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UsingDeconvolutiontoImprovePETSpatial

ResolutioninOSEMIterativeReconstruction

G.Rizzo1,I.Castiglioni1,G.Russo2,M.G.Tana2,F.Dell’Acqua1,M.C.Gilardi1,F.Fazio1,

S.Cerutti2

1IBFM-CNR,UniversityofMilano-Bicocca,SanRaffaeleScientificInstitute,Milan,Italy

2DepartmentofBiomedicalEngineering,PolytechnicUniversity,Milan,Italy

Summary

Objectives:AnovelapproachtothePETimage

reconstructionispresented,basedontheinclusion

ofimagedeconvolutionduringconventionalOSEM

reconstruction.Deconvolutionishereusedtoprovide

arecoveredPETimagetobeincludedas“apriori”

informationtoguideOSEMtowardanimproved

solution.

Methods: Deconvolutionwasimplementedusing

theLucy-Richardson(LR)algorithm:Twodifferent

deconvolutionschemesweretested,modifyingthe

conventionalOSEMiterativeformulation:1)Webuilt

aregularizingpenaltyfunctionontherecoveredPET

imageobtainedbydeconvolutionandincludedit

intheOSEMiteration.2)Aftereachconventional

globalOSEMiteration,wedeconvolvedtheresulting

PETimageandusedthis“recovered”versionasthe

initializationimageforthenextOSEMiteration.

Testswereperformedonbothsimulatedand

acquireddata.

Results:ComparedtotheconventionalOSEM,both

thesestrategies,appliedtosimulatedandacquired

data,showedanimprovementinimagespatial

resolutionwithbetterbehaviorinthesecondcase.

Inthisway,smalllesions,presentondata,couldbe

betterdiscriminatedintermsofcontrast.

Conclusions:Applicationofthisapproachtoboth

simulatedandacquireddatasuggestsitsefficacy

inobtainingPETimagesofenhancedquality.

Keywords

Deconvolution,OSEMreconstruction,PETspatial

resolutionrecovery

MethodsInfMed2007;46:231–235

1.Introduction

Positronemissiontomography(PET)iscur-

rentlyuniversallyrecognizedinmanyfields

(such as oncology or neurotransmission

studies) as a molecular imaging modality

able to provide unique insight into the bio-

logical aspects of the process under exami-

nation.ThepotentialofPETmethodologyis

however limited by its poor spatial resol-

ution and low signal to noise ratio (SNR),

which reduce visibility of the details and

quantification accuracy.

In recent years, many efforts have been

madetoimprovethequalityofPETimages,

resulting in both hardware and software

development.Thishasledtotheavailability

of commercial PET scanners, where the

shields among the detector rings (present in

the conventional 2D acquisition modality)

can be removed to perform 3D acquisition;

in 3D, detection efficiency increases, con-

sequently improving SNR [1]. 3D scanners

are typically provided with statistical iter-

ative ordered subset expectation maxi-

mization (OSEM) reconstruction, which is

known to produce images of higher quality,

in terms of both resolution and contrast,

than the more classical analytical algo-

rithms, based on filtered back projection

(FBP) [2].

However, within an OSEM reconstruc-

tionscheme,thequalityofthereconstructed

PET images depends on a variety of imple-

mentationchoices:thedefinitionofthesys-

tem matrix, which models geometrical as-

pects of signal generation and can also ac-

count for signal degradation (e.g. detector

normalization, system blurring etc.), the

presence of regularization terms aimed at

controlling noise propagation, and the in-

clusion of Metz restoration filters to im-

prove spatial resolution, all of which con-

tribute to the final quality of the recon-

structed images.

In this work, we propose a novel ap-

proach to the PET image reconstruction,

which is based on the well-assessed OSEM

approach and includes deconvolution as a

specific factor to improve the spatial reso-

lutionofthefinalPETimages.Essentially,a

recovered version of the PET image, as ob-

tained by deconvolution, is used during

OSEM as “a priori” information in order to

addressthefinalOSEMsolutiontoanimage

withimprovedspatialresolutionproperties.

The proposed approach is applied to

simulated and real data in order to test its

efficacy.

