EM Reconstruction Algorithms for Emission and ...

Journal of Compuler Assisted Tomography 8(句:306-316， ApriI @ 1984 Rav巳n Press , New York

EM Reconstruction Algorithms for Emission and Transmission Tomography Kenneth Lange and Richard Carson

$

Abstract: Two proposed likelihood models for emission and transmission image reconstruction accurately incorporate the Poisson nature of photon counting noise and a number of other relevant physical features. As in most algebraic schemes , the region to be reconstructed is divi由d into small pixels. For each pixel a concentration or attenuation coe主Ícient must bεesti岳ated. In the maximum like1ihood approach these parameters are estima主ed by maximizing the 1ikelihood (probability of the observations). EM algorithms are iterative techniques for findíng maximum likelihood esti韶ates 穆 In this paper we discuss the general principles behind all EM algorithms and 过erive in detail the specific aIgorithms for emission and transmission tOP1ography.τhev挝u毯 。f the EM algorithms include (a) accurate incorporation of a good p主ysical model , (b) automatic inclusion of non-negatìvity constraints on all parameters , (c) anexcellent measure of the quality of a reconstruction , and 括)主10己al convergence to a single vector of parameter estimates. 暂e~discuss the specification of necessary physical features such as source and detector geometries. Actual reconstructions are deferred to a later time. Index Terms: Image reconstruction-Emission computed tomography - Maximum likelihood-EM algorithm.

.. -

e

~

imum likelihood approach. 有hat t主ey lacked was a feasible method for computing maximum Iikelihood estimates for pro桂ems 有i自 thousands of parameters to estimate. The EM algorithms discussed here supply the missing 呈n主 in the 理aximum likelihood approach. Our optimism a色。ut maxi垣um likelihood methods stems from a number of considerations. First妻 maximum Iikelihood has exce主ent theoretical properties (5) and an excellent 'track re己ord in diverse application arεas. Secon在 the maximum likelihood approach all。有s modeling of the physics of emission and transmission in a realistic 暂ay. A key modeling assumption concerning the Poisson nature of photon counting 主ts readily into the maximum likelihood framework. This is not true for most 单­ gebraic methods. Furthermo窍， the physical 基iffer­ ences between transmission an注 εmission can 色e carefully maintained. Some emission reconstruction methods blur these differences in an attempt to carry over wholesale the techniques of transmission tomography. The combination of a good statistical model and a superior physical 黯odel sho曰:注 pro­ duce superior reconstructions. The emission simu-

Image reconstruction methods fall broadly into one of two categories (1, 2). Fourier methods approximate explicit deterministic inversion formulas for reconstructing a function from its line integrals. On the otherhand, algebraic reconstruction techniques can capture' stochastic variation in photon counts and, in theory, yield more accurate recon.:... structions or provide equivalent reconstructions with lower patient radiation dose. Because algebraic techniques are usually iterative, and therefore slower, the single step Fourier-based methods are generally preferred in practice. With the advent of more powerful computers, the arguments favoring algebraic reconstruction become more' compelling. In this paper we present two new reconstruction algorithms based on maximum likelihood techniques. [One of these, the emission algorithm , was derived independently by Shepp and Vardi (3).J Rockmore and Macovski (.啡， among others , recognized the value of the maxFrom the Department of Biomathematics. University of California at Lo s Angeles , School of Medicine , Los Angeles , CA 90024. Address correspondence and reprint requests to Dr. Lange. 306

307

EM RECONSTRUCTION ALGORITHMS

J气

lation studies of Shepp and Vardi (3) support this contention. EquaIl y impressive are the further studies of Snyder and PoIitte (6.), who incorporate time-of-flight information based on reference 3. In this paper, we first summarize the theory be缸ind the EM aIgorithm. We then mention briefly some relevant properties of the Poisson distribution. These ideas are applied in deriving the EM algorithm for emission computed tomography (CT). We proceed to the EM aIgorithm for transmission tomography. This second algorithm is mathematically more difficult to derive; in fact , an exàct analytical solution cannot be specified. However, we propose some highly accurateapproximations that should overcome all practical obstacles. Throughout the paper, we try to maintain as much commonaIity of notation as possible to aid in understanding the many similarities between the two aIgorithms. The discussion spelI s out in detail the characteristics of the EM algorithms , with particular emphasis on some of the modeling possibilities inherent in their application to emission (p ositron and single photon) and transmission tomography. The appendix contains a unified treatment of some convergence results 岛r both algorithms. At a later time we plan to provide detaited simulation studies of both aIgorithms as well as reconstmctions from actual scan data. D臼cription

"

..

