PET Clin 2 (2007) 173 190 Advances in PET Image … · 2018-10-03 · parametric images from the...

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Advances in PET ImageReconstructionAndrew J. Reader, PhDa,*, Habib Zaidi, PhD, PDb

PET image reconstruction usually involves thegeneration of three-dimensional (3D) images of a ra-diotracer’s concentration to estimate physiologic pa-rameters for volumes of interest in vivo. To enhancethe functional information, often a time sequence ofthese 3D images is required so that time-activitycurves for each particular volume of interest can beobtained. These time-activity curves can then befitted with a kinetic model from which functionalparameters (such as the metabolic rate of glucoseor blood flow) can be estimated. The key point tonote is that a PET scanner does not measure thespace-time–dependent radiotracer concentration di-rectly. Instead, the annihilation photon pairs arisingfrom the positron emissions are externally detectedby the PET scanner and recorded as event histograms(sinograms or projection data) or as a list ofrecorded photon-pair events (list-mode data). Insome cases, the data may even be backprojected to

form backprojected images, as a means of data com-pression, with only minimal loss in spatial preci-sion. Figs. 1 and 2 illustrate the basic formats ofdata that can be acquired by a PET scanner, indicat-ing how the measured data are indirect measure-ments of the unknown activity distribution.Indirect measurement is the problem that image re-construction seeks to solve. Projections/sinogramsare the most popular data format, but as more attri-butes for each PET event are recorded (ie, not onlythe coordinates of the two detected photons butalso energy and timing measurements), list-modedata can become more practical for data storagewithout loss of information. Although the problemof image reconstruction has conventionally beenone of determining the space-time–dependentradioactivity concentration in vivo, there is alsoa move toward the direct estimation of functionalparametric images from the raw PET data.

P O S I T R O NE M I S S I O N

T O M O G R A P H Y

PET Clin 2 (2007) 173–190

This work was supported by grant SNSF 3100A0-116547 from the Swiss National Foundation.a School of Chemical Engineering and Analytical Science, The University of Manchester, PO Box 88, Manches-ter, M60 1QD, UKb Division of Nuclear Medicine, Geneva University Hospital, CH-1211 Geneva, Switzerland* Corresponding author.E-mail address: [email protected] (A.J. Reader).

- Foundations for image reconstruction:representation of the object and thedata

Object representationData representation

- Analytic image reconstruction methods- Iterative image reconstruction methods

The system matrix and the parametersObjective functionsDirect algorithms

Iterative algorithmsUpdating methodsRegularizationDecomposition of the system matrix,

including attenuation, normalization,scatter, and randoms

Four-dimensional methods- Summary- References

173

1556-8598/07/$ – see front matter ª 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.cpet.2007.08.001pet.theclinics.com


Initially this article concerns one-step analyticimage reconstruction techniques, which operateon the measured PET projection data to estimatethe radiotracer concentration within cubic volumeelements (voxels). These methods are applied foreach time frame of acquired data, thus providinga time-series of 3D images that can subsequentlybe used for postreconstruction kinetic analysis toestimate functional parameters. These analyticmethods have the advantage of being relativelyfast to use and linear and, as a result, they havea good track record for producing reliable quantita-tive PET images. The article then considers iterativeimage reconstruction methods, which have the ad-vantages of complete flexibility in modeling the PETdata acquisition process and the corresponding

freedom in specifying the parameters to estimate.Conventionally, the parameters to estimate are thevalues (representing radiotracer concentration) invoxels, and the PET data acquisition model is thatof taking sets of line-integrals to form projections(ie, the Radon or 3D x-ray transform [1,2]). If voxelvalues are chosen as the parameters to estimate andif the line-integral imaging model is used, then iter-ative reconstruction methods (when run to conver-gence on high-statistics data) often give resultsbroadly comparable to analytic reconstruction.There is an increasing trend, however, towardmore fully exploiting the capacities of iterative re-construction techniques through the use of moreaccurate and complex modeling of the PETacquisition process. Such modeling can be based

Fig. 1. The three main formats of PET measured data, shown for the simplified case of 2D PET. (Top left) Anyradioactivity distribution can be regarded as a collection of point source emitters of different intensities (a singlepoint source example is shown for clarity). (Top middle) The radioactivity distribution emits back-to-back photonpairs that may be detected by the PET detectors surrounding the field of view (FOV). Each event is usually asso-ciated with a pair of detection points outside the FOV (eg, r1, r2), and these two points define a line that can beparameterized by a distance s from the origin, and an angle f. (Top right) Each detected event can be recordedin a list-mode file (often the raw detection coordinates are recorded, along with any other information such asphoton energy). (Bottom left) Each detected event can also be accumulated into a sinogram, which has samplingintervals (‘‘bins’’) in s and in f to histogram the events. (Bottom middle) Note that after many events, a pointsource results in a sine wave on the sinogram (hence the name) and that each point (s,f) on the sinogram isapproximately equal to a line integral through the radioactivity distribution. (Bottom right) An alternative isto backproject the events along their lines of detection. Note that s corresponds to the y’ used in Fig. 4.

Reader & Zaidi174


on analytic methods or direct measurements(eg, see Refs. [3–7]) or on Monte Carlo methods([9–11]). In addition, iterative methods easilyaccommodate different choices of the parametersto be estimated. (Rather than pixel or voxel values,one can consider coefficients for spherically sym-metric ‘‘blob’’ basis functions [12,13],‘‘natural pixels’’[14–16], or even directly specifying kinetic parame-ters as the unknowns to be estimated [17,18].)

This article covers the foundations of imagereconstruction, with an emphasis on advancesand future directions for PET image reconstruction.Recent reviews by Lewitt and Matej [19] and Qi andLeahy [20] and the textbooks by Barrett and Myers[21] and Zaidi [22] also offer excellent comprehen-sive coverage of image reconstruction algorithmsand the foundations of image science appropriateto PET.

