Research Article Evaluation of the Feasibility and ...

Hindawi Publishing CorporationISRN Biomedical ImagingVolume 2013 Article ID 943051 11 pageshttpdxdoiorg1011552013943051

Research ArticleEvaluation of the Feasibility and Quantitative Accuracyof a Generalized Scatter 2D PET Reconstruction Method

Hongyan Sun12 and Stephen Pistorius123

1 Department of Physics and Astronomy University of Manitoba Allen Building Winnipeg MB Canada R3T 2N22 CancerCare Manitoba Winnipeg MB Canada R3E 0V93Department of Radiology University of Manitoba Winnipeg MB Canada R3E 0V9

Correspondence should be addressed to Hongyan Sun hongyansuncancercarembca

Received 21 December 2012 Accepted 16 January 2013

Academic Editors N Belcari Y Chen and F Rannou

Copyright copy 2013 H Sun and S Pistorius This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Scatter degrades the contrast and quantitative accuracy of positron emission tomography (PET) images and most methods forestimating and correcting scattered coincidences in PET subtract scattered events from the measured data Compton scatteringkinematics can be used to map out the locus of possible scattering locations These curved lines (2D) or surfaces (3D) whichconnect the coincidence detectors encompass the surface (2D) or volume (3D) where the decay occurs In the limiting case wherethe scattering angle approaches zero the scattered coincidence approaches the true coincidenceTherefore both true and scatteredcoincidences can be considered similarly in a generalized scatter maximum-likelihood expectation-maximization reconstructionalgorithm The proposed method was tested using list-mode data obtained from a GATE simulation of a Jaszczak-type phantomFor scatter fractions from 10 to 60 this approach reduces noise and improves the contrast recovery coefficients by 05ndash30compared with reconstructions using true coincidences and by 30ndash245 with conventional reconstruction methods The resultsdemonstrate that this algorithm is capable of producing images entirely from scattered photons eliminates the need for scattercorrections increases image contrast and reduces noise This could be used to improve diagnostic quality andor to reduce patientdose and radiopharmaceutical cost

1 Introduction

Compton scattering degrades image contrast and compro-mises quantitative accuracy in positron emission tomography(PET) [1 2] Scattered coincidences are typically consideredas noise which reduces PET image quality This issue is moreserious when operating in 3D mode without slice-definingsepta and in large patients where the scatter fraction can be ashigh as 40ndash60 [3ndash6] Consequently a number of approachesfor estimating and correcting scattered coincidences in PETdata have been proposed [3 7ndash19] Most of these techniquesestimate a scatter sinogram which is used to subtractthe scatter from the projection data [20] in precorrectionmethods [21] or as a constant additive term incorporatedin a reconstruction algorithm [22ndash24] Inaccuracy in theestimation of the scatter sinogram will introduce significant

biases in the activity distribution [6] The subtraction-basedcorrection methods destroy the Poisson nature of the datareduce the systemrsquos sensitivity and amplify image noise [325]

With list-mode acquisitions in modern PET and im-proved detector technology the use of the energy of indi-vidual photons becomes feasible and some authors haveproposed new approaches that include the energy infor-mation in the estimation of the scatter distribution andimage reconstruction process [6 26ndash28] These approachesattempt to improve the accuracy of the rejection of scatteredcoincidences from measured data but are limited by theenergy resolution of the detectors [29]

Since the energy of detected photons carries some prob-abilistic information about the spatial distribution of theannihilation and by taking advantage of Compton scattering

2 ISRN Biomedical Imaging

scattered coincidences are also a potential source of latentinformation and can be included in PET image reconstruc-tion Incorporating scattered coincidences directly into thereconstruction algorithm eliminates the need for scattercorrection and could improve both image quality and systemsensitivity since a lower energy threshold can be used In thiswork we propose a reconstruction method similar to that ofConti et al [30] that directly includes scattered photons inthe image reconstruction algorithm instead of correcting forthem as do conventional emission imaging methods

An important difference between our work and that ofConti et al [30] is that the method proposed by Conti etal makes use of both time-of-flight (TOF) information andthe energy of individual photons to reconstruct scattered andunscattered PET coincidences [31] This paper focuses on anon-TOFPET algorithmwhich does notmake use of the timedifference between the two detected annihilation photonsAnother difference is that we use Compton kinematics topredict the locus of possible scattering positions to confinethe annihilation position instead of using a pixel-drivenapproach which can reduce the computational workloadThis work has been presented in part at theWorld CongressonMedical Physics and Biomedical Engineering in 2012 [32]

In this paper we evaluate the feasibility of this approachthe effects of the scatter fraction and the choice of a new coin-cidence selection threshold which uses both the scatteredphoton energy and detector position for each coincidenceon the image quality obtained using a generalized PETreconstruction algorithm

2 Materials and Methods

21 Reconstruction Theory In clinical PET systems morethan 997 of the scattered events undergo Compton interac-tions in water at 511 keV [33] Coherent scattering is neglectedbecause its contribution to the total cross-section is smallfor the energies of interest in nuclear medicine [25] Inapproximately 80 of detected scattered events only one ofthe two annihilation photons undergoes a single Comptoninteraction [5 8 34]This assumption has been validatedwithboth Monte Carlo (MC) simulation and experiment mea-surements in the single-scatter simulation (SSS) technique[35 36]

Figure 1 illustrates a scatter coincidence in a patient Thetwo annihilation photons are generated at source X and theunscattered photon is detected at A while the second photonundergoes a scatter at S and is detected at B with energy 119864

1015840The scattering angle 120579 is given by the Compton equation

cos 120579 = 1 +1198981198901198882

119864minus

1198981198901198882

1198641015840 (1)

where the scattering angle 120579 is defined as the angle betweenthe direction of the incident and the scattered photon119898

1198901198882 is

the electron restmass energy equal to 511 keV119864 is the incidentphoton energy and 119864

1015840 is the scattered photon energy in keVThe energy of the scattered photon decreases as the scatteringangle increases with the energy of the scattered photonranging from sim170 to 511 keV corresponding to backscatter

S

119903

120579

Detector A

Detector B

X

Figure 1 A diagram of a Compton scattering event in a patientThetwo antiparallel photons are generated at the annihilation positionX shown by the green dot The unscattered photon is observed bydetector A while the other photon suffers a Compton scatteringevent at S deviates from its initial path by an angle 120579 and is detectedby detector BThis figure also illustrates the two circular arcs (TCA)shown as blue dotted curves which describe the locus of possiblescattering locations for the scattered photon and which enclose theannihilation position

and forward scatter respectively By taking advantages of thekinematics of Compton scattering a 2D surface described bytwo circular arcs (TCA) connecting the coincidence detectors(rather than the conventional straight line assumption whichis true only for unscattered photons) which scribes thepossible scattering locus and encompasses the annihilationposition can be identified as shown in Figure 1 The locusdefined above ignores the positron range and the accuracy isclosely related to the energy resolution of the detectors Ourpreliminary investigation assumes that the detectors haveperfect energy resolution and data was simulated with theMonte Carlo code