.

of the EM Algórithm

The EM algorithm is an iterative technique for computing maximum IikeIihood estimates. Dempster et al. (7) were the frrst to state the algorithm in its fu11 generality, although particular examples were known many years previously. Baum et al. (:凹， Hartley and Hocking (到， Or­ chard and Woodbury (1 0) , and Sundberg (1 1) a11 anticipated various parts of the theor孔 The basic idea is fairly simple. Suppose the observed data in some experiment or sequence of experiments is a random vector Y. Let Y have density function g( Y， 的， where 6 is some vector of parameters to be estimated. (For purposes of this paper, we made no distinction between density functions and likelihoods.) In general , it may 起e difficult to maximize g(Y， 的 with respect to 6. A possi也，le solution to thîs predicament is to perform the thought experiment of embedding the sample space for Y in a richer or larger sample space where optimization problems are easier to solve. Thus , the EM algorithm postulates a "complete data" random vector X such that Y is a function h(X) of X. Furthermore , X is supposed to have a density function f{X， 的 with respect to some measure μ(X). Under these assumptions , g(Y,6) can be recovered by the integration g( 瓦的 = Jf(X， 的dμ(X). {X:h(X) = η

(1)

Each iteration of the E坦 algori毡m consists of t暂O steps. In the E-step垄 one forms the con岳泣ional expec毡， tion E苍白主义叫王§号， 飞~here

lndenotes natural loga丘吉m ， -and 伊 denotes 在e current vector of parameter estimates. In 白s 茧-step 雯在is conditionaI expectation is 戳在xi黯ized 有it主 respect to 号， 6 n held fIxed , to give t主e new parameter estimates 6报刊­ W起en f{X， 的 belongs to a regular exponential family, this description can be 1仅ast using the sufficient statistic 岳r x 口， 11).

The essence of the EM algorithm can be explained considering the function e → H(61铲)

E(lnf(X， 现 Y，铲') - ln 挺立的每

Regardless ofthe value of6n , H(6挂号 attains its maximum 驴. This fo11ows f旨。m a varian主 of Jensen's when 6 inequality (5). The E盟 algorithm exploits 白is Jact 峙 choosing 6 = 6n + 1 to m在ximize E(ln 丘之前jY， 的­

It fo11ows readily that

In

g(Y，铲 +1) 二三 ln g(Y，6号，

with strict inequality in most circumstances. In 0毛主er words. the EM algorithm is designed 量。 increase 白e log likelihood at each iteration. 暂eno垣 for later use th延

土 E(ln f(X , 6)1 Y， 6哇 =11n=iin zmn} 8鸟 电 for any component 6 j of 告暂hen 号 is not on a boun垂ary. Unfortunately, the- convergencè results in (7) are seriously flawειBoyles (1 2) and Wu (13) atte理ptto remedy the situation. Althougl豆豆草orous ， their ar.辜uments do not cover the present problems. T主 ese problems have boundalγfeatures that complicate 在e analysis of convergence. We sho暂 in the appen垂在在at convergence occurs regardless of the choice of 白e initiaI estimate 6气 Ourproof 垂epends strongly on t主e fact 也就§→ lng(Y二号 is strictly concave for both reconstruction models. The art in using the E班 algorlt主应茧es in choosing an appropria主e complete data specification X. Al主hough thεre canbe 血any 暂ays of embeddin主 Y in a larger sample space , often physical considerations suggest a naturaI defInition of X. This happens to 主e the case for both image reconstruction algorithms 岳eveloped here. Some Properties of the Poisson Distribution A

non-negative乡 integer-valued

random

varial主le

Z fol-

löws a Poisson distribution if 飞 k

P(Z = 每

e 百

also is a Poisson

一一

Z

mZ妇

for some λ> O. The Poisson distribution has some very useful properties. 1. Z has mean E(Z) = λ 2. If Zl' .莓. , Zm are in垂ependent Poisson variates , then

In the present context , a11 density 直lßctions are discrete and Eq. 1 reduces to the summ剖ion g( 丑的 = Lf(X. 的. {X:h(X) = 另

=

主y

z

'k

v在riate (1每.