Foundations for image reconstruction:representation of the object and the data

Object representation

The unknown space-time radiotracer distributioncan be regarded as a continuous function f(r,t),specifying the radiotracer concentration (in unitssuch as MBq/mL or mCi/mL) for all spatial locationsr 5 [x y z]T at time t. It is not practical to measure andreconstruct such a continuous function; this wouldrequire essentially infinite levels of sampling andquantities of measured data. Since both samplingand data quantities are limited, a resolution-limitedrepresentation of the function f is usually consid-ered, through the use of a set of j 5 1.J spatial basisfunctions aj(r) and a set of b 5 0.B-1 temporal ba-sis functions bb(t). In the first instance, consider thetime-independent case such that the continuousfunction f is approximated by

f ðrÞzXJ

j 5 1

cjajðrÞ ð1Þ

where c 5 cj Jg�

is a J-dimensional vector, holdingthe coefficients for each of the j 5 1.J spatial basisfunctions that cover the field of view (FOV) of thePET scanner. The most common choice for thespatial basis function a(r) is the voxel such thatthe vector c simply holds the voxel values and canbe interpreted directly (after appropriate rearrange-ment of the elements) as a two-dimensional (2D)or 3D image. To extend Equation 1 to the time-dependent case, the vector c can be extended toinclude the coefficients for the temporal basisfunctions such that f is now represented by

f ðr; tÞzXB�1

b 5 0

XJ

j 5 1

cj1bJajðrÞbbðtÞ ð2Þ

where c is now a JB-dimensional vector, holding thecoefficients for each spatial basis function for eachof the temporal basis functions. The most commonchoice for the temporal basis function is the top-hatfunction such that the function f is composed ofdistinct time frames, each of which is considered in-dependently from the others. Note that Equation 2assumes that the function f can be created from fac-torized spatiotemporal basis functions, rather thanmore general nonfactorizable spatiotemporal basesg(r,t). For the case of orthogonal basis functions(eg, pixels/voxels or the Fourier basis), the elementsof the JB-dimensional vector of coefficients c arespecified by

cj1bJ 5

Z Zf�r; t�aj

�r�bb

�t�drdt ð3Þ

for which the time-independent case (setting b 5 0)is simply

cj 5

Zf�r�aj

�r�dr ð4Þ

It should be emphasized that Equations 3 and 4only hold for orthogonal basis functions (eg, vox-els for the spatial functions a(r), top-hat functionsfor the temporal functions b(t), or the Fourier ba-sis functions, and so forth). The use of nonorthog-onal basis functions is possible, but in such a case,even finding the coefficients vector c when thefunction f is actually known is in itself an inverseproblem. This, however, reveals yet another advan-tage of iterative reconstruction methods: nonor-thogonal basis functions can be easily used andaccounted for by the system model such that thecoefficients vector c is estimated directly. The sys-tem model is considered in detail in the followingsections.

Data representation

The acquired PET data are usually projections (sino-grams or 2D parallel projections) or a list-mode file(although image-based storage such as 3D backpro-jected images can also be used, see Figs. 1 and 2).List-mode data and projection data can be regardedas a single IT-dimensional vector m, holding thenumber of counts detected by each of the i 5 1.Ielements in the measurement space (often theseelements are PET detector pairs, commonly referredto as ‘‘lines of response’’ [LORs]) at each of thet 5 1.T time sample points. For the case of list-mode data, there is no grouping or resampling(‘‘binning’’) of the data: information on the exactdetector pair i that detected each event, along withthe precise time sample point t, is retained by list-mode data (along with additional informationsuch as energy and difference in arrival time of

Advances in PET Image Reconstruction 175


the photons, in which case the number I of mea-surement elements would be correspondinglylarger). Such precision can be equaled throughthe use of finely sampled projection data bins(one for each possible event that could be

detected), but this often becomes impractical dueto the large number of detector pairs employed bystate-of-the-art PET systems (eg, I 5 4.5 � 109 forthe high-resolution research tomograph (HRRT)[23]).

Reader & Zaidi176


Considering first the time-independent case, eachelement i of the mean q of the noisy PET measureddata m is modeled by

qi 5 E

�mi

�5

Zf ðrÞsm

i ðrÞdr ð5Þ

where sim(r) is a function describing the PET scan-

ner sensitivity (for all spatial locations r) to a radio-tracer distribution for a given detector pair i, andalso for a given linear attenuation coefficient distri-bution m(r). The functions si

m (r) can in fact beregarded as probability images, whereby the func-tion value is related to the probability of a positronemission from location r giving rise to an eventbeing detected by detector pair i (see the laterdiscussion on the system matrix, and Fig. 3). Forthe time-dependent case, the mean q of the noisy(ie, count-limited) PET measured data m is mod-eled by

qi1tI 5 E�mI1tl

�5

Z ZDtt

f ðr; tÞdt

smi

�r�dr ð6Þ

where the t (time) integral is over the time intervalDtt related to the temporal resolution of the timesample t. Conventionally, the set of functions si

m

(i 5 1.I) used in Equations 5 and 6 are tubes ofresponse (or LORs) through the FOV, which iswhy the i 5 1.I detector pairs are also referred toas LORs. Using LORs makes Equation 5 equivalentto a line-integral; see Equation 7. In addition, con-ventionally, the set of spatial basis functions aj(r) inEquations 1 to 4 are voxels, with the temporal basisfunctions bb being top-hat functions. Such a choicefor the four-dimensional (4D) volume elementsisolates a unique subspace of the space-time vol-ume; however (as is considered later), given thecount-limited nature of the acquired data m, thereare advantages in considering basis functionsthat are nonorthogonal and overlap in space-time. Finally, it is worth noting in Equation 5that if the spatial basis functions aj(r) correspondto si(r) and if these functions are orthogonal,then there is no longer a spatial inverse problem

because Equation 4 is immediately obtained, giv-ing all that is necessary to estimate the function fin Equation 1. A similar argument holds forEquations 6 and 3. Hence, it is the difference be-tween the functions a(r) and s(r) (or if theymatch but are not an orthogonal set of functions)that encapsulates the image reconstructionproblem.