2.Methodology

2.1 UsingDeconvolutioninOSEM

Reconstruction

OSEM algorithm is an accelerated version

of the maximum likelihood (ML) EM algo-

rithm. ML-EM performs PET image recon-

struction, according to the following itera-

tive formula [2]:

(1)

with: λjnthe j-th image voxel at the n-th

iteration, i the detector pair index, yithe

measuredcountsatdetectorpairi.Aijrepre-

sents the probability that annihilation

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photon pairs emitted from voxel j are de-

tected in the detector pair i. The set of aij

composes the so-called system matrix,

which typically describes the geometrical

model of the acquisition system. In fact,

OSEM updates the λjvoxel values, using a

weight factor equal to the ratio between the

measured projection data (yi) and an esti-

mate of them ( ).

With respect to ML-EM, OSEM imple-

ments each iteration as a number of sub-

sequent sub-iterations, during which only a

subset of the acquired data is used. This

scheme results in an accelerated algorithm

convergence [2].

The relation in (1) is often used in a

modified version (OSEM One Step Late,

OSEM-OSL), which includes a penalty

function to regularize the solution as the

iterations proceed (see Eq. 2 in Fig. 1),

where U(λn, j) is a potential function and β

a weight factor [3].

Startingfrom(2)and(1)respectively,we

implemented two different ways to include

deconvolution in the OSEM iterative

schemes:

1)Priorterm:in(2)webuiltthepotential

function U(λn, j) on the recovered PET

image obtained by deconvolution (fdeconvj),

by modifying the relation (see Eq. 3 in

Fig. 1), where now the penalty function

weighs the difference between each voxel

value and the corresponding “recovered”

value.

2) Inter-iteration deconvolution: after

eachconventionalglobalOSEMiterationas

in (1), we deconvolved the resulting image

andusedthe“recovered”versionofthePET

imageastheinitializationimageforthesub-

sequent OSEM iteration.

2.2 DeconvolutionAlgorithm

We know that, if the Point Spread Function

(PSF) of an imaging system, h(x,y,z), is

space invariant and linear, the spatial dis-

tribution of any physical quantity, f(x,y,z),

anditsrepresentation,g(x,y,z),acquiredby

that imaging system, are related as

g(x,y,z)=f(x,y,z)⊗h(x,y,z)+n(x,y,z) (4)

where n(x,y,z) is the superimposed noise

and ⊗ the convolution operator. In decon-

volution,f(x,y,z)isrecoveredfromg(x,y,z),

once h(x,y,z) is known.

In this work, we implemented decon-

volution using the Lucy-Richardson (LR)

algorithm.LRalgorithmperformstheesti-

mation of f(x,y,z), once the PSF of the

system is known, by the iterative formu-

lation (see Eq. 5 in Fig. 2), where rk(x,y,z)

= fdeconv

ation step and ∗ is the correlation operator

[4].

In our case, as the PET scanners space

variance of h(x,y,z) had to be taken into

account, Equation 4 was reformulated as a

superposition integral (see Eq. 6 in Fig. 2).

Consequently, in (5) h(x,y,z) was in-

tended as h(x,y,z;α,β,γ), changing shape

at each {x,y,z} voxel point.

k(x,y,z) ⊗ h(x,y,z), k is the iter-

It can be noted that the LR algorithm

closely resembles the ML-EM algorithm,

bothbeingderivedfromtheBayes’theorem

in the case of Poisson noise. Therefore, the

formulation in (5) is similar to the formu-

lation in (1), with the difference that in (5)

the factor that weighs the ratio between

measuresandestimatesisapplieddirectlyto

theimage,ratherthantotheprojectiondata.

3.ExperimentalProtocol

3.1 DeterminationofthePET

ScannerPSF

The PET scanner considered in this work is

the ECAT EXACT HR+. It consists of

32 BGO detector rings, acquiring, in 3D

modality, a total of 1024 sinograms

(288 × 288 matrices, bin size = 2.25 mm).

Forthissystem,

h(x,y,z;α,β,γ) was calculated, starting

from the values of Full Width Half Maxi-

mum(FWHM),measuredbyAdamandco-

workers, using point sources (64Cu small

spheres) located in different positions with

respect to the center of the scanner field of

view (FOV)[5]. We linearly interpolated

FWHM values, in radial, tangential and

axial directions, in order to express FWHM

as a continuous function in the 3D space.

H(x,y,z;α,β,γ) was then modeled as a 3D

Gaussian function, centered at each voxel

value {x–,y–,z–} and obtained, by convol-

ution, from the three mono-dimensional

FWHM Gaussians along the radial, tangen-

tial and axial directions.