The size and shape of the area encompassed by theTCA are a function of the scattering angle and the detectedpositions of the coincidence photons Figure 2 illustratesthat in the limit where the scattering angle approacheszero and the energy of the scattered photon approaches511 keV the shape approaches the line of response (LOR)for unscattered photons Thus the true coincidences may beconsidered as a subset of the scattered coincidences with zeroscattering angle We therefore propose a generalized scatter(GS) approach which generalizes the conventional meaningof scatter to include both true and scattered events and willshow in Section 22 that they can be used to reconstruct animage in a similar way

The work in this paper is carried out in two dimensionsalthough the same approach can in general be carried out inthree dimensions where it will ultimately be of greater value

22 The Generalized Scatter Algorithm 119875ab(120579) representsthe mean number of coincidences where an unscatteredphoton is observed at A while the other annihilation photonundergoes a Compton scattering through an angle 120579 and is

ISRN Biomedical Imaging 3

Detector A

Detector B

0∘30∘60∘90∘120∘ 150

∘

Figure 2 The TCA shapes versus the scattering angles for the sametwo detectors Each TCA consists of two symmetrically circular arcsThe inner to the outer arcs correspond to scattering angles of 0 3060 90 120 and 150 degrees respectively When the scattering angleis smaller than 90 degrees the TCA is made up of two minor arcswhen the scattering angle is 90 degrees the TCA is a circle whenthe scattering angle is larger than 90 degrees the TCA is made up oftwo major arcs As the area encompassed by the TCA increases withscattering angle the ability to accurately determine the annihilationposition decreases

observed at B with an energy 1198641015840 (see Figure 1) 119875ab(120579) can be

calculated using the following equation

⟨119875ab (120579)⟩ = 120591∮

B

Aint

S

A

119891119909

2120587119889119909 lowast 120588

119890(S) lowast

119889120590KN119862

119889120579lowast

1

2120587

lowast 119890minus(int

SA 120583119889119897+int

BS 120583119889119897)119889119904

(2)

The inner integral in the above expression calculates the totalphoton flux that can reach one of the possible scatteringpoints (say S) due to the source 119891

119909lying on the line segment

AS If the effect of the noncollinearity of the annihilationphotons is neglected the photons are emitted isotropicallyand no distinction can bemade between the two annihilationphotons Thus the probability of the annihilation photonstraveling along AS is 12120587 instead of 14120587 Once thesephotons arrive at one of the possible scattering positions Sthe possibility of photons suffering a Compton interactionwith a scattering angle 120579 is linearly proportional to thedifferential Klein-Nishina electronic cross-section 119889120590

KN119862

119889120579

and the electron density 120588119890at this point The second 12120587

takes into account that only a portion of the photons scatteredinto the solid angle around 120579 are detected at B The outerintegral sums the scattered coincidences over all the possiblescattering positions along the TCA Tau (120591) is the acquisitiontime and 120583 is the linear attenuation coefficient

To reduce the computational load the electron densitywas assumed to be uniformly distributed and for thisgeometry the attenuation was taken to be negligibly smallThus 119875ab(120579) is approximately proportional to the integral ofthe source intensity within the area confined by the TCA and119875ab(120579) can be written as

⟨119875ab (120579)⟩ = 119862 lowast119889120590

KN119862

119889120579∬

B

A119891119909119889119909 119889119910 (3)

where 119862 is a constant normalization factor Consideringthe sparse number of coincidences contributing to each⟨119875ab(120579)⟩ we derived the maximum-likelihood expectationmaximization (ML-EM) [37 38] in a list mode in which eachcoincidence will be stored and calculated successively Thus

119894

119899+1

= 119891119899

119894

1

sum119873

1198951015840=1

1198861198951015840119894

119873

sum

119895=1

119886119895119894

1

sum119901

1198941015840=1

1198861198951198941015840119891119899

1198941015840

119894 = 1 119901 (4)

where 119901 is the total number of pixels in the image 119873 isthe total number of detected coincidences and 119886

119895119894is the

element of system matrix representing the probability thatthe two annihilation photons emitted from pixel 119894 will bedetected as the 119895th coincidence (whether scattered or not) ina list mode entry weighted by the Compton differential crosssection In practice 119886(119894 119895) is proportional to the Comptoncross-section if the 119894th pixel falls within the 119895th TCA and iszero if 119894 is outside the TCA The main difference betweenthis GS-method-based ML-EM algorithm (GS-MLEM) andthe conventionalML-EM algorithm (LOR-MLEM) is that thesummation for each coincidence is over the area confined bythe TCA instead of being along the LORThis algorithm alsocan account for random coincidences 119877 by adding them tothe projector as follows sum

119901

1198941015840=1

1198861198951198941015840119891119899

1198941015840 + 119877

119895 The GS-MLEM

was implemented in MATLAB as shown in Figure 3 Theratio of the measured number of coincident events (which is1 with a list-mode algorithm) to the sum of the pixel valueswithin each TCA for the current estimate is backprojectedinto a ldquoratiordquo matrix This backprojection is repeated forall coincident events The ldquoratiordquo matrix is used to modifythe current estimate This process is repeated in an iterativefashion updating the current image until a good convergenceis obtained

23 The GATE Platform and Phantom Measurements Theproposed algorithm was evaluated by simulating a 2D smallanimal PET system using GATE [39] GATE is an object-oriented Monte Carlo simulation platform based on Geant4libraries (a generic Monte Carlo code) [40] The simulatedPET scanner had a 24 cm diameter detector ring made upof 42 detector blocks Each detector block is arranged ina 9 times 5 lutetium oxyorthosilicate (LSO) crystal array Eachcrystal element had a surface area of 2 times 2mm2 and was18mm thick The energy resolution of the simulated scannerin this preliminary investigation was 01 full width at halfmaximum (FWHM) at 511 keV and a 170ndash511 keV energywindow in which all single scattered coincidences can beselected was used Noise and dead time were not includedin the simulation process in order to focus on the role of thatscattered coincidences played in the image quality

A simplified Deluxe Jaszczak phantom [41 42] having3 hot disks and 1 cold disk with radii of 15mm 3mm45mm and 6mm within a cylindrical water phantom witha 40mm radius and a hot-to-background ratio of 4 as shownin Figure 4 was used

Images were reconstructed within a 99 times 99 pixel matrixwith a pixel size of 1 times 1mm2 The contrast recovery ratio ofthe reconstructed images was analyzed using the mean value


Initialize image matrix

Convergenceobtained

Yes

No

Output image

Update image by multiplying ratio matrix with image matrix

Reconstruction algorithm based on MLEM (but equally could use OSEM FBP etc)initialize ratio matrix

Calculate the scattering angle for each pair and identifyTCA (2D) or volume of response (3D) for each pair

Sum pixel valuse encompassed by TCA (2D) or volumes of response (3D) defined aboveNormalize and back project into a ratio matrix

Sum for all coincidences

Figure 3 The GS-MLEM algorithm flow chart The additions and changes to the conventional MLEM (LOR-MLEM) algorithm arehighlighted by the boxes with a black thick outline

of the disks relative to the local background while the noisewas calculated using the relative standard deviation (RSD) ofa 10mm diameter circular region of interest (ROI) locatedin the center of the phantom image Two concentric circularROIrsquos with diameters of 6mm and 8mmwere defined withinthe 3 and 4 disks respectively Annular ROIrsquos with a 20mmouter diameter and an inner diameter of 9mm and 12mmrespectively were drawn in the background around disks 3and 4The local contrast recovery coefficient (CRClocal) wasdefined by