Analytic image reconstruction methods

For analytic reconstruction, the approximation ofEquation 1 is not assumed; the inverse problem isformulated in a continuous framework and thepractical implementation is performed as a discreteapproximation of the continuous solution. Also,time-dependent reconstruction is handled as a seriesof independent ‘‘static’’ reconstructions, so only thetime-independent case is considered here. The fun-damental assumption of analytic inversion formu-lae is that a 2D parallel projection representinga set of LORs pðs; buÞis equal to a set of line integralsthrough the radioactivity distribution f(r) (the 3Dx-ray transform) [1]:

p�s; bu�5

Z1N

�N

f�s1x0bu�dx0 ð7Þ

where the 3D vector r in the imaging volume isdecomposed into the 2D parallel projection posi-tion vector s 5[y’ z’]T and the one-dimensional(1D) orientation unit vector x0bu (Fig. 4). Thisline-integral equation can be seen as a special caseof Equation 5, whereby the continuous functionssm

i(r) are now lines through the FOV, with each de-tector-pair regarded as an LOR i (specified by a dis-placement s from the center of the FOV and anorientation bu5ðf; qÞ) and with the vector q re-placed by the continuous function p. Effects suchas photon attenuation and normalization fordetector nonuniformities can be accounted for asmultiplicative corrections to these line integrals,and photon scatter and random events need to besubtracted from the measured data before the

Fig. 2. Four examples of activity distributions (left column) and their corresponding sinograms (middle column)and backprojected images (right column) for the simplified case of 2D PET. (First row) A positron emitting pointsource gives rise to a single sine wave of corresponding intensity on the sinogram, and a (1/r) distribution (thepoint response function (PRF)) in the backprojected image. (Second row) A collection of point sources gives riseto a corresponding number of sine waves on the sinogram, and a set of PRFs in the backprojected image. (Thirdrow) A more general radioactivity distribution can be regarded as a collection of point sources of various inten-sities; hence, the sinogram is a superposition of sine waves of corresponding intensities for each of these points,and the backprojected image is a superposition of PRFs of corresponding intensities. (Fourth row) A brain activ-ity distribution [8]. Since in this case of 2D PET the point response function in the backprojected image is shift-invariant, the simple model of convolution can be used and reconstruction can be achieved by deconvolution.Annotations using the vector c and the matrix A are added for reference in later sections of the text.

:



approximation of Equation 7 can be considered asappropriate. Hence, the measured data m 5{mi.mI} are approximately regarded as propor-tional to Equation 7 according to

KAiNiðmi � ri � siÞzZ1N

�N

f�s1x0bu�dx0 ð8Þ

where Ai is the attenuation correction factor for LORi (calculated by taking the exponential of a 3D x-raytransform of the attenuation map or from the ratioof blank to transmission scans); Ni is the normaliza-tion correction factor for LOR i; and ri and si areestimates of the randoms and scatter on the projec-tions. The global scale factor K accounts for correc-tions such as radioactive decay and detector deadtime. Equation 8 assumes that the positron-electronannihilation event occurred exactly along the LOR

that joins the two detectors in detector pair i, thus ig-noring photon acollinearity, positron range, andparallax error.

The inversion of Equation 7 (ie, to find f, given p)can be found through the central section theorem,which relates the projection data p to the image f[24]: a central plane of the 3D Fourier transformF(k) of the 3D true activity f(r) is equal to the 2DFourier transform P(k) of the 2D parallel projectiondata at the same orientation bu5ðf; qÞ. Consideringthe case of 2D reconstruction, the central section the-orem corresponds to the 1D Fourier transform ofa 1D parallel projection being equal to a single linethrough the 2D Fourier transform of f. It becomes ap-parent that the 2D Fourier transform of f can be con-structed by the superposition of these 1D Fouriertransforms (one from each projection angle), butthat this superposition of the many lines at many

Fig. 3. Illustrative example of rows and columns of the system matrix A for different imaging models (the simplecase of using pixels with 2D PET is considered). (Top row) Three examples of a single row i of the matrix A (a rowis equivalent to an image) are shown for the case of the line integral model, a model using time-of-flight infor-mation, and incorporating scatter into the system matrix. Forward modeling of a given radioactivity distribution(represented by c) to obtain an element qi of the mean data q merely involves taking a pixel-by-pixel product ofone of these row images with the c image and summing all values (ie, performing a scalar product). (Bottom row)Three examples of a single column j of the system matrix (in forward modeling, to get mean data q from c, thecolumns are weighted by each point source intensity [ie, each pixel value in the c image] and summed to producethe overall expected data q). A simple and approximate way of including an image-space resolution modelwould be to convolve the row images, or for a simple detector-space resolution model, one could convolvethe column data. Note that the image values are not to scale and are for illustrative purposes only.

Reader & Zaidi178


angles will give rise to a 1/jrj density of contributionsto the 2D Fourier transform. The ramp filter is used(equal simply to jrj) to compensate for the varyingcontributions in the frequency domain. Ratherthan performing a direct reconstruction by way ofthe Fourier domain, it is often more straightforwardto filter the projections and then backproject them.

Various commonly used analytic reconstructionmethods are described in the literature [25,26], in-cluding the simple backprojection method (whichresults in a blurred version of the object distribution,as was shown in Fig. 2) [27]; backprojection fol-lowed by filtering (BPF) [28]; and filtered backpro-jection (FBP)/convolution backprojection. The BPFmethod usually relies on angle-limited backprojec-tion of the list-mode or projection data to producea shift-invariant point response function (PRF) thatcan then be deconvolved by Fourier methods.

In the particular case of 3D PET, because there issufficient information to reconstruct the 3D activitydistribution using only the direct projections(ie, those resulting from LORs within the same de-tector ring: q 5 0), the oblique projections (jqj>0)are redundant and are not required to reconstructthe activity distribution. Given the count-limited

nature of the acquired data, however, using theseoblique projections has the potential to effectivelyincrease the sensitivity of the scanner and thus re-duce the noise in the reconstructed image. Whenthe oblique projections are not truncated, theycan be reconstructed by FBP using the Colsher filter[29]. This algorithm can be implemented as FBP orBPF (the backprojection operation is 3D in bothcases). However, for real PET systems that havea limited axial extent, the 3D reconstruction prob-lem is complicated by the fact that all the oblique(jqj>0) 2D projections are truncated (ie, they areincompletely measured). One way to handle thisis by recognizing that the direct (q 5 0) projectionsallow complete reconstruction of the activity distri-bution. The activity distribution reconstructed fromthese direct projections can then be reprojected toobtain the missing portions of the oblique projec-tions. In practice, as much as 40% of the totalreconstruction time can be spent in estimatingand then backprojecting the unmeasured projec-tion data. The algorithm combining this forwardprojection step with the Colsher filter is a 3D FBPalgorithm referred to as the 3D reprojection (3DRP) method [30].

