc high― frequency component of statistical noisc, although the
reconstructed ilnages are sharpened too much at early iterations.
For the FIRA method,
we have the frcedoIII to select the paralnetcrs
aS tO
,β and theαnlter h(s)in (9),SO
eq。
minimize degradation of the image quality.Further studies on FIRA will bc reported
elscwhcrc〔
.
34〕
CONCLUS10N
For a noisc― free prdcctiOn sct, all the methods reported here prOvide stable convegcnce;
that is, the three functions, m(x), N2(x)and l(X), approach their inal valucs monotonically
as the itcrations procced.Thc thrcc CONVO equivalcnt itcration numbers,n(m),n(N2)and
n(1)WCrc used to evaluatc thc rate of convergcnce.
From these quanfitative comparisons,
the fastcst convergence can be obtained with thc FIRA mcthod.
Discrepancies in the values of n(N2)and n(1)arC ObServed with some mcthodsi namely,
FIRA,CONGRW,EM and ASIRT givc smaller valucs of n(1)than n(N2),while MsIRT
and ISRA give larger values of n(1)・
ThiS iS beCause thc former nicthods incorporate larger
weights fOr matching low count prttCCtiOns than for high count prttectiOttS,comparcd to thc
latter methods.
The formcr mcthods arc superior in tcrlns of noiso enhanccmcnt in the arca
outsidc the ottcct,
Wc intrOduced theヽ
在SIRT mcthod as a candidate for lhe recOnstruction of volume images.
ed algorithm of Gilbert's SIRT and has a similar algebraic expression to ISRA.
This is a modif■
It was shown that ヽ
在SIRT provides fastcr convcrgence and bcttcr imagc quality than ISRA,
CSPCCially in the area outsidc the ottect.
In practical usc, the FIRA mcthOd is a good choicc due to its fastcst convcrgcnce, its iow
nOisc On the images with the automatic non―
negativity constraints, and comparativcly stabic
image quality for iterations after thc optimunl at 2-4 iteratiOns.
is also useful in sOme practical cases because it provides adeq■
numbers.
The CONGRW ttethod
ate images at several iteratiOn
脱IEDICAL Iヽ
7:332
在AGING TECHNOLOGY Vol.7 No.3(1989)
ACKNOWLEDGEMENT
在ctroロ
The authors wish to thank T,Tollitani, M.Yamamoto of NIRS and H. Toyama of irokyo 、
politan lnstitutc of Gcrcntology for thcir uscful discussions. Vc予 wish to cxprcss apprcciation to K,
Fukuhisa of NIRS for his tcchnical assitancc.
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_
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〔
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