J Comput Assist Tomogr, Vol. 8, No. 2 , 1984

308

K. LANGE AND R. CARSON

3. Suppose that a random number Z ofparticles is generated according to a Poisson distribution with meanλ­ Let each particle be independently distributed to one of m categories , the 缸主 of which occurs with probability Pk' If we let Zk represent the random number of particles falling in categ。可 ι Zk follows a Poisson distribution with mean λPk' Furthermore , the collection of random varia坦les ZI' . . . , Zm is independent (1 4). Since m

Z = 2:马， k=l

In emission CT the goal is to estimate the local intensities of photon emission in the object of interest. The emitted photons can be captured or deflected en route to a detector. Thus , there must be some correction for photon attenuation. With positron emitters , two photo~s are generated on the annihilation of an emitted positron and a neighboring electron. These two photons retreat from each other at approximately a 180 angle. For an emission event to be counted , both photons must regìster nearI y simultaneously at two opposite detectors.τhe principal d i:fference between positron emitters and single photon emitters is that the attenuatiori path for positron emitters amounts to -the whole line segment between the pair of detectors. For single photon emitters the attenuation path is just the 险le segment between the source and the single detector. With this distinction c1early in mind , it is possible to formulate a single reconstruction algorithm that covers both kinds of emitters. As with a坦 algebraic reconstruction algòrithms ，社 is necess町Y to divide the plane 缸ound and including the o埠ect of interest into pixels. These pixels typic a11 y are sma11 squares defmed by a rectangular grid. In Thble 1 we list necessary notation for both the emission and transmission models. Somè of the assumptions behind these defmitions deserve coIimlent. First，与 and 0

J Comput Assist Tomogr, VoJ. 8, No. 2 , 1984

(2)

To accommodate radioactive decay of the source material this expression shou挝岳e modified to 在一 -1位 -13 tH 二 e 问}马马 =

C

ü

慧

Emission Computed Tomography

A毛与马 = Cü 马，

. ，人严

note the consistency between 2 and 3. 4. In the same vein as 3, suppose Z is given. Then the conditional distribution of (Z1' . . . • Zm) given Z is multinomial with ceI1 probabilities PI' . . . 'P m and total Z (1 4). Examples of 3: Suppose m = 2. Category 1 particles are detected by some device , and category 2 particles are lost. The number of detected particles Zl isPoisson with mean ÀPl' As another example , suppose the particles are generated over a time inte凹al 屿，马). The categories can correspond to a partition of (句， tJ into di电joint subintervals. Yet another example is furnished by particles leaving at a random angle from a fixed point of generation. The categories can correspond to a subdivision of a unit circle centered at the point of generation. The relevance of these examples to image reconstruction should be obvious. For instance , the Iast two examples together imply that the numbers of partic1es detected for the various projections are independent random variables; these pr'叶ections either occur over disjoint time intervals or involve different angular sectors for each point source of partic1es (photons). Because of this independence property, the likelihood over all projectioÌls reduces to the-product of the separate projection likelihoods.

f.ti represent averages 百1t主in a pixeL Second ，单位ough we aI1。有 for the Poisson process of photon emission to be nonhomogeneous in spacε，有e re哇uire i生 to be homogeneous in ti理e ， exclu岳ing 在e easily corrected process of radioactive decay for the source. Thir岳，主主e probabilities bij summarize all remaining p呈ysical fea生ures such as detector geometry a摇 摇tenuation. Further specification of 挂lese constants is deferred t。在e discussion section. With the notation of 1油le 1 ，至t is clear 在就在e mean num岳er of detected photons 争岳窍。f photons) originating from pixel j during the 主h projection is

^:i,

where ß is the decay ratεand (缸，毛主 is the time interval for the ith projection. It is important to note 边at the C ii are considered kno暂n constants. The intensitiesλ: are parameters to bεestimated. For notational conveniènce , 有8 暂ill arrange À; into a single vectorλ No飞w let Xii, jEIi• be 白e random number of photons (pairs of photons) 白at arεemitte岳七y pixel j and contribute to projection i. As mentioned 必ove ， the mean of Xi山 is CifÀ;. Also let Yi be the total number recorde吾 for projection i. It is obvious 也at

拿

雪

乓=2:x.苔' j E1i

In view of our earlier re理arks 动。ut Poisson distril黯­ tions , X ij and Yí have Poisson distributions. lj are the observed data for projection i. 主y 过efinition ， the X u constitute the unobserved but complete 也钮 Before specifying the EM algo挂在m let us check the strict concavity of 在e log likelihood In g( Y， 治事 Since the Yi are independent Poisson variates , ln g(瓦 λ) =三~ - 2:句马÷乓 ln( l

j E1i

2; cüÀj J -

VE1i

ln 乓!