Fig. 4. A 2D parallel projection (2D PP), at orientation bu5ðf; qÞ where f is the azimuthal angle and q is the co-polar angle. The y’–z’ plane (s 5 [y’ z’]T) holds the LORs, each characterized by a distance s from the center of theFOV, and by an orientation bu5ðf; qÞ, along which the 3D radioactivity distribution f(r) is considered to be in-tegrated. For 3D PET, the direct (q50) 2D PPs are complete and sufficient to reconstruct f(r), but the oblique(jqj>0) 2D PPs are truncated, due to the limited axial extent of 2D PET scanners. One of the sinograms fromFig. 2 would be obtained by considering the case of q 50 and z’ 50, leaving just y’ and f as the two coordinatesfor a single, direct sinogram.



An alternative exact approach to solve the prob-lem of truncated 2D projections is the Fast VolumeReconstruction (FAVOR) algorithm [31]. FAVOR,however, presents different noise propagationproperties compared with other approaches (ie, itdoes not uniformly weight the data, unlike theColsher filter). With FAVOR, simple ramp filteringcan be used but at the expense of suboptimallyweighting the data, attributing a much greaterweight to the direct projections than to the obliqueones.

Although 3D RP works well and has served asa ‘‘gold standard’’ for 3D analytic reconstruction al-gorithms, it requires significant computational re-sources (compared with, for example, a series of2D FBP slice reconstructions). As a result, a numberof approximate methods have been proposed. Mostof the well-known approaches involve rebinningthe data from the oblique projections into 2D directsinogram data sets, thus allowing the use of 2D re-construction techniques and resulting in a signifi-cant decrease in computation time. Thesealgorithms rebin the 3D data into a stack of ordi-nary 2D direct sinograms, and the image isreconstructed in 2D slice by slice, for instance usingthe standard 2D FBP or iterative algorithms. Themethod used to rebin the data from the obliquesinograms into the smaller 2D data sets is the cru-cial step in any rebinning algorithm and has a signif-icant influence on the resolution and noise in thereconstructed image.

The simplest rebinning algorithm is single-slicerebinning [32]. It is based on the simple approxi-mation that an oblique LOR can be projected tothe direct plane halfway between the two endpoints of the LOR. Hence, single-slice rebinningassigns counts from an oblique LOR (in whichthe detectors are on different axial PET detectorrings) to the sinogram of the transaxial slice lyingmidway axially between the two rings. This approx-imation is acceptable only near the central axis ofthe scanner and for small apertures. For largerapertures and for greater displacements from theaxis of the scanner, single-slice rebinning resultsin position-dependent axial blurring.

Several alternative methods have been suggestedto provide more accurate rebinning. Lewitt andcolleagues [33] proposed a more refined rebinningapproach, the multislice rebinning algorithm, thatis more accurate than single-slice rebinning but suf-fers from instabilities in the presence of noisy data.The Fourier rebinning (FORE) algorithm is a moresophisticated algorithm in which oblique rays arebinned to a transaxial slice using the frequency-distance relationship [34] of the data in Fourierspace [35]. Although rebinning can be done exactly(using, for example, the FOREX or FOREPROJ

methods) [36], a considerable speed advantage isgained through using the approximate FOREapproach. Another exact rebinning approachreferred to as FORE-J was proposed and is basedon the fact that the 3D x-ray transform of a functionis a solution to a second-order partial differentialequation (John’s equation) [37]. FORE-J is easy toimplement, given that it has the same structure asFORE with a small modification, although it in-volves adding a small correction to each obliqueprojection before rebinning. Matej and Kazantsev[38] revived the interest for the direct inversion ofthe 3D x-ray transform by proposing sophisticatedinterpolation techniques in the Fourier domain,provided that the oblique projection data are nottruncated. More recently, the approach derived byDefrise was extended to derive a 3D exact rebinningformula in Fourier space, leading to an iterativereprojection algorithm (iterative FOREPROJ) thatallows estimation of unmeasured oblique projec-tions using the whole set of measured projections[39].

Iterative image reconstruction methods

The primary limitation of the analytic inversionmethods just described is their reliance on theline-integral Equation 7 (ie, the 3D x-ray trans-form), rather than the more general Equation 5. It-erative methods not only accommodate morecomplex (and hence realistic) models of theacquired PET data but also easily allow the use ofnonorthogonal basis functions.

The system matrix and the parameters

Two of the fundamental components to an iterativereconstruction algorithm are (1) the definition ofthe parameters to estimate (usually a set of valuesthat provides a representation of the radiotracerconcentration, such as the vector c in Equation1, and (2) the definition of the system model A(which describes the relation between the radio-tracer distribution and the mean of the measureddata). Note that it is the mean of the measureddata that is normally modeled, and with iterativealgorithms, there is substantial flexibility in howthis mean is modeled and how it is parameterized.Because the system model (mapping from theparameters to the mean) is usually time invariant,this case is considered here. The elements of thesystem matrix (or the system model) A 5 {aij}I�J

are defined by

aij 5

Zsmi

�r�aj

�r�dr ð9Þ

Reader & Zaidi180


Considering the previous Equations 1 and 5along with Equation 9, the matrix-vector relation

q 5 Ac ð10Þ

is obtained, which allows a mean data set q to be‘‘predicted’’ for any given set of parameters c (wherea set of parameters c represents a radioactivity distri-bution). A mean data set q can never be observed inpractice but can be regarded as the average numberof counts that would theoretically be obtained ineach detector pair i by repeating precisely thesame PET scan (of a radioactivity concentrationspecified by c) an infinite number of times.

It is perhaps helpful to visualize the contents ofthe matrix A for a case in which the vector c con-tains the coefficients of voxels (ie, voxel valuessuch that c can be simply rearranged to form a 2Dor 3D image). In such a case, the rows of the matrixA can be regarded as images, with values propor-tional to the probability of a positron emission ineach of the j 5 1.J voxels giving rise to an event be-ing detected by detector pair (or LOR) i (see Fig. 3).Hence, for the case of the x-ray transform (line inte-grals), the probability image is simply an image ofa single line passing through the FOV (the linecould be found by any of the many ray-tracingmethods; eg, see Refs. [40–42]). If positron rangewere to be included, then the line would need tobe broader; for example, by convolving the ray-traced line with a positron-range kernel if theapproximation of shift-invariant blurring is made(there are fast methods for achieving such ap-proaches; eg, see Ref. [43]). If time-of-flight infor-mation were to be included, then a Gaussianfunction (with mean equal to the estimated annihi-lation location and with variance linked to the tim-ing resolution [44]) can be used to modulate thevalues along the line. In general, for the most accu-rate system modeling, essentially every element ofthese probability images that make up the rows of Awould be nonzero. (If, for example, scatter were tobe included in the system model [7], then there isstill a finite probability that positron emissionsfrom voxels remote from the LOR i will result inphotons ultimately detected along i.)