CRClocal =(119867119861 minus 1)

119877 minus 1 (5)

where 119867 is the average values within the ROI in the hot disk119861 is the average value in the annulus around the disk and 119877

is the experimentally simulated hot-to-background ratio Forthe cold source CRClocal is defined by

CRClocal = 1 minus119862

119861 (6)

where 119862 was the average value of the ROI in the cold disk

For the purpose of evaluation a point on the CRC curvewas determined which was the shortest distance from theCRC curve to the point defined by an ideal CRC of 1 and aRSD of 0

231 The Feasibility Test of Images Reconstructed Using OnlyScattered Coincidences To illustrate that scattered coinci-dences can indeed be used to reconstruct images 106 scat-tered coincidences were used to reconstruct images usingthe GS-MLEM method By comparison the conventionalML-EM (LOR-MLEM) method which takes scattered coin-cidences as true was used to reconstruct 106 scattered coin-cidences as well as 106 true coincidences generated using theGATE simulation

232 Comparison of Images Reconstructed by GS-MLEMand LOR-MLEMwith Different Scattering Fraction Data Thescatter fraction is a function of the geometry of scannerthe density and the size of the patient as well as the energy


1

2

3

4

Figure 4 A simplified Deluxe Jaszczak phantom The three yellowcircles represent sources with radii of 15mm 3mm and 45mmrespectively The black circle represents a cold area with radius6mm The red area represents the 40mm water phantom The hot-to-background ratio is 4

window employed To compare images reconstructed withdifferent scatter fractions not affected by the above factorswe simulated 3 times 10

5 true coincidences and added scatteredcoincidences to obtain the required datawith scatter fractionsranging from 0 to 60 using the same phantom Thesimulation data we used is more artificial than real howeverit serves the goal of this research to investigate the relativevalue of the proposed algorithm for different fractions ofscattered coincidences to the image quality while at the sametime removing unwanted complexities resulting from morerealistic simulations

233 Evaluating the Different Energy Thresholds of ScatteringCoincidences on Image Quality The scatter fraction affectscontrast and noise of images As more scattered coincidencesare included in the image reconstruction the noise willdecrease but so will the contrast However the strength of theproposed method lies in its ability to recover contrast whilekeeping noise low The tradeoff between contrast and noise(for a particular scatter fraction) can be adjusted by varyingthe scattered photon energy threshold Photons scatteredthrough large angles may still be of value close to the periph-ery of the phantom In this paper those coincident photonswhich were scattered in an area defined by the intersectionof the area encompassed by the TCA and the image matrixwhich was less than some predetermined threshold wereused This is similar to using a scattered energy thresholdbut it enables both energy and detector position to be usedto full advantage A total of 6 times 10

5 coincidences with a50 scattering fraction were used to test how the varyingthresholds enabled the GS method to trade off betweenthe contrast and noise By setting different thresholds andcalculating the intersection areas between TCA and imagematrix the data was generated and used for reconstruction

3 Results

31 Feasibility Test of Reconstructing Images Only by ScatteredCoincidences Figure 5(a) displays a typical image producedfrom true (unscattered) coincidences using a conventionalML-EM (LOR-MLEM) algorithm Figure 5(b) shows thatthe same algorithm was incapable of producing recognizableimages from scattered photons only since the LOR assignedfor each scattered coincidence no longer passes through theannihilation point This illustrates why scattered photonsare traditionally considered contributing only noise to theresultant image and are where possible typically eliminatedOur initial experiments demonstrated that the proposedalgorithm was capable of producing an image entirely fromscattered photons and Figure 5(c) while not as good as theimage in Figure 5(a) still adequately represents the activitydistribution There is still blurring evident in the scatteredphoton reconstruction and image converges at least 2-3 timesslower From the profiles under the corresponding imageswe can see that the image produced using only scatteredevents has a decreased contrast and noisier backgroundcompared with that using the same number of true coinci-dences Figure 5(c) was used to illustrate the feasibility ofreconstructing an image from only scattered coincidencesHowever in practice an image would be reconstructed byincorporating both true and scattered events instead of onlyusing scattered coincidencesWe hypothesized that includingscattered coincidences in the reconstruction provides moreadvantages than simply rejecting them and the results will beshown in Section 32

32 Comparison of Images Reconstructed by GS-MLEM andLOR-MLEMwithDifferent Scattering FractionData Figure 6shows the CRC curves for the 3 (largest hot disk) versusthe relative standard deviation of the background obtainedby varying the number of iterations Figure 7 is similarexcept that it shows the results for 4 (the cold) sourceThe CRC curves for images reconstructed using the GS-MLEM algorithm were always above those created by theLOR-MLEM algorithm Unlike the CRC curves for the LOR-MLEM algorithm which decrease with increasing scatterfraction the CRC curve for the GS-MLEM algorithm gen-erally increased with increasing scatter fraction For the hotdisk this trend reversed beyond the point where the CRCcurve for the GS-MLEM intersected with that of the CRCfor reconstructions using only true coincidences For thecold disk this reversal was not observed This reductionin CRC was small and occurred only for iterations beyondthe evaluation point For scatter fractions from 10 to 60the evaluation points on the CRC curve for the hot diskwere 05ndash30 greater than those obtained using only truecoincidence data only and were 30ndash245 greater than thecorresponding curves reconstructed using the LOR-MLEMalgorithm The noise was reduced by 0ndash17 in comparisonwith the true coincidences data and was 20ndash120 less thanthat produced by LOR-MLEMmethod with the same scatterfraction For scatter fractions between 10 and 60 theevaluation points for the cold disk had a CRC 0ndash35 greaterthan the curves calculated using only true coincidences and


10

20

30

40

50

60

70

80

90

20 40 60 80

1090807060504030201

00 50 100

0908

07

06

05

04

03

02

010

0 50 100

Imag

e cou

nts

Horizontal pixels Vertical pixels

Imag

e cou

nts

(a)

10

20

30

40

50

60

70

80

90

20 40 60 80Im

age c

ount

s09

08

07

06

05

04

03

02

01

00 50 100

Horizontal pixels

07

06

05

04

03

02

01

00 50 100Vertical pixels

Imag

e cou

nts

(b)

10

20

30

40

50

60

70

80

90

20 40 60 80

Imag

e cou

nts

090807060504030201

00 50 100


Imag

e cou

nts

0 50 100

1 08

07

06

05

04

03

02

01

0

(c)

Figure 5 (a)The image created by reconstructing 106 true coincidences using an LOR-MLEMwith 36 iterations (b)The image reconstructedusing 106 scattered coincidences and an LOR-MLEM approach which assumes that these photons are true coincidences (c) It uses the samescattered coincidences dataset as in (b) but reconstructs them using a GS-MLEM approach with 62 iterations The second row plots theprofiles through the above images in the horizontal and vertical direction passing through the center of the images respectively

were 40ndash240 greater than the LOR-MLEM method Thenoise at the evaluation point was 0ndash16 less than thatobtained using only true coincidences and was 05ndash83 lessthan that calculated using the LOR-MLEM method withthe same scatter fraction This trend is consistent with theposition that the scattered coincidences contribute to noisein the conventional PET image reconstruction and that as thescatter fraction increases the image quality deteriorates