I

t

J

。}

A typical second partial derivative of ln

g(Y， 均 is

TABLE 1. Notationfor the models both e黯ission 缸ld transmission i , projection subscript j , pixel subscript l i> set of pixels contributing to projection i J ù set of pr!唔εctions to which pixelj contri垣utes If:. t" length of time over which 在e ith projection is

lected Unique to the emission case λ:. source intensitv of Dixel i

•

磊毫

" d岳

col-

photolÌs) leaving' detector (pair of

Unique to the transmission case Cti' source intensity for pr，唔εction i 屿， attenuation coε茸茸ient of pixel j (pro检abi1ity of -photon capture per unit length) f泣， length of pr司ection line i that 亟主ersects P住elj

雹

EM RECONSTRUCTION ALGORITHMS

瓦￡￡忐古古 击东杂 :3 i云IE♂ =云-1… n时μ( 州山 g(吭瓦

First，在e non-negativity constraints 巧主 o are 乒utomat­ ically satisfied. Howeve1二 note th单位e initialλj must be positive; otherwise all su已se哇uent 巧=在 Second， at each iteration it is easy to verify that

τE

‘

(2: Cij λ:; )2

t

I

飞 jEli

where a ,l,

= ~ Cjk

2:2: Cü 与 =L i j El

The associated quadratic form ~2

L.; Vk一」一:-ln k ,t

g(Y， λ) VI

naλkδλ_..

,

,.

is strictly negative definite whenever the matrix A = (aik) has at least as many rows (projections) as columns (pixels) and is of full rank. Under these conditions -In g(Y,J.l,) is strictly concave. To specify the E-step of the EM algorithm we need the complete data log likelihood

ln other wor缸， the current expec垣d number of photons (pairs of photons) stays fixed at the total 0七 served number. Certain equality constraintsalso become trivial to implement in the above 丘ame百ork.For instance , any Întensity Ài can be held at a constant value. The update 主4 6 is simply ignored at each iteration for pixel j. Neighboring pixels can be force岳 to have 白e same 岳tensities. If S is a set of pixels 有ith common intensity ÀSJ 在e 搓， step of the algorithm is mo过运edso 在at Às is updated by

LLNi川

夫S

λ

ln JtX , À) =车~ {-CijÀ,i + Xij In(cyÀ) - ln X ü !}'

E(ln JtX ,À)! Y，λn) = LL{-Cij λ:; + N Ü ln(cij À}} + R , i jdi

(4)

where

c"

À~ 巨

E(Xij! 乓，的=号ι二f μC政 Àk

kEli

and R does not depend on the new λ. Note that 背e have the fact that , for fixed i, the Xij are multinomial given

ll~ed

YÚ

The M-step is now trivial. Simply take partial derivatives ,

土 E(ln JtX , À)! Y，的= - LCii 十二 Nii λ -1 ðÀj

均

s - LLC亩，

'J

吨!J

J

。}

Transmission Computed Tomography The goal in transmission Cτis to compute local attenuation coefficients in the 0与ect of interest. ln contrast with emission CT the source of photons Îs extema1 to 毡e o坦ject. The parameters to be estimated are the attenuation coefficientsμi for each pixelj. Corresponding to eac.主 projection i, there is also an associated 纽ean source intensity αj' (See 主运ble 1 for notation.) T主ε fX.i depends on the geometries öf the source and 白c 王letector and can be estimated empirically simply by removing 白εobject to be reconstructed. Hence ， αi is taken as kno暂且 For-each pr乓jectioD i , let Wj be the t。但1 num己er of photons leaving the source and headed toward 桂le detector二 ln the absence of ~呈ttenuation ， all 曹i photons would be detected, and by our earlier remarks 在e 在'i are independent. Let Yi be the actual nUI由er of photons 垂b tected. The Yj constitute 白eobserv，εd data. Assumin喜 an infinitely thin beam , each of the 曹i photons leaving the source has the same probability

set them to zero , and solve. The result is the very pleasing method of moments estitnate

Z 叫f ι1:

、n

ι，

坠'

••

L Cü t吨

LCü 骂 2: Cik λZ

i吨

kElj

(6) 尊

e

一.乏与芝i

(7)

Jelj

of reaching the detector二It follo有S 在at li 主as a Po~son distribution 有ith mean the product of the 血ean of辑'i， dj = åt卢þ t ÌInes the probability (至q. 7). As a conse号uence of the independence of the 1亨j， the Yj are also in命pen­

V

\. n+l 一一一一」与_ V_二ι二L

丁

4

jESi.哼

Taking the conditional expectation of X;; with respect to Yj and the current vector of parameter estimates λn yields

=

j，哼

一一一一­

1 Jelj

N jj

乓­

j

keIj kfT.