Finally,foreveryvoxel{x–,y–,z–},thecor-

responding H(x–,y–,z–;α,β,γ) was discret-

ized into a 3 × 3 × 3 voxel mask.

the3DPSF,

3.2 PETImageReconstruction

The 3D PET sinograms were scatter-

precorrected in 3D, using MC-based cor-

rection [6].

Image reconstruction was carried out

using the STIR Software Package v. 1.3

(http://stir.sourceforge.net). After recon-

struction, a set of 63 contiguous tomo-

graphic slices was obtained, imaged as

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Fig. 1

Equations2and3

(2)

(3)

Page 3

128 × 128 matrices (pixel size = 4.0 mm,

slicethickness=2mm).OSEMwasapplied

by using the conventional attenuation-

weighting version (AWOSEM), which in-

cludes the attenuation map within the sys-

tem matrix, in order to preserve the statisti-

cal properties of the sinograms.The OSEM

optimized configuration set-up proposed in

[7] was adopted, consisting of three global

iterations, 12 subsets, and Gaussian low-

pass filtering (4 × 4 × 2 FHWM) every six

subiterations as a regularization step.

OSEM reconstruction was also per-

formedbyincludingthepriortermbasedon

deconvolution (PRIOR-DEC) and by ad-

ding an inter-iteration

(INTER-DEC),asdescribedinSection2,in

order to compare the final image quality of

these three different approaches. Both

PRIOR-DEC and INTER-DEC schemes

wereimplementedbyinterfacingSTIRwith

homemade modules, developed in MAT-

LAB 6.5 (The Mathworks Inc., Sherborn,

MA, USA).

The iteration number of the LR algo-

rithm was set to 3, as a trade-off between

processing time and recovery level. The

choiceofalownumberofiterationalsopro-

tects against an excessive noise increase,

duetothenon-Poissonnoise,superimposed

onto the PET image, which could degrade

the algorithm performance.

PRIOR-DEC was performed with the β

factor equal to 0.5.

deconvolution

3.3 SimulatedData

The three reconstruction strategies were

tested on 3D data generated by the Monte

Carlo code PET-EGS, which simulates the

ECATHR+scannerwithhighaccuracy[8].

Three radioactive linear sources, posi-

tioned at 1 cm, 10 cm, and 20 cm out of the

scanner FOV were simulated, in order to

permit,inasimplecondition,theestimation

of the quantitative FWHM values of the re-

constructed PET images.

An anthropomorphic phantom, derived

from a real whole-body tumor-staging

PET/CT18F-FDG study, was also used to

evaluate the global image quality in a more

compleximagepattern.Thephantomrepre-

sents the abdominal district, in which small

high-activity spots (4, 8, 12 mm) were in-

serted as simulation of neoplastic lesions.

3.4 AcquiredData

The “Jaszczak” phantom with cold spots,

conventionally used for the evaluation of

image spatial resolution and contrast, was

acquired with the ECAT HR+ PET scanner.

The phantom consists of a Lucyte cylinder

with six triangular arrays of solid bars, with

diameters equal to 12.7 mm, 11.1 mm, 9.5

mm,7.9mm,6.4mm,4.8mm,respectively.

The phantom was filled with a water sol-

utionof18F(400µCiin5640cc)andatotal

of 14 MCounts was collected. Before re-

construction, the 3D sinograms were pre-

corrected for normalization to compensate

differences in detection efficiency among

detectors.

4.Results

4.1 SimulatedData

Tables 1 and 2 show FWHM values calcu-

lated for the linear sources reconstructed

followingthethreeapproaches,fortheradi-

al and the tangential components, respec-

tively. FWHM were derived, drawing an in-

tensityprofileonthelinesourceimagesand

fitting it by Gaussian function. Values ob-

tained for all the tomographic planes con-

taining the line sources were averaged. It

can be noted that the use of either PRIOR-

DEC or INTER-DEC improves spatial res-

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UsingDeconvolutiontoImprovePETSpatialResolution

MethodsInfMed2/2007

Fig. 2

Equations5and6

Reconstructionconfiguration Linesourceposition

1cm

4.56±0.06

4.43±0.06

4.39±0.05

AWOSEM

PRIOR-DEC

INTER-DEC

10cm

4.69±0.07

4.14±0.06

4.25±0.06

20cm

6.59±0.1

3.73±0.07

4.07±0.06

ReconstructionconfigurationLinesourceposition

1cm

5.13±0.06

4.40±0.05

4.56±0.05

AWOSEM

PRIOR-DEC

INTER-DEC

10cm

5.41±0.07

4.85±0.05

4.81±0.05

20cm

5.43±0.1

4.27±0.05

4.43±0.06

Table 1

RadialFWHM(mm)

Table 2

TangentialFWHM(mm)

(5)

(6)

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olution, especially in the radial direction.