33 Evaluating the Different Energy Thresholds of ScatteringCoincidences on Image Quality Figure 8 plots the CRC of3 (the largest source) for different TCA area thresholdsas a function of the relative background standard deviationobtained by varying the number of iterations Figure 9 issimilar except that it shows the results for the 4 (cold)sourceThe results show that the value of the CRC for imagesreconstructed using the GS-MLEM method were alwaysgreater than those obtained with a conventional LOR-MLEMapproach which assumed that all coincidences that fell intothe typical 350ndash511 keV energywindowwere true For theGS-MLEM method the evaluation point (point 1 in Figure 8)

for the source had a CRC that was 25 better than imagesreconstructed with true coincidence data only (which is anidealistic upper limit on existing algorithms) (point 2 inFigure 8) and was 130 greater than images reconstructedusing the conventional LOR-MLEM algorithm (point 3 inFigure 8)The image noise was 20 less (point 1 in Figure 8)than the ideal reconstruction using true coincidences (point2 in Figure 8) and was 70 less than the LOR-MLEMapproach (point 3 in Figure 8) For the cold source theevaluation point for the GS-MLEM approach (point 1 inFigure 9) had a CRC that was 50 greater than the CRCobtainedwith the ideal true coincidence calculation (point 2in Figure 9) and 180 greater than the LOR-MLEMmethod(point 3 in Figure 9) For the cold source the noise at theevaluation point using the GS-MLEM method (point 1 inFigure 9) was 20 less than the ideal reconstruction usingtrues (point 2 in Figure 9) and was 80 less than theLOR-MLEM approach (point 3 in Figure 9) Virtually nodifference was found between the CRC curves for thresholdsof 918 and 100 The evaluation CRCnoise point for thesource occurred for thresholds close to 100 of the matrixsizewhile the evaluation point for the cold sourcewas optimal


Ideal

Evaluation point

06

03

00 03 06

Relative standard deviation

Hot

cont

rast

reco

very

coeffi

cien

t

GS 10LOR 10GS 30LOR 30GS 40

LOR 40GS 50LOR 50GS 60LOR 60

Figure 6 The CRC curves for 3 disk versus the relative standarddeviation of the background The CRC curves shown with filledsymbols represent the images reconstructed by GS-MLEM withdifferent scattering fractions The lines in the same style withempty symbols correspond to images with the same scatter fractionreconstructed by LOR-MLEM The CRC for true coincidences isshown as a solid line for comparison The diamonds reflect theoptimal points on the CRC-RSD curves

when the threshold was set to 102 of the matrix size Theevaluation contrast recovery point for the source approachedthe true value as the threshold approaches zero but theCRC for the cold source still showed an improvement forthresholds approaching 100

Figure 10(a) displays the image reconstructed from 3times105

true coincidences using a conventional LOR-MLEMmethodFigure 10(b) shows the image reconstructed using the sametrue 3 times 10

5 coincidences plus 3 times 105 scattered coincidences

that fall into the 350ndash511 keV energy window and recon-structed using a conventional LOR-MLEM algorithm as acomparison Figure 10(c) shows the image produced by 6times10

5

coincidences with 50 scatter fraction and a threshold of100 reconstructed using the GS-MLEMmethod

4 Discussion

We have described a novel PET reconstruction algorithmwhich can incorporate both true and scattered coincidencesinto the reconstruction process Our initial results haveshown that under the ideal conditions used in this studyincluding scattered coincidences in the reconstruction ismore advantageous than simply rejecting them

It has been shown that an image of the activity dis-tribution can be reconstructed from scatter data only asshown in Figure 5(c) The image in Figure 10(c) is still

Ideal

Evaluation point

06

03

00 03 06


Col

d co

ntra

st re

cove

ry co

effici

ent


LOR 40GS 50LOR 50GS 60GS 60

Figure 7 The CRC curves for the 4 (cold) disk versus the relativestandard deviation of the background The CRC curves shown withfilled symbols represent the images reconstructed using the GS-MLEM with different scattering fractions The lines in the samestyle with empty symbols correspond to images with the samescatter fraction reconstructed by LOR-MLEMTheCRC for the truecoincidences is shown as a solid line for comparison The diamondsreflect the optimal points on the CRC-RSD curves

idealistic and far from optimized but these results showthat the proposed method produces an image with a morehomogeneous background sharper edges reduced noise andimproved contrast (for both hot and cold disks) relative toboth conventional algorithm (which degrade in the presenceof scatter) and the idealistic situation (only true eventsused) Algorithms which incorporate scatter correction willmove closer to the idealistic response curve but will neverexceed it Thus including scattered photons directly into thereconstruction could eliminate the need for (often empirical)scatter corrections required by conventional algorithms andincrease image contrast and signal to noise ratio (SNR) Thiscould be used to either improve the diagnostic quality of theimages andor to reduce patient dose and radiopharmaceuti-cal cost

The results in Section 33 show that an additional advan-tage of the GS-MLEM algorithm is that a threshold canbe used to adjust contrast and noise in order to optimizethe role that the scattered coincidences play in the imagereconstruction process At the optimal threshold addingscattered coincidences into imaging reconstruction appearsto result inminimal (if any) loss in contrast while significantlyimproving the SNR and contrast recovery The results showthat by appropriately including a greater number of scatteredphotons in the reconstruction improvements in both con-trast and noise can be achieved In addition this threshold


True102

3061

5102

918

100

LOR-MLEMEvaluation point

0

02

04

06

0 02 04 06


1

2

3

Hot

cont

rast

reco

very

coeffi

cien

t

Figure 8 The CRC curves for the 3 (largest hot) disk calculatedwhen only trues are present and with 50 scatter fraction calculatedusing the conventional LOR-MLEM approach and with the GS-MLEM approach using different thresholds defined by the ratio ofthe area of intersection between the TCA and image matrix to thetotal image matrix area Points 1 2 and 3 identify the evaluationpoints for reconstructions using GS-MLEMmethod only trues andthe LOR-MLEM respectively

also can remove partial coincidences scattered within detec-tors or in the gantry which usually have relatively largerscattering angles and unusually short distances betweenscattering positions and detector positions The TCA forthese situations usually covers the whole image space and canbe removed from the reconstruction dataset by employingthe proposed threshold Our current work has assumed thatintercrystal scattering is small when compared to scatteringwithin the patient While LSO crystals have been used inthis simulation in future we will use a dual layer detector tominimize inter-crystal scattering

Similar to conventional energy-based scatter correctionmethods the accuracy of the proposed method is limitedby the energy resolution of the detector In conventionalscatter-correction-based methods the ability to distinguishand reject scattered coincidences is limited by energy res-olution In the proposed method the energy resolution ofdetectors determines the confidence with which the TCAcan be defined for each scattered coincidence which in turnaffects the locus of the possible scattering positions andhence the annihilation position The locus is sensitive to theenergy of the detected photons and a small uncertainty inthe energy of the scattered photon can result in a significantdifference in the shape of the TCA and the position of thesource For example if the detectorrsquos energy resolution is4 at 511 keV the maximum uncertainty in the calculatedarea between the TCA is approximately 21 for a scattered