In a similar manner, the columns of the matrix Acan be visualized as ‘‘data images’’ that show the en-tire system response (ie, the mean data q) to a singlepoint source emitter. Thus, for the case of the 3D x-ray transform, each column of A would be a set ofsinograms corresponding to the mean set of 3D si-nograms that would be obtained from a pointsource acquisition. Consequently, there is consider-able impetus to design system matrices that moreaccurately model the detection process within A,through Monte Carlo simulation [45] or even

from actual point source measurements [5]. (Therecent high-definition PET (HD-PET) [46] relieson this kind of improved system modeling to de-liver marked improvements in image quality.)

Objective functions

It is the vector c (which represents the radioactivityconcentration, Equation 1) that needs to beestimated from the noisy measured PET data m.An iterative image reconstruction algorithm aimsto find a vector ck, which produces a vector qk (byway of the system matrix A) that matches in someway with m (where the superscript k on c and q de-notes an estimate number, usually corresponding toan iteration number). Most algorithms seek to min-imize a distance measure between the measureddata m and the vector qk, which can be specifiedas a function of ck. This function is called the objec-tive function. The two most widely used objectivefunctions in PET are least squares (LS) and maxi-mum likelihood (ML). The LS objective concernsfinding the vector ck, which minimizes the follow-ing function:

OLS

�ck�

5XI

i 5 1

�mi � qk

i ðckÞ�2 ð11Þ

where, as previously specified in Equation 10,

qki

�ck�

5XJ

j 5 1

aijckj ð12Þ

The ML objective is based on a statistical modelof the noisy data vector m, and is therefore oftenpreferred in PET. The probability of obtaininga measurement mi (ie, a number of counts m in de-tector pair i) if the mean value was qi

k (ie, if the vec-tor of parameters was ck) is given by the Poissondistribution:

Pr

mi

��qki

5

�qk

i

�mi

mi!exp�� qk

i

�ð13Þ

The value of qik that maximizes this probability is

simply qik 5 mi. For a given ck, a whole set of qi

k

values is obtained (ie, the entire vector qk, by wayof Equation 10 or 12). The likelihood of obtainingthe vector m if the mean was qk is defined by theproduct of these Poisson probabilities.

Pr

m

��qk

5YI

i 5 1

�qk

i

�mi

mi!exp�� qk

i

�ð14Þ

In order, to maximize Equation 14, each of theterms in the product must individually be maxi-mized. Therefore, the concept of the ML estimatebecomes evident: it corresponds to the choice ofck, which gives a vector qk by way of Equation 10,which gives a maximum value for the likelihood



Equation 14. Note that if the measured data hadbeen direct measurements of the parameters of in-terest (ie, if A was the unit matrix I), then the trivialresult of ck 5 qk 5 m is obtained as the ML estimate.It is only the introduction of a nontrivial A that cre-ates the reconstruction problem. The logarithm ofEquation 14 is usually taken to simplify the mathe-matics (but taking the logarithm will not alter thelocation of the maximum of the function):

lnPr

m

��qk

!5XI

i 5 1

milnqki �

XI

i 5 1

qki �

XI

i 5 1

lnmi! ð15Þ

Equation 15 is called the log likelihood. Maxi-mizing the log likelihood also corresponds to min-imizing the Kullback-Leibler distance measurebetween the noisy vector m and the estimate ofthe mean qk. Many other objective functions exist[20], but the main two encountered in PET are theML and the LS objectives as mentioned earlier. Itis also worth noting that if a Gaussian model ofthe measured data statistics is used, then the ML ob-jective corresponds to the LS objective.

Direct algorithms

Before exploring further the iterative methods thatachieve minimization or maximization of a particu-lar objective function, it is worth noting the exis-tence of direct methods for estimating thecoefficients vector c. To clarify the discussion, cwill be regarded as holding the values for pixel orvoxel basis functions such that any estimate bc canbe interpreted directly as a 2D or 3D image. Inessence, a solution vector cs to the set of linearequations

m 5 Acs ð16Þ

is sought, although for many reasons, no such solu-tion vector cs exists (eg, A is rarely square and m isnoisy). Singular value decomposition of A intothe three matrices USVT, however, can allow a pseu-doinverse for a direct LS estimate of c:

bc 5 VS�1UTm ð17Þ

where U and Vare orthogonal matrices and S is a di-agonal matrix (holding the singular value spec-trum). This approach has been successfullydemonstrated in 2D PET [47]; however, for 3DPET, the size of A is typically of the order of 109 �107 elements, which makes such an approach com-putationally impractical at the present time [48](but worthy of consideration in the future). Alter-natively, applying the transpose of matrix A toboth sides of Equation 16 results in

ATm 5 ATAcs ð18Þ

where the overall matrix ATA is square, with col-umns corresponding to the PRF for each possiblepoint source location, and ATm can be interpretedas a backprojected image. Consequently, theestimate

bc 5�ATA

��1ATm ð19Þ

can be found, which corresponds to the familiarnormal equations for LS fitting. The inverse of ATAcould be found by eigenvector decomposition:�ATA

��15�EDET

��15 ED�1ET ð20Þ

where E is the orthogonal matrix of eigenvectorsand D is a diagonal matrix holding the eigenvalues.As with singular value decomposition, the problemis that ATA is a huge, nonsparse matrix (typically>1014 elements), which does not allow straightfor-ward computation of the eigenvectors. If, however,the case of an angular constrained backprojectionalong LORs is considered (as in the BPF method;see the Analytic Image Reconstruction Methodssection), then ATA corresponds to a convolution(ie, its columns contain shifted copies of the PRFor kernel) and, hence, the matrix of eigenvectorscontains the Fourier basis functions, and the diag-onal matrix D simply holds the modulation trans-fer function (the Fourier transform of the PRF/kernel) along its diagonal. This means that an LSsolution can be directly found by Fouriermethods:

bc 5 ED�1ETATm ð21Þ

Equation 21 corresponds to (1) backprojection ofthe measured data (ATm); (2) Fourier transformingthe image (applying the transpose of the eigenvectormatrix E); (3) multiplying point by point in theFourier domain by the inverse of the transfer func-tion (D�1); and finally, (4) inverse Fourier trans-forming (applying E). Noise can be controlled bymodulating, truncating, or adding an offset to theFourier domain filter contained in the diagonal ofD�1.