10

IK

309

for the jth component of the new parameter vector Àn + 1. Since E(l n JtX，λ)!Y， λn) is concave inλ ， Eq. 6 must represent the maximum. Equation 6 has a simple intuitive interpretation. N;; is the best current estimate of the number of detected photons (pairs of photons) that originate_ from pixelj during the itt1projection.The smiENu is the estimmma且ted 阳 nu E口lm

dent. ln summary, the duces to

log 主k注lood ovεra丑严'ojections 妃，

r - ~ 1;;各i ln g( 瓦叫= 2:~ -dl! j哇 " J

一乓2 主岛÷乓ln d i

-

ln 乓寸， J

jdi

J

ber that originate from pixel j summed over all projections 由at pass_ through pixel j. This estimated nllmber is set equal to the expected number .4Cfi Ài and the resulting equation solved forλ俨 Z气 Equation 6 is also remarkable Ior two other reasons.

(8) 暂here Yandμare vectors 有hose and J.t j , respectively.

τb

components

éheckthe strict concavity ofln

g(Y， 抖)

are 在elj

note that

J Comput Assist Tomogr多 Vol. 8 , No. 2, 1984

K. LANGE AND R. CARSON

310

ικ

ιε

zia

~Z

一--=-::-ln g( Y， 同= ð /-1fÍJμt

然 -1

与吱j e-z j Eli ail'

-

a:L

= ~ Ifk keIi

沾一

lo

The conditions on the matrix A quadratic form

P(耳Xm)

=

"IJ寺

k Éli •

= (aik) for the associated

I "1m\ X mI . "1m\号-Xm = 口车\X:)\ ~~-) 1 一~~") "Ii号I X i \

LVk一ι一k， ! δ/-1kð 抖

\

and the conditional proba主运ty

-亏空/鸟飞("1m )Xm (

~2

ln g(Y,/-1)Vl

"1m 号-马

1 _

主j! 气Xm/ 飞毛/飞

P(~1Xm) =

to be strictly negative definite are the same as in the emisSlOn case. To derive the E-step of the EM algorithm we focus on a single projection. It simplifies 四atters to abandon te血­ porarily our previous notation and drop the projection subscript i. We also relabel pixels , st盯ting wÎth pixel 1 adjacent to the source and ending with pixel m - 1 adjacent to the detector. N ote the irrelevance of pixels disjoint from the projection line. Natural candidates for the complete da阳 are the numbeI's of photons entering each pixel along the projection. Let X j> 1 ::::; j ::::; m - 1, denote the number of photons entering pixelj. X 1 = W is the number of photons emitted by the source and has mean d = Atα . X m we identify with 瓦 the number of photons detected. τ主e 主ey to writing the correct complete data likelihood is to note that the number of photons X i + 1 leaving pixel j depends only on the' number X; entenng and on thè prodllct~/-1j of the projection path length and attenuation coefficieñt: The number leaving should therefore follow' a binomial distribution with probability of success e-~lLj and number of trials X j ' By an appropriate conditioning argument which induciively adds pixels from source to detector, it is possible to show that the complete data log likelihood is

+

1主ILk

Hence, the joint probability

where now

一d

Z

e 主=j

毛/

Xm! e

一 ι-~

\

'm'

飞 1)

(-毛 - "1m) 号-Xm {乓 - X m )!

In other words , conditional on X w X j - X m fo11ows a Poisson distribution ，世岳拇指亏 - "Im-= E(写一 E(Xm).