True102

3061

5102

918

100


08

07

06

05

04

03

02

01

00 02 04 06


1

2

3

Col

d co

ntra

st re

cove

ry co

effici

ent

Figure 9 The CRC curves for the 4 (cold) disk calculated whenonly trues are present and with 50 scatter fraction calculatedusing the conventional LOR-MLEM approach and with the GS-MLEM approach using different thresholds defined by the ratio ofthe intersection area between the TCA and image matrix to thetotal image matrix area Points 1 2 and 3 identify the evaluationpoints for reconstruction using GS-MLEMmethod only trues andthe LOR-MLEM respectively

photon energy of 450 keV Thus developing a detector withhigh energy resolution and weighting the spatial probabilitydistribution of the annihilation position by using both thepatient outline [43] and the energy resolution of the detectorwill be important future work

Guerin et al indicated that neglecting the effect ofmultiple scattered coincidences was not a serious sourceof error In practice multiple scattered events cannot bedistinguished from single scattered events on the basis of theenergy of the scattered photons Even though the relationbetween scattered energy and scattering angle cannot beconnected by Compton scattering equation for multiple scat-tered coincidences we propose to use a synthetic scatteringangle for multiple scattered events based on the scatteredphoton energy In addition the uncertainty in the energyof the detected photon as a result of the limited energyresolution of the detector is more challenging for multiplescattered events when trying to determine the locus ofpossible scattering positions Corrections for randoms canbe addressed by adding a factor to the forward projection inthe algorithm as do most existing methods The data modelused to derive the reconstruction algorithm can be improvedby including the electron density and tissue attenuation Theimpact of these on the quality of the reconstructed imagesis beyond the scope of this paper and will be the focus ofa separate study Since the TCA is a function of scatteringangle and detector positions which will be calculated foreach detected coincidences in the reconstruction algorithm


10

20

30

40

50

60

70

80

90

20 40 60 80

1090807060504030201

00 50 100

07

06

05

04

03

02

01

00 50 100

Imag

e cou

nts

Imag

e cou

nts


(a)

10

20

30

40

50

60

70

80

90

20 40 60 80

09

08

07

06

05

04

03

02

01

00 50 100

08

07

06

05

04

03

02

01

00 50 100

Imag

e cou

nts

Imag

e cou

nts


(b)

10

20

30

40

50

60

70

80

90

20 40 60 80

1090807060504030201

00 50 100

07

06

05

04

03

02

01

00 50 100

Imag

e cou

nts

Imag

e cou

nts


(c)

Figure 10 (a)The image using 3 times 105 true coincidences reconstructed using an LOR-MLEM reconstruction algorithm (b)The image using

the same true 3 times 105 coincidences plus 3 times 10

5 scattered coincidences that fall into the 350ndash511 keV energy window reconstructed using theconventional LOR-MLEM algorithm as a comparison (c)The image using 6 times 10

5 coincidences with a 50 scattering fraction reconstructedusing theGS-MLEMalgorithmThe second row shows the profiles of the above images along horizontal and vertical direction passing throughthe center of the images respectively

the computational time is currently 3-4 times longer whencompared with a conventional MLEM algorithm Howeverthis algorithm still needs to be optimized for calculationefficiency

The uniform attenuation phantom used in this work issimple in comparison to anthropomorphic phantoms andmay not reflect the complexity of scatter in humansHoweverthe purpose of this work was to compare the results ofthe proposed algorithm as a function of the scatter fractionwhich would be difficult to analyze with an anthropomorphicphantom

5 Conclusion

A new method which includes scattered photons directlyinto the reconstruction has been presented and evaluated inthis paper The results of the phantom study in this paperhave shown that PET images can be reconstructed fromscattered coincidences Including scattered coincidences intothe reconstruction eliminates the need for scatter corrections

while increasing image contrast and reducing noise A thresh-old which depends on both energy and detector positionis used adjust noise and contrast was also evaluated in thiswork The optimal threshold was different for hot and coldsources but the variation in CRC for the cold source wasonly weakly dependent on the threshold Improvements inthe CRC and noise for both hot and cold sources could beobtained by maximizing the use of all scattered photons

Acknowledgments

This work was supported in part by CancerCare Mani-toba Foundation University of Manitoba and the Natu-ral Sciences and Engineering Research Council of Canada(NSERC)

References

[1] C H Holdsworth C S Levin T H Farquhar M Dahlbomand E J Hoffman ldquoInvestigation of accelerated Monte Carlotechniques for PET simulation and 3D PET scatter correctionrdquo


IEEE Transactions on Nuclear Science vol 48 no 1 pp 74ndash812001

[2] K S Kim and J C Ye ldquoFully 3D iterative scatter-correctedOSEM for HRRT PET using a GPUrdquo Physics in Medicine andBiology vol 56 no 15 pp 4991ndash5009 2011

[3] H Zaidi and M-L Montandon ldquoScatter compensation tech-niques in PETrdquo PET Clinics vol 2 no 2 pp 219ndash234 2007

[4] A Konik M T Madsen and J J Sunderland ldquoGATE simu-lations of human and small animal PET for determination ofscatter fraction as a function of object sizerdquo IEEE Transactionson Nuclear Science vol 57 no 5 pp 2558ndash2563 2010

[5] D L Bailey D W Townsend P E Valk and M N MaiseyPositron Emission Tomography Basic Sciences 2005

[6] B Guerin and G El Fakhri ldquoNovel scatter compensationof list-mode PET data using spatial and energy dependentcorrectionsrdquo IEEE Transactions on Medical Imaging vol 30 no3 pp 759ndash773 2011

[7] D L Bailey ldquoQuantitative procedures in 3D PETrdquo inTheTheoryand Practice of 3D PET pp 55ndash109 1998

[8] H Zaidi and K F Koral ldquoScatter modelling and compensationin emission tomographyrdquo European Journal of Nuclear Medicineand Molecular Imaging vol 31 no 5 pp 761ndash782 2004

[9] H Zaidi ldquoScatter modelling and correction strategies in fully3-D petrdquo Nuclear Medicine Communications vol 22 no 11 pp1181ndash1184 2001

[10] S Grootoonk T J Spinks D Sashin NM Spyrou andT JonesldquoCorrection for scatter in 3D brain PET using a dual energywindow methodrdquo Physics in Medicine and Biology vol 41 no12 pp 2757ndash2774 1996

[11] B Bendriem R Trebossen V Frouin and A Syrota ldquoA PETscatter correction using simultaneous acquisitions with lowand high lower energy thresholdsrdquo in Proceedings of the IEEENuclear Science Symposium amp Medical Imaging Conference pp1779ndash1783 San Francisco Calif USA November 1993

[12] R L Harrison D R Haynor and T K Lewellen ldquoDual energywindow scatter corrections for positron emission tomographyrdquoin Proceedings of the IEEE Nuclear Science Symposium andMedical ImagingConference pp 1700ndash1704 Santa FeNMUSA1992