To summarize, A and ATA are huge matrices thatdo not readily lend themselves to pseudoinversionmethods, and Fourier domain methods can beapplied only when the columns of ATA correspondto shifted copies of the same PRF (ie, a convolu-tion). Therefore, although such direct approacheswould be desirable (to avoid issues of conver-gence), they pose a significant computational chal-lenge. Iterative techniques, on the other hand, oftenonly need row or column access of A and oftenavoid explicit storage of the matrix A through on-the-fly calculation, through factorization of A intomanageable components (see later discussion), orboth.

Reader & Zaidi182


Iterative algorithms

Most iterative reconstruction algorithms in PET relyon finding the gradient of the objective function,which is a basic principle of optimization. Forexample, if the objective function were quadratic,then the gradient would be linear. Any estimate ofthe location of the maximum (or minimum) ofthe objective function could then be improved bysimply adding the value of the gradient at thecurrent location. A key question is how much ofthe gradient to add on (known as the step size).Hence, the basic form of an iterative reconstructionalgorithm is often

ck11j 5 ck

j 1lkvOðckÞvck

j

ð22Þ

where lk is the step size. For the case of c corre-sponding to voxel values, the gradient can be re-garded as an image. For the log-likelihoodobjective, the gradient image is given by differenti-ating Equation 15:

v

vckj

O�ck�

5XI

i 5 1

aij

mi

qki

�XI

i 5 1

aij ð23Þ

Equation 23 is just a backprojection of what canbe considered as correction factors for each i (themi/qi

k ratio, which indicates whether the current es-timate qk of the mean of the data is above or belowthe noisy data m) minus the sensitivity image(PI

i51 aij). The sensitivity image corresponds to thebackprojection of every possible system LOR; thatis, the summation of all the ‘‘images’’ that make upall the rows of the matrix A. Referring again toFig. 4, for the line-integral model, it would meanadding together every possible line through theFOV. There is effectively a choice in how much ofthis gradient image to add to obtain the nextestimate of c. If a step size of

lk 5ck

jPIi 5 1

aij

ð24Þ

is chosen and substituted into Equation 22 alongwith Equation 23 to obtain

ck11j 5 ck

j 1ck

jPIi 5 1

aij

XI

i 5 1

aij

mi

qki

�XI

i 5 1

aij

!ð25Þ

then the well known expectation maximization(EM) algorithm is obtained [49,50]:

ck11j 5

ckjPI

i 5 1

aij

XI

i 5 1

aij

mi

qki

ð26Þ

In matrix-vector form, this can be written as

ck11 5ck

AT1AT m

qkð27Þ

where 1 is a vector with all elements set to one, andelement-by-element vector multiplication and divi-sion is assumed for simplicity of notation, butwhere matrix-vector multiplication remains con-ventional (eg, see Ref. [51]). The notation of Equa-tion 27 becomes useful when data corrections anddecomposition of the system matrix are used (seethe Decomposition of the System Matrix, IncludingAttenuation, Normalization, Scatter, and Randomssection) to prevent excess writing of summationsand subscripts. Hence, the core iterative process ofthe EM algorithm of Equation 26 involves (1) for-ward projection of the current estimate (qk 5 Ack)to model the mean; (2) taking the ratio of the mea-sured data to this mean (m/qk); (3) backprojectingthis ratio [AT(m/qk)] to form a multiplicative correc-tion image; and (4) multiplying the correction im-age by the current estimate of ck to obtain the newestimate ck11. It is also necessary to normalize forthe number of contributions to the correction im-age, hence the division by AT1, which representsthe backprojection of every possible LOR (ie, the su-perposition of every row of the system matrix A, aspreviously mentioned, to obtain the sensitivityimage).

Variants on Equation 26 include over-relaxation[52–54] and methods that perform a line search(a 1D optimization of l to find the step size thatmaximizes the log likelihood). This latter approachcorresponds to the steepest ascent algorithm [55].Another important modification is to divide thedata vector m into L subsets S1.SL, whereby fromthe L subsets of the vector m, L updates of the esti-mate of c can be performed:

ck;l11j 5

ck;ljP

i˛Sl

aij

Xi˛Sl

aij

mi

qk;li

ð28Þ

This modification leads to a much more compu-tationally efficient update because the amount ofprocessing per update is approximately a factorL smaller. This block-iterative version of the EMalgorithm is referred to as ‘‘ordered subsets’’ EM(OSEM) [56]. The problem with OSEM is that con-vergence to an ML estimate is in general lost, andinstead, a limit cycle corresponding to the numberof subsets is encountered. This loss of convergencecan be overcome through consideration of previ-ous corrective images in each update [57] orthrough the use of a relaxation parameter Lk as



is done in the row-action ML algorithm (RAMLA)[58,59]:

ck;l11j 5 ck;l

j 1Lkck;l

jPi˛Sl

aij

Xi˛Sl

aij

mi

qk;li

�Xi˛Sl

aij

!ð29Þ

This algorithm is simply Equation 25 but with theuse of subsets and relaxation. Another interestingexample is to combine over-relaxation with subsets(such as the high–over-relaxation single-projectionmethod [60]).

Turning now to the gradient of the LS objective,

v

vckj

OLS

�ck�

5 2XI

i 5 1

aij

mi � qk

i

�ck�

ð30Þ

it can be seen that algorithms such as the algebraicreconstruction technique [61] use a form of thisgradient:

ck;l11j 5 ck;l

j 1lkaij

�mi � qk;l

i

�Pj

�aij

�2 ð31Þ

This algorithm is equivalent to taking each LOR ias a subset of the vector m for each update and us-ing the gradient of the LS objective with a relaxedstep size.