Thus 萝

E(乓 - XmlXm)

=

E(乓J

- E(Xm),

= Xm +

E(乓J

- E(Xm).

or E(主jlXm)

在街

This is precisely the con岳itional expectation nee岳εdin the E-step. To perform the M-step of the EM algorithm wεrevert to our previous notation. Let Mii 部哇乓民出e expec泌 number of photons entering and leaving pixel j , respectively, for pr唔ection i. T主ese expectations are conditional on the number of photons Yí 0七serve岳 and 在ec出Tent parameter vector 扩. They are 主iven by E哇. 10 above. Summing the values (主哇，窍 derived from t主e 主-ste苦 over all pr司ections no曹 yields L L{Nij ln(e-1ij与)

+ (Mij -

毒

害

斗也

在

Nij)ln口一 e- t，的n +R,

j e/j

X 1 1n

d- 叫 +21h(二)

(11)

+鸟A 一切) +吁鸟牛川+刊 +1) 川} (9)

where the term R includes a主哇lose quantities 白atdo not depend on the ne有 para黯eters /-1j' The M-step is carried out by computing the partial derivc说ive of E导 11 wi在 respect to /-1k and equating it to zero for each k. τ主is yields

To perform the E-step we must compute the expectations

o = iLk

E(~iXm' μn) ，

eJ

of the complete data ~二 conditional on the incomplete data X m and the current parameter vector /-1 n• The con-

ditional 目pectations of In X 1! and ln( ..~j )ar，盯翩al 碰11 飞 LLj+ll

irrelevant in the M-step. Just as in the case X mm = Y , the random variables X;. 1 S三 j :::; m - 1~ fo11ow Poisson distributions with mean-s

E(乓) =可

-

j-l z 也ILk

de k=l

也'k

呈-

e

-lIf;.r-/Ii:

三〈主4.-R-K}fik ieJk 反 eti在均

-

tk

馨。2) 1

ZZ 毛飞

1=

、盔，，，

The transcendental 主哇. 12 cannot be solved exactly. However, the right ha至ld si岳e of Eq. 12 Îs monotone in /-1k and has one positive sol吕立ion. Furthermore, an easy convexity 在rgument sho有s that the solutions correspo至ld 部 a maximum of Eq. 11. A丑 practical applications will have su茧ciently. s黯a主 pixels so that terms such as lík /-1k in E哇. 12 飞可望 be s部a茧， τhis suggests using 在e approximation 1-s 1-2 s-2 (13) ÷ + S -e 。

J Comput Assist Tomogr, Vol. 8, No. 2 , 1984

t

= L- N，点÷

-ez豆

n章

ieJk

一一

Furthermore , the distribution of X ,.. conditional on X; is binomial with success probability

p-1i主11聋

I. ,

-Ni~ik + 三 (M政 - Nik) 章 'ZK-JLL ·毒，

EM RECONSTRUCTION ALGORITHMS with s Ijk fJ..k' (To derive Eq. 13 expand sl(eS - 1) in a Taylor series.) Retaining only the 118 term in Eq. 13 and substituting into Eq. 12 gives the approximate _soIution 主(Mik

=

μ.2+ 1

-

N ik)

..

(1 4)

…~

NiJ放

ieJk

A better approximate solution can be obtained by retaining the frrst two terms of Eq. 13. Substitution of 1 1 二~=;-2， 8

缸冉，

obviously accuracy, and the other, computatìonal s垃1plicity. On the basis of both these cri主eria，有e prefer 白e second approximate solution (主导 I窍. But it must be recognized that close to the maximum, an inaccurate approximation may not lead to an increase in ln g( 瓦时 from one iteration to the next. In theory，也is difficulty can be circumvented by switching approximations in the course of iterating. It should be reemphasized 岳at all three estimates sati玛F μ'k+l > 0 飞;vheneverμ'k > O. If fJ.-k = 0委主lik = N ik in E皂 11 , and the only 暂ay to maxìmize'E号 11 is 主o take 抖γz o also. Finally, the presence of simpleequality constraints on the f.Lk can be handle过 just 怒坦白e emissìon case.

into Eq. 12 and rearrangement yields

DISCUSSION

,2: (Mik - NiJ

1+I=

ieJk

This method of moments estimate can be derived heuristically by equating the expected number of phötons captured by pixel k , ~1/2(MJ政十 Njk)likfJ.-k' ieJk

to the current estimated number of captured photons , ~(M拔一 NiJ)， ieJk

and sol ving for fJ.-k' The two Eqs. 14 and 15 actuaIly furnish upper and lowerbounds , respectively, to the true solution fJ.-Z+ 1. This follows from the two elementary inequaIities 1

1

-一-~一-一一罢王 -.8> 8 2 - eS 一 1 s'

O.