[13] V Sossi J S Barney and T J Ruth ldquoNoise reduction in thedual energy window scatter correction approach in 3D PETrdquoin Proceedings of the Nuclear Science Symposium and MedicalImaging Conference pp 1512ndash1515 Norfolk UK November1994

[14] L-E Adam J S Karp and R Freifelder ldquoScatter correctionusing a dual energy window technique for 3D PET withNaI(Tl) detectorsrdquo in Proceedings of the IEEE Nuclear ScienceSymposium Conference Record pp 2011ndash2018 Toronto CanadaNovember 1998

[15] L Shao R Freifelder and J S Karp ldquoTriple energy windowscatter correction technique in PETrdquo IEEE Transactions onMedical Imaging vol 13 no 4 pp 641ndash648 1994

[16] M Bentourkia P Msaki J Cadorette and R Lecomte ldquoEnergydependence of scatter components in multispectral PET imag-ingrdquo IEEE Transactions on Medical Imaging vol 14 no 1 pp138ndash145 1995

[17] C S Levin M Dahlbom and E J Hoffman ldquoA Monte Carlocorrection for the effect of Compton scattering in 3-D pet brainimagingrdquo IEEE Transactions on Nuclear Science vol 42 no 4pp 1181ndash1185 1995

[18] O Barret T A Carpenter J C Clark R E Ansorge and TD Fryer ldquoMonte Carlo simulation and scatter correction of theGE Advance PET scanner with SimSET and Geant4rdquo Physics inMedicine and Biology vol 50 no 20 pp 4823ndash4840 2005

[19] S Basu H Zaidi S Holm and A Alavi ldquoQuantitative tech-niques in PET-CT imagingrdquo Current Medical Imaging Reviewsvol 7 no 3 pp 216ndash233 2011

[20] D L Bailey and S RMeikle ldquoA convolution-subtraction scattercorrectionmethod for 3DPETrdquoPhysics inMedicine and Biologyvol 39 no 3 pp 411ndash424 1994

[21] F J Beekman C Kamphuis and E C Frey ldquoScatter com-pensation methods in 3D iterative SPECT reconstruction asimulation studyrdquo Physics in Medicine and Biology vol 42 no8 pp 1619ndash1632 1997

[22] M Tamal A J Reader P J Markiewicz P J Julyan and D LHastings ldquoNoise properties of four strategies for incorporationof scatter and attenuation information in PET reconstructionusing the EM-ML algorithmrdquo IEEE Transactions on NuclearScience vol 53 no 5 pp 2778ndash2786 2006

[23] J Qi R M Leahy C Hsu T H Farquhar and S R CherryldquoFully 3D bayesian image reconstruction for the ECAT EXACTHR+ lrdquo IEEE Transactions on Nuclear Science vol 45 no 3 pp1096ndash1103 1998

[24] J E Bowsher V E Johnson T G Turkington R J JaszczakC E Floyd and R Edward Coleman ldquoBayesian reconstructionand use of anatomical a priori information for emission tomog-raphyrdquo IEEE Transactions on Medical Imaging vol 15 no 5 pp673ndash686 1996

[25] H Zaidi ldquoRelevance of accurate Monte Carlo modeling innuclear medical imagingrdquoMedical Physics vol 26 pp 574ndash6081999

[26] R Levkovitz D Falikman M Zibulevsky A Ben-Tal and ANemirovski ldquoThe design and implementation of COSEM aniterative algorithm for fully 3-D listmode datardquo IEEE Transac-tions on Medical Imaging vol 20 no 7 pp 633ndash642 2001

[27] H T Chen C-M Kao and C T Chen ldquoA fast energy-dependent scatter reduction method for 3D PET imagingrdquo inProceedings of the IEEE Nuclear Science Symposium andMedicalImaging Conference pp 2630ndash2634 October 2003

[28] L M Popescu R M Lewitt S Matej and J S Karp ldquoPETenergy-based scatter estimation and image reconstruction withenergy-dependent correctionsrdquoPhysics inMedicine andBiologyvol 51 no 11 pp 2919ndash2937 2006

[29] L M Popescu ldquoPET energy-based scatter estimation in thepresence of randoms and image reconstruction with energy-dependent scatter and randoms correctionsrdquo IEEE Transactionson Nuclear Science vol 59 pp 1958ndash1966 2012

[30] M Conti I Hong and C Michel ldquoReconstruction of scatteredand unscattered PET coincidences using TOF and energyinformationrdquo Physics in Medicine and Biology vol 57 pp 307ndash317 2012

[31] M Conti ldquoWhy is TOF PET reconstruction a more robustmethod in the presence of inconsistent datardquo Physics inMedicine and Biology vol 56 no 1 pp 155ndash168 2011

[32] S Pistorius and H Sun ldquoA novel scatter enhanced PETreconstruction algorithmrdquo in World Congress Medical Physicaland Biomedical Engineering Beijing China 2012

[33] H ZaidiM-LMontandon and SMeikle ldquoStrategies for atten-uation compensation in neurological PET studiesrdquoNeuroImagevol 34 no 2 pp 518ndash541 2007


[34] E Van Uytven S Pistorius and R Gordon ldquoAn iterativethree-dimensional electron density imaging algorithm usinguncollimated Compton scattered x rays from a polyenergeticprimary pencil beamrdquo Medical Physics vol 34 no 1 pp 256ndash265 2007

[35] C C Watson D Newport M E Casey R A Dekemp RS Beanlands and M Schmand ldquoEvaluation of simulation-based scatter correction for 3-D PET cardiac imagingrdquo IEEETransactions on Nuclear Science vol 44 no 1 pp 90ndash97 1997

[36] C C Watson ldquoNew faster image-based scatter correction for3D PETrdquo IEEE Transactions on Nuclear Science vol 47 no 4pp 1587ndash1594 2000

[37] L A Shepp and Y Vardi ldquoMaximum likelihood reconstructionfor emission tomographyrdquo IEEE Transactions on Medical Imag-ing vol 1 no 2 pp 113ndash122 1982

[38] G Kontaxakis and L G Strauss ldquoMaximum likelihood algo-rithms for image reconstruction in Positron Emission Tomog-raphyrdquo in Radionuclides For OncologymdashCurrent Status andFuture Aspects pp 73ndash106 1998

[39] httpwwwopengatecollaborationorg[40] K Assie V Breton I Buvat et al ldquoMonte Carlo simulation

in PET and SPECT instrumentation using GATErdquo NuclearInstruments and Methods in Physics Research A vol 527 no 1-2pp 180ndash189 2004

[41] R J Jaszczak K L Greer C E Floyd Jr C C Harris and RE Coleman ldquoImproved SPECT quantification using compen-sation for scattered photonsrdquo Journal of Nuclear Medicine vol25 no 8 pp 893ndash900 1984

[42] httpwwwspectcompubFlanged Jaszczak Phantomspdf[43] H Sun and S Pistorius ldquoImprovements in image quality when

using patient outline constraints with a generalized scatter PETreconstruction algorithmrdquo in Proceedings of the IEEE NuclearScience Symposium amp Medical Imaging Conference vol M16-1pp 16ndash11 Anaheim Calif USA 2012