Faster gradient methods include the precondi-tioned conjugate gradient method, which usuallytakes the form of the following steps for PET[20,55,62,63]. First, a modified gradient image isobtained by multiplication of the current gradientimage vector gk by a preconditioning matrix C:

dk 5 Ckgk ð32Þ

where the preconditioning matrix is normally takento be diagonal, with elements inspired by the stepsize of the gradient used in the EM algorithm (thestep size given in Equation 24), such as

Ck 5 diag

(�ck

j 1d .X

i

aij

)ð33Þ

where d is a small constant. Another step size is thenfound through

bk�1 5ðgk � gk�1ÞTdk

gk�1T dk�1ð34Þ

which is used for the updating step of a vector a de-fined by

ak 5 dk1bk�1ak�1 ð35Þ

which is finally used to update the vector of param-eters to be estimated:

ck11 5 ck1akak ð36Þ

Equation 36 then requires a 1D optimization tofind the ak that maximizes the objective function;this is nontrivial for the ML case and, often,a Newton-Raphson method (with a positivity con-straint) is employed. More recently, even fastergradient techniques using subsets have beendevised [64].

Updating methods

It is unfortunate that the choice of algorithm is oftendictated by the type of data to reconstruct from: list-mode data or projection data. For list-mode data,a so-called RAMLA (such as EM and its variantsOSEM and row-action ML algorithm) is often essen-tial for practical reasons. A row-action method is onethat accesses the rows of the system matrix one ata time, which corresponds directly to accessing indi-vidual events in the list-mode data. Furthermore, forlist-mode data, it is impractical to consider algo-rithms that require every possible LOR to be ac-cessed for each update [65]. As a result, methodssuch as ART (Equation 31) are impractical for list-mode data because the zero values in the vector mneed to be accounted for in each update. Likewise,column-based (ie, voxel-based) methods areimpractical for list-mode data because the entiretyof the randomly ordered list-mode file needs to beconsidered for each voxel update. For projectiondata, the choice of a column-based algorithm,whereby a whole column of the system matrix isrequired for a single update of a voxel, is practical.Examples of column-based algorithms includespace-alternating generalized EM [66], iterativecoordinate ascent, grouped-coordinate ascent, andsuccessive over-relaxation [67] and related methods.As stated, these approaches can be readily imple-mented for projection data but tend to beimpractical for list-mode data.

Regularization

A key problem often encountered in iterative recon-struction is the ill conditioning of the inverse prob-lem such that a trivial perturbation in the measureddata m gives rise to a nontrivial perturbation in theimage estimate. This issue can be understoodthrough consideration of the singular value spec-trum of the matrix A (or in simpler terms, the mod-ulation transfer function for the diagonalization ofATA in the very special case in which this is a convo-lution matrix). The values in the spectrum decay, in-dicating the decreasing efficiency with which thevarious singular vector components of the imageare measured by the PET scanner. Any algorithmthat progresses toward effectively achieving anytype of inversion of A will be performing the equiv-alent of inverting the decaying spectrum of singularvalues (or inverting the transfer function in the

Reader & Zaidi184


special case of Fourier), and division by the increas-ingly small values in the singular value spectrum cancause significant noise amplification. Hence, regula-rization methods such as postsmoothing, early ter-mination of the iterative process, or bayesianpriors (used to modify the ML objective to a maxi-mum a posteriori [MAP] objective) are often usedto counteract this. Of these possibilities, early termi-nation of the iterative process is by far the most pop-ular choice, but this can lead to problems (ifquantification is essential) because spatially variantand object-dependent convergence occurs withnonlinear updating algorithms such as EM. MAPmethods or postsmoothing (after reaching conver-gence) can counteract these problems. Qi and Leahy[20] provide a good review of MAP regularization. Asignificant amount of prior information on imagesmoothness, however, can be accounted for in themodeling of the mean of the data—that is, a choiceof spatial basis function that is no longer a voxel butrather a smooth function (such as spherically sym-metric basis functions; for example, ‘‘blobs’’[12,68], or even a cluster of voxels [69]).

Decomposition of the system matrix,including attenuation, normalization,scatter, and randoms

As previously mentioned, the system matrix A for3D PET can be prohibitively large. To make thematrix manageable, on-the-fly calculation can beperformed (ie, the system matrix elements are cal-culated as and when they are required) or symmetryand compression techniques can be exploited (eg,see Refs. [70,71]). A popular approach is to decom-pose the matrix A into components that are individ-ually easy to calculate or easy to store (eg, see Refs.[72,73]). An example of this approach would be

A 5 NDLXH ð37Þ

where X is a geometric projection matrix (eg, thiscould perform line-integrals such that each row ofX can be interpreted as an image of a line throughthe FOV); where H models image space resolution(eg, positron range); D models resolution effects inthe detector space; L is a diagonal matrix accountingfor attenuation along each LOR; and N is a diagonalmatrix giving the inverse of the normalization cor-rection factor for each LOR. Specific examplesinclude using a convolution operation for H(whereby the columns of the matrix contain shiftedcopies of the resolution kernel), a ray tracing algo-rithm (such as the Siddon method [41] or that ofJoseph [40]) for X, and a convolution matrix forD. Note that the matrix X can be modified toaccount for time-of-flight information, throughreplacement of the implicit uniform weightingalong each line by a Gaussian probability density,

with a mean and variance corresponding to the pho-ton pair detection time difference and the timingresolution (see Fig. 3). The advantage of the decom-position of the form shown in Equation 37 is thateach matrix can be stored (in the case of the diagonalmatrices N and L and possibly X if symmetry is ex-ploited) or easily calculated on the fly (in the caseof D, X, and H). Inclusion of scatter and randomsis normally done in an additive way such that themean of the data is modeled by

q 5 NDLXHc1s1r ð38Þ

where s and r are estimates of the mean of the scat-ter and randoms, respectively, in the measurementspace. Scatter could be included in the matrix[7,71], which is strictly the correct way to handlescatter [74,75] (because s is normally not knownwithout a first estimate of the true activity distribu-tion). Using the model of Equation 38, the EM al-gorithm of Equation 27 becomes

ck11 5ck

HTXTLTDTNT1HTXTLTDTNT m

NDLXHck1s1r

ð39Þ

Equation 39 handles factors such as attenuationand scatter by means of modeling (rather than byexplicit corrections such as subtraction of the scat-ter) and, as such, the statistical form of the data vec-tor m is preserved as Poisson. Hence, an algorithmsuch as Equation 39 is often referred to as an ‘‘ordi-nary Poisson’’ version of the EM algorithm. Thereare important advantages to such an approach; forexample, the attenuation is included as a forwardmodeling step rather than as a precorrection (aswas done for Equation 8), which more optimallyweights the acquired data (hence the name ‘‘attenu-ation weighted’’ OSEM [76]).