Our third and most accurate solution is based on using aIl three terms of Eq. 13. Substitution into Eq. 12 and

rearrangement gives the quadratic equation

。 =μ.12: (Mik

General Characteristics of the E坦 Reconstruction AIgorithms

士(15) 1/2 2.; (M，放 + Nik)lj故

…

Nik)年

- -

马阳盹

311

12

The EM reconstruction algorithms are 切se岳 on an exact stochastic model of the projection 摇钱­ surements. Fourier/convolution techni罕les (1 5) are deterministic in nature and completely ignore the stochastic element of the data. Iterative approach郎 。) can accommodate the presence of noise in the observations by means of the definition of criteria for an acceptable reconstruction. For instance , some algebraic techniques implement a 有eigh捂住 least squares approach wÎth weights proportional to the estimated variance of each measurement (1 6). In scanning protocols in which count垣g statistics are high霎 ignoring 白e true statistical nature of the observations is not a ser拎出 limitation because Poisson counting noise is only a co理ponent in total system noise. This is the case currently 暂i在 most transmission tomography and 前出 some applications of emission tomograph}人 For example雯 ahigh statistics fluorodeoxyglucose scan on the positron tomograph NeuroECAT (1万 might collect from 2 to 5 million coun毡， di轧过ed 在mong 8 , 448 pr吃jecti01毡， for an average value of 300 to 600 counts/pr司ection. In emission tomography, ho百ever， some protocols are limited by statistical noise and the use of an inappropriate model is detrimental. τ主is 有ill 色epar­ ticularly true for future brain ligand tracers (1 8). It is precisely in these 1。有 count cases that the E坦 algorithms are expected to provide the greatest i血­ provement in reconstruction quality. Furthermore , better reconstructions at 1。有er count rates 生ould allow reduction in pf过ient radiation dose. tecl主1n 垣i哇uεs ， alge主raic U nlike the Fou呈r垃r-base 0 is 3, suppose 9j = 0 a时

usin~

~: L(俨) =

8鸟

0

O. Furthermore , there are only a finite number of limit points. Proof: Let 9nk be a subsequence that approaches 俨. Also , let j be an index with 咛> O. Since 119nk + 1 - 9呵! 0, whenever 的>

•

l~O

fJj=在

a1rea垂y

taken care of 己yLe器ma

ð 丘吉穹 >0雹

δ乌

By continuity,

土L(9号=主Q甜的i章=♂ >0 B鸟鸣

holds for a l1 n large enough. But

土豆2(919号 码 z is strictly decreasing in 导 This means 号 < 9t 1 for all large n. a contradiction to the assumptión 号→告. The proof is complete.

,

,

,

J Comput Assist Tomogf Vol. 8 No. 2 1984

/

K.LANGEAND 丑

316

Acknowledgment: The authors wish to thank Dr. Sungcheng Huang for offering helpful suggestions and Avis Will且ms for preparation of the manuscript. This work was supported in part by the U niversity of Califomia at Los Angeles , NIH Research Career Development Award K04 HDOO307 , and NRSA 古aining Grant GM719 1.

REFERENCES 1. Herman GT, ed. Topics in applied physics. vol. 32. Image reconstruction Irom projections: implementation and applications. Ber1 in: Springer-Verlag. 1979. 2. Herman GT. Image reconstruction Irom projections: the lundamentals 01 computerized tomography. New York: Academic Press, 1980. 3. Shepp LA , Vardi Y. Maximum lìkelìhood reconstruction for emission tomography. IEEE Trans Med Imaging 1982;MI-