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



scattered coincidences are also a potential source of latentinformation and can be included in PET image reconstruc-tion Incorporating scattered coincidences directly into thereconstruction algorithm eliminates the need for scattercorrection and could improve both image quality and systemsensitivity since a lower energy threshold can be used In thiswork we propose a reconstruction method similar to that ofConti et al [30] that directly includes scattered photons inthe image reconstruction algorithm instead of correcting forthem as do conventional emission imaging methods

An important difference between our work and that ofConti et al [30] is that the method proposed by Conti etal makes use of both time-of-flight (TOF) information andthe energy of individual photons to reconstruct scattered andunscattered PET coincidences [31] This paper focuses on anon-TOFPET algorithmwhich does notmake use of the timedifference between the two detected annihilation photonsAnother difference is that we use Compton kinematics topredict the locus of possible scattering positions to confinethe annihilation position instead of using a pixel-drivenapproach which can reduce the computational workloadThis work has been presented in part at theWorld CongressonMedical Physics and Biomedical Engineering in 2012 [32]

In this paper we evaluate the feasibility of this approachthe effects of the scatter fraction and the choice of a new coin-cidence selection threshold which uses both the scatteredphoton energy and detector position for each coincidenceon the image quality obtained using a generalized PETreconstruction algorithm

2 Materials and Methods

21 Reconstruction Theory In clinical PET systems morethan 997 of the scattered events undergo Compton interac-tions in water at 511 keV [33] Coherent scattering is neglectedbecause its contribution to the total cross-section is smallfor the energies of interest in nuclear medicine [25] Inapproximately 80 of detected scattered events only one ofthe two annihilation photons undergoes a single Comptoninteraction [5 8 34]This assumption has been validatedwithboth Monte Carlo (MC) simulation and experiment mea-surements in the single-scatter simulation (SSS) technique[35 36]

Figure 1 illustrates a scatter coincidence in a patient Thetwo annihilation photons are generated at source X and theunscattered photon is detected at A while the second photonundergoes a scatter at S and is detected at B with energy 119864

1015840The scattering angle 120579 is given by the Compton equation

cos 120579 = 1 +1198981198901198882

119864minus

1198981198901198882

1198641015840 (1)

where the scattering angle 120579 is defined as the angle betweenthe direction of the incident and the scattered photon119898

1198901198882 is

the electron restmass energy equal to 511 keV119864 is the incidentphoton energy and 119864

1015840 is the scattered photon energy in keVThe energy of the scattered photon decreases as the scatteringangle increases with the energy of the scattered photonranging from sim170 to 511 keV corresponding to backscatter

S

119903

120579

Detector A

Detector B

X

Figure 1 A diagram of a Compton scattering event in a patientThetwo antiparallel photons are generated at the annihilation positionX shown by the green dot The unscattered photon is observed bydetector A while the other photon suffers a Compton scatteringevent at S deviates from its initial path by an angle 120579 and is detectedby detector BThis figure also illustrates the two circular arcs (TCA)shown as blue dotted curves which describe the locus of possiblescattering locations for the scattered photon and which enclose theannihilation position

and forward scatter respectively By taking advantages of thekinematics of Compton scattering a 2D surface described bytwo circular arcs (TCA) connecting the coincidence detectors(rather than the conventional straight line assumption whichis true only for unscattered photons) which scribes thepossible scattering locus and encompasses the annihilationposition can be identified as shown in Figure 1 The locusdefined above ignores the positron range and the accuracy isclosely related to the energy resolution of the detectors Ourpreliminary investigation assumes that the detectors haveperfect energy resolution and data was simulated with theMonte Carlo code

The size and shape of the area encompassed by theTCA are a function of the scattering angle and the detectedpositions of the coincidence photons Figure 2 illustratesthat in the limit where the scattering angle approacheszero and the energy of the scattered photon approaches511 keV the shape approaches the line of response (LOR)for unscattered photons Thus the true coincidences may beconsidered as a subset of the scattered coincidences with zeroscattering angle We therefore propose a generalized scatter(GS) approach which generalizes the conventional meaningof scatter to include both true and scattered events and willshow in Section 22 that they can be used to reconstruct animage in a similar way

The work in this paper is carried out in two dimensionsalthough the same approach can in general be carried out inthree dimensions where it will ultimately be of greater value

22 The Generalized Scatter Algorithm 119875ab(120579) representsthe mean number of coincidences where an unscatteredphoton is observed at A while the other annihilation photonundergoes a Compton scattering through an angle 120579 and is


Detector A

Detector B

0∘30∘60∘90∘120∘ 150

∘




⟨119875ab (120579)⟩ = 120591∮

B

Aint

S

A

119891119909

2120587119889119909 lowast 120588

119890(S) lowast

119889120590KN119862

119889120579lowast

1

2120587


SA 120583119889119897+int

BS 120583119889119897)119889119904

(2)




KN119862

119889120579




⟨119875ab (120579)⟩ = 119862 lowast119889120590

KN119862

119889120579∬

B

A119891119909119889119909 119889119910 (3)


119894

119899+1

= 119891119899

119894

1

sum119873

1198951015840=1

1198861198951015840119894

119873

sum

119895=1

119886119895119894

1

sum119901

1198941015840=1

1198861198951198941015840119891119899

1198941015840

119894 = 1 119901 (4)


119895119894is the


119901

1198941015840=1

1198861198951198941015840119891119899

1198941015840 + 119877

119895 The GS-MLEM







Convergenceobtained

Yes

No

Output image









119877 minus 1 (5)




119861 (6)






1

2

3

4






3 Results




10

20

30

40

50

60

70

80

90

20 40 60 80

1090807060504030201

00 50 100

0908

07

06

05

04

03

02

010

0 50 100

Imag

e cou

nts


Imag

e cou

nts

(a)

10

20

30

40

50

60

70

80

90

20 40 60 80Im

age c

ount

s09

08

07

06

05

04

03

02

01

00 50 100

Horizontal pixels

07

06

05

04

03

02

01


Imag

e cou

nts

(b)

10

20

30

40

50

60

70

80

90

20 40 60 80

Imag

e cou

nts

090807060504030201

00 50 100


Imag

e cou

nts

0 50 100

1 08

07

06

05

04

03

02

01

0

(c)






Ideal

Evaluation point

06

03

00 03 06


Hot

cont

rast

reco

very

coeffi

cien

t









5


4 Discussion



Ideal

Evaluation point

06

03

00 03 06


Col

d co

ntra

st re

cove

ry co

effici

ent







True102

3061

5102

918

100


0

02

04

06

0 02 04 06


1

2

3

Hot

cont

rast

reco

very

coeffi

cien

t




True102

3061

5102

918

100


08

07

06

05

04

03

02

01

00 02 04 06


1

2

3

Col

d co

ntra

st re

cove

ry co

effici

ent





10

20

30

40

50

60

70

80

90

20 40 60 80

1090807060504030201

00 50 100

07

06

05

04

03

02

01

00 50 100

Imag

e cou

nts

Imag

e cou

nts


(a)

10

20

30

40

50

60

70

80

90

20 40 60 80

09

08

07

06

05

04

03

02

01

00 50 100

08

07

06

05

04

03

02

01

00 50 100

Imag

e cou

nts

Imag

e cou

nts


(b)