Furthermore, the final spatial resolution

presents more uniform values in the tomo-

graphic plane.

Figure 3 shows a tomographic image of

the anthropomorphic phantom, recon-

structed with the three reconstruction strat-

egies.Aprogressiveimprovementofspatial

resolution could be noted from AWOSEM

to PRIOR-DEC to INTER-DEC. Some of

the simulated small lesions are visible in all

three reconstructed images, with better

contrast when INTER-DEC was used.

4.2 AcquiredData

Figure 4 shows a slice of the “Jaszczak”

phantom, obtained with the three recon-

struction strategies. Progressive improve-

ment in spatial resolution can be observed

from AWOSEM to PRIOR-DEC and

INTER-DEC. From the quantitative point

ofview,thebehaviorcanbeobservedinFig-

ure 5, showing the intensity profile of the

threeimages,drawnacrossfourholesofthe

first sector (largest holes).The background

signal, theoretically equal to zero, becomes

progressively lower for PRIOR-DEC and

INTER-DEC.

Fig. 3

ing the inserted simulated neoplastic lesions (a); the corresponding images obtained with

AWOSEM(b),PRIOR-DEC(c)andINTER-DEC(d)

Anthropomorphicphantom:originalPETslice,usedasinputinMCcodeandshow-

Fig. 4

respondingimagesobtainedwithAWOSEM(b),PRIOR-DEC(c)andINTER-DEC(d)

“Jaszczak” phantom: A CT slice showing the geometry of the object (a); the cor-

Fig. 5

obtainedbydividingeachvoxelcountforthemaximumcountoftheprofile.

“Jaszczak”phantom:Theintensityprofile,drawnacrosstheholesofthesectors1:AWOSEM(darkgray),PRIOR-DEC(middlegray),INTER-DEC(lightgray).Normalizedcountswere

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5.Discussion

In3DPET,theiterativeOSEMalgorithmis

now the most widely used approach in

image reconstruction, yielding a better

imagequality,comparedtotheconventional

FBP.

The spatial resolution of the recon-

structed PET images could be further im-

proved, by slightly modifying OSEM, in

ordertotakeintoaccountthePSFofthesys-

tem during the estimation. A recent ap-

proach has been to include all the degrada-

tion factors within the OSEM system ma-

trix, in order to “recover” projection data,

before using them [9]. Here we proposed a

differentwayoftakingintoaccountthesys-

temPSF,byprovidinga“recovered”version

of the reconstructed PET image, obtained

by deconvolution, during OSEM iterations.

The idea is to guide OSEM solution, using

therecoveredPETimageas“apriori”infor-

mation.

Weproposedandcomparedtwodifferent

implementations. In the first case, we

adopted the classic OSEM-OSL scheme, in

which a penalty function is added to guide

thesolution:thepenaltyfunctionweighsthe

difference between each estimated image

pixel value and the corresponding pixel

value “recovered” by deconvolution. In the

second case, we applied deconvolution to

theimageobtainedaftereachOSEMglobal

iteration,inordertoprovidearecoveredver-

sion of it to be used as the initialization

image for the next iteration step.

Compared to the conventional OSEM,

both these strategies, applied to simulated

and acquired data, show an improvement in

image spatial resolution with better beha-

viorinthesecondcase,thusconfirmingthe

efficacy of our idea. Further improvements

can be found in the optimization of the LR

algorithm, for instance relative to the

number of iterations and the noise model.

A very frequent situation in a clinical

contextisa3DPETacquisitionprotocolas-

sociated to 2D reconstruction after sino-

grams rebinning, in order to speed up pro-

cessing time. The application of our ap-

proach in 2D is straightforward, requiring

only easy modification in the definition of

PSF. Furthermore, in the case of 2D recon-

struction the efficacy of our strategies

should be even more enhanced, given the

more pronounced spatial resolution degra-

dation induced by the rebinning procedure.

6.Conclusions

Within OSEM reconstruction, the use of

image deconvolution is effective in improv-

ing image spatial resolution, as shown by

simulatedandacquireddata.Theeaseofim-

plementationandpositiveresultssupportits

application in a clinical context.

Acknowledgments

The authors wish to thank Michael John for English

revision and Lorena Bonaldi for figure editing.

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