27:1368-74军

21. Herman GT , Len主 A ， Lutz PH. 主elaxation method主 for image reconstruction. Comm Assoc Comput 主lach 1978;21:153-8. 22. Gordon 豆， Bender 丑， Herman 舌T. Algebraic reconstruction techniques (ART) for three-dimensional electron 黯Ìcros­ copy and X-ray photography. J Theor BioI1970;29:471-8 1. 23. Gull SF, Daniell GJ. Image reconstruction from incomplete and noisy 曲也 Nature 1978;272:岳86-90. 24. Herman GT, Lent A. Iterative reconstruction algori在ms. Comput Biol Med 1976;6:273-94. 25. Budinger T F. Physical attributes of single-photon tomography. J Nucl Med 19宰。 ;21:579-92. 26. HoffmanEJ , Phelp主挝E. An ana1ysis of some of the physical aspects of positron transaxial tomography. Comput Biol Med 1976;6:345-60. 27. Phelps 班E 章 Ho宫man EJ, Huang SC, Kuhl DE. Design 号。n­ siderations in positron computed tomography (PC写 .IEEE Trans Nucl Sci 1979;NS-26:2746-51. 28. Phelps ME. Huang SC. Hoffi黯an EJ, Plu嚣黯er D , Carson R. An analysi主 of signal amplification u主ing small detectors in positron emission tomography. J Comput Assist TOI究ogr 1982;6:551-65. 29. Hoffman EJ , Huang SC , Plum血er D , Phelp主革E. Q注anti­ tation in positron emission compu给d tomography: 6. E吉ect of nonuniform resolution. J Comput Assist Tomogr 1982;6:987-99. 30. Phelps ME. Emission computed tomogr在phy. Semin Nucl Med 1977;7:337-65. 31. Lewis 垣L，暂illerson JT, Lewis SE 董 Bonte 窍， Park句去育， Stokely E盟. Attenuation compensation in single-photon emission tomography: a comparative evalua主ion. J Nucl Med 1982;23: 1121-7. 32. Huang SC, Hoffman EJ , Phelps ME，主uhl DE. Quantitation 垣 positron emìssion compute岳 tomography: 2. E主εcts of inaccurate attenuation correction. J Comput Assist Tomogr 1979;3:804-14. 33. Huang SC , Carson RE , Phelps 茧E， Hoffi里an EJ , Schelbert HR, Kuhl DE. A boundary method for attenuation correction in positron computed tomograph歹• J Nu è/ Med 1981;22:627-37. 34. Phelps 班E ， Ho主man EJ , Huang SC , Ter-Pogo主sian M应. E茸ect of positron range on spatial resolution. J Nucl f.led 1975;16:649-52. 35. Ter-Pogossian M盟霎挝ullani NA , Fid王e DC , MarkhafI1 J , Snyder DL. Photon time-of-t1ight assisted positron emi重sion tomography. J Comput Assist Tomo,

aEj •

3

亮d ，、

,

20. Huang SC. Phelps 盟E ， Hoffman EJ. 豆豆豆hl DE. Cubic 主p监les for fdter desÌgn in CT. IEEE Trans Nucl Sci 19吉。 ;NS-

、茸

专了

、

iJ

专亏

J Comput Assis/ Tomogr, Vo l. 8, No. 2, 1984

J4' 手

1:113-22. 4. Rockmore AJ , Macovski A. A maximum likeIihood approach to image reconstruction. Proc Joint Automatic Control ConI1977;782-6. 5. Rao CR. Linear statistical in庐rence and its applications. 2nd ed. New York: Wiley, 1973:59. 6. Snyder DL, Politte DG. Image reconstruction from list-mode data in 缸1 emission-tomography system having time-of-flight measurements. IEEE Trans Nucl Sci 1983;NS-30:1843-9. 7. Dempster AP, L剖rd NM , Rubin DB. Maximum likelìhood 齿。m incomplete data via the EM algorithm. J R Stat Soc Series B 1977:39:1-38. 8. Baum LE警 Petrie T, Soules G , Weiss N. A maxìmizatìon technique occurring in the statistical analysis of probabilistic functions of Markov chains. Ann Math Stat 1970;41: 164-7 1. 9. Hartley HO , Hocking RR.τhe analysis of incomplete data. Biometrics 1971 ;27:783-808. 10. Orchard T WoodburY MA. A mìssing information principle: theory and applications. In: Proceedings of the 6th Berkeley Symposium on Mathematical Statistics and Probability. Univ. Calif. Press: 1972;1:697-715. 11. Sundberg R. Maxìmum likelihood theory for incomplete data ftom ßn exponential family. Scand J Stat 1974;1:49 …58. 12. Boyles RA. On the convergence of the EM algorithm. J R Sla(SOc Series B 1983;45:47-50. 13. Wu CF. On the convergence properties ofthe EM algorithm. Ann Slat 1983;11:95-103. 14. FelIer W. An introduction 10 probabiU市的eoη and its applications. Vol. 1. 3rd ed. New York: Wi1ey, 1968:216-37. 15. ~hepp LA , Logan BF. τhe Fourier reconstruction of a head section. IEEE Trans Nucl Sci 1974;NS-21:21-43. 16. Budinger TF, Gullberg GT. 古ansverse section reconstruction of gamma-ray emìtting radionuclides in patients. ln: TerPogossian MM , P

CARSON