Patient motion can also be included througha system modeling approach (eg, using the Polarissystem to supply a feed of movement informationsynchronized with the acquired PET data) [77].

Four-dimensional methods

4D image reconstruction methods aim to moreoptimally use the acquired PET data to reconstructa time series of images or even to directly estimatethe kinetic/functional parameters of interest. Thereare two basic approaches: (1) estimation of coeffi-cients for temporally extensive basis functions toinclude the time dimension in the reconstructionand (2) estimation of the time-dependent func-tional parameters (eg, blood flow) directly fromthe data. The motivation behind both approachesis to avoid the independent time-frame reconstruc-tions used for conventional dynamic PET, which arecharacterized by relatively poor signal-to-noise



ratio arising from the very limited number ofcounts available for each time frame.

Reconstruction of independent time framesignores temporal correlations, whereas the use oftemporally extensive basis functions allows eachtime point in the time series reconstruction todraw from more, if not all, of the acquired data.Examples of these types of method include Nichols[78], who used spline temporal basis functions; Mat-thews [79], who used singular value decompositionto derive a set of basis functions; Reader [69], who es-timated the temporal basis functions from the dataduring reconstruction; and Verhaeghe [80], whoconsidered five-dimensional basis functions (thefurther dimension coming from gating the data).The principle of such methods when applied totime-dependent PET data m can be represented by

ck11 5ck

BTATt 1

BTATt

m

AtBck1bð40Þ

where the system matrix At, often assumed to betime independent, holds t 5 1.T copies of thetime-independent matrix A previously considered(see The System Matrix and the Parameters section);the matrix B holds the temporal basis functions;and b holds the time-dependent estimates of thescatter and randoms. An example result from thiskind of algorithm is shown in Fig. 5.

There is also an increasing move toward task-oriented image reconstruction such that the finalpurpose of the images is accounted for within thereconstruction process (eg, to deliver higher resolu-tion and better quality images suited to the task(Fig. 6) or to directly estimate the kinetic parame-ters of interest). One specific example is for the

case of [18F]fluorodeoxyglucose (FDG), wherebyimages of the metabolic rate of glucose are sought,rather than images of the time course of FDG con-centration in tissue. The work of Kamasak [17] isa good example of this, which estimated the kineticparameters for the two-tissue compartment modelat each voxel by using a parametric iterative coordi-nate descent algorithm (each voxel’s parameter setis updated to maximize the likelihood). Suchapproaches are based on the original idea of Carson(the EM parametric image reconstruction algo-rithm) [81]; methods that extend this approachfurther (eg, to include estimation of the input func-tion) have followed [82]. Another recent approachis that of Fu and colleagues [18], whereby theparameters of a Patlak plot [83] (a plot normallyperformed as a postreconstruction analysis to deter-mine the irreversible radiotracer uptake) are esti-mated directly within a preconditioned conjugategradient reconstruction.

Summary

Image reconstruction for PET has progressed fromthe use of FBP algorithms based on analytic inver-sion formulae (reliant on a simple line-integralmodel) to the use of iterative algorithms that allowmore accurate modeling of the PET data (in termsof the acquisition process and the statistical noise).Iterative methods (which are often terminatedbefore convergence and often use a positivity con-straint) usually deliver visually improved imagequality that perhaps cannot be matched by directmethods, even when they use the same systemmodel (eg, pseudoinversion by singular value

Fig. 5. Impact of 4D reconstruction methodology for one 5-minute frame of a 60-minute [11C]flumazenil studyusing the HRRT. (Left) Sagittal slice, reconstructed using an EM algorithm based only on the line-integral model(ie, A 5 X). (Middle) The same slice reconstructed with list-mode EM including a resolution model with the line-integral model (ie, A 5 XH). (Right) The same slice reconstructed using a 4D method that is able to benefit fromall 60 minutes of data without compromising temporal resolution (although it is dependent on the type of tem-poral basis functions chosen or estimated). (The raw HRRT PET data are courtesy of CEA–Service Hospitalier Fred-eric Joliot, Michel Bottlaender, Orsay, France.)

Reader & Zaidi186


decomposition). It is well known, however, thatnonlinear algorithms such as EM demonstratespatially variant and object-dependent convergenceproperties, and any reconstruction based on earlytermination of the iterative process must be usedwith caution. The final purpose of the images needsto be carefully considered when selecting a recon-struction algorithm and its associated parameters(iterations, regularization, and so forth). If thepurpose of the PET images is to determine whethertumors are present (lesion detection), then incom-pletely converged EM images may serve well in thisrole. On the other hand, if quantitative measures ofblood flow or receptor density were needed, thena converged reconstruction (or even an FBP recon-struction) would be more appropriate.

There is still further scope for progress in PETimage reconstruction, which can perhaps besummarized by the following three points:

1. Hardware developments mean that the acquiredPET data vector m is increasingly rich in infor-mation and can now include high-precisionmeasurements of photon detection positions,timing, photon arrival time differences (time offlight), and photon energy. In addition, mea-surements of patient movement and respiratoryand cardiac gating information are increasinglyavailable as parallel streams of information[77]. To use these data in their richest formwould suggest the need for list-mode data stor-age and reconstruction, rather than binning ofdata into projections (which can sometimesimpose a level of sampling that compromisesthe original measurement information).

2. Improved accuracy in the definition of the prob-abilistic system transfer matrix A is needed. Thismay involve accurate Monte Carlo modeling ofthe positron emission, photon emission, trans-port and detection processes [45], with themodeling making use of CT or MR images for

anatomic information. The photon interactionsneed to be considered not only in the individualpatient but also for the detection hardware.Ideally, there needs to be a comprehensive prob-abilistic mapping relating the radioactivity dis-tribution to each possible list-mode event.

3. More careful definition of the parameters to beestimated is needed. Are images of radiotracerconcentration required or should images ofblood flow, receptor density, or metabolic ratesbe directly estimated from the raw PET data?There is no reason why PET reconstructioncannot be tailored to address specific clinicalresearch questions through direct estimation ofthe pertinent parameters of interest.

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