10

20

30

40

50

60

70

80

90

20 40 60 80

1090807060504030201

00 50 100

07

06

05

04

03

02

01

00 50 100

Imag

e cou

nts

Imag

e cou

nts


(c)







5 Conclusion



Acknowledgments


References

















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Detector A

Detector B

0∘30∘60∘90∘120∘ 150

∘




⟨119875ab (120579)⟩ = 120591∮

B

Aint

S

A

119891119909

2120587119889119909 lowast 120588

119890(S) lowast

119889120590KN119862

119889120579lowast

1

2120587


SA 120583119889119897+int

BS 120583119889119897)119889119904

(2)




KN119862

119889120579




⟨119875ab (120579)⟩ = 119862 lowast119889120590

KN119862

119889120579∬

B

A119891119909119889119909 119889119910 (3)


119894

119899+1

= 119891119899

119894

1

sum119873

1198951015840=1

1198861198951015840119894

119873

sum

119895=1

119886119895119894

1

sum119901

1198941015840=1

1198861198951198941015840119891119899

1198941015840

119894 = 1 119901 (4)


119895119894is the


119901

1198941015840=1

1198861198951198941015840119891119899

1198941015840 + 119877

119895 The GS-MLEM







Convergenceobtained

Yes

No

Output image









119877 minus 1 (5)




119861 (6)






1

2

3

4






3 Results




10

20

30

40

50

60

70

80

90

20 40 60 80

1090807060504030201

00 50 100

0908

07

06

05

04

03

02

010

0 50 100

Imag

e cou

nts


Imag

e cou

nts

(a)

10

20

30

40

50

60

70

80

90

20 40 60 80Im

age c

ount

s09

08

07

06

05

04

03

02

01

00 50 100

Horizontal pixels

07

06

05

04

03

02

01


Imag

e cou

nts

(b)

10

20

30

40

50

60

70

80

90

20 40 60 80

Imag

e cou

nts

090807060504030201

00 50 100


Imag

e cou

nts

0 50 100

1 08

07

06

05

04

03

02

01

0

(c)






Ideal

Evaluation point

06

03

00 03 06


Hot

cont

rast

reco

very

coeffi

cien

t









5


4 Discussion



Ideal

Evaluation point

06

03

00 03 06


Col

d co

ntra

st re

cove

ry co

effici

ent







True102

3061

5102

918

100


0

02

04

06

0 02 04 06


1

2

3

Hot

cont

rast

reco

very

coeffi

cien

t




True102

3061

5102

918

100


08

07

06

05

04

03

02

01

00 02 04 06


1

2

3

Col

d co

ntra

st re

cove

ry co

effici

ent





10

20

30

40

50

60

70

80

90

20 40 60 80

1090807060504030201

00 50 100

07

06

05

04

03

02

01

00 50 100

Imag

e cou

nts

Imag

e cou

nts


(a)

10

20

30

40

50

60

70

80

90

20 40 60 80

09

08

07

06

05

04

03

02

01

00 50 100

08

07

06

05

04

03

02

01

00 50 100

Imag

e cou

nts

Imag

e cou

nts


(b)

10

20

30

40

50

60

70

80

90

20 40 60 80

1090807060504030201

00 50 100

07

06

05

04

03

02

01

00 50 100

Imag

e cou

nts

Imag

e cou

nts


(c)







5 Conclusion



Acknowledgments


References

















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Convergenceobtained

Yes

No

Output image









119877 minus 1 (5)




119861 (6)






1

2

3

4






3 Results




10

20

30

40

50

60

70

80

90

20 40 60 80

1090807060504030201

00 50 100

0908

07

06

05

04

03

02

010

0 50 100

Imag

e cou

nts


Imag

e cou

nts

(a)

10

20

30

40

50

60

70

80

90

20 40 60 80Im

age c

ount

s09

08

07

06

05

04

03

02

01

00 50 100

Horizontal pixels

07

06

05

04

03

02

01


Imag

e cou

nts

(b)

10

20

30

40

50

60

70

80

90

20 40 60 80

Imag

e cou

nts

090807060504030201

00 50 100


Imag

e cou

nts

0 50 100

1 08

07

06

05

04

03

02

01

0

(c)






Ideal

Evaluation point

06

03

00 03 06


Hot

cont

rast

reco

very

coeffi

cien

t









5


4 Discussion



Ideal

Evaluation point

06

03

00 03 06


Col

d co

ntra

st re

cove

ry co

effici

ent







True102

3061

5102

918

100


0

02

04

06

0 02 04 06


1

2

3

Hot

cont

rast

reco

very

coeffi

cien

t




True102

3061

5102

918

100


08

07

06

05

04

03

02

01

00 02 04 06


1

2

3

Col

d co

ntra

st re

cove

ry co

effici

ent





10

20

30

40

50

60

70

80

90

20 40 60 80

1090807060504030201

00 50 100

07

06

05

04

03

02

01

00 50 100

Imag

e cou

nts

Imag

e cou

nts


(a)

10

20

30

40

50

60

70

80

90

20 40 60 80

09

08

07

06

05

04

03

02

01

00 50 100

08

07

06

05

04

03

02

01

00 50 100

Imag

e cou

nts

Imag

e cou

nts


(b)

10

20

30

40

50

60

70

80

90

20 40 60 80

1090807060504030201

00 50 100

07

06

05

04

03

02

01

00 50 100

Imag

e cou

nts

Imag

e cou

nts


(c)







5 Conclusion



Acknowledgments


References

















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










1

2

3

4






3 Results




10

20

30

40

50

60

70

80

90

20 40 60 80

1090807060504030201

00 50 100

0908

07

06

05

04

03

02

010

0 50 100

Imag

e cou

nts


Imag

e cou

nts

(a)

10

20

30

40

50

60

70

80

90

20 40 60 80Im

age c

ount

s09

08

07

06

05

04

03

02

01

00 50 100

Horizontal pixels

07

06

05

04

03

02

01


Imag

e cou

nts

(b)

10

20

30

40

50

60

70

80

90

20 40 60 80

Imag

e cou

nts

090807060504030201

00 50 100


Imag

e cou

nts

0 50 100

1 08

07

06

05

04

03

02

01

0

(c)






Ideal

Evaluation point

06

03

00 03 06


Hot

cont

rast

reco

very

coeffi

cien

t









5


4 Discussion



Ideal

Evaluation point

06

03

00 03 06


Col

d co

ntra

st re

cove

ry co

effici

ent







True102

3061

5102

918

100


0

02

04

06

0 02 04 06


1

2

3

Hot

cont

rast

reco

very

coeffi

cien

t




True102

3061

5102

918

100


08

07

06

05

04

03

02

01

00 02 04 06


1

2

3

Col

d co

ntra

st re

cove

ry co

effici

ent





10

20

30

40

50

60

70

80

90

20 40 60 80

1090807060504030201

00 50 100

07

06

05

04

03

02

01

00 50 100

Imag

e cou

nts

Imag

e cou

nts


(a)

10

20

30

40

50

60

70

80

90

20 40 60 80

09

08

07

06

05

04

03

02

01

00 50 100

08

07

06

05

04

03

02

01

00 50 100

Imag

e cou

nts

Imag

e cou

nts


(b)

10

20

30

40

50

60

70

80

90

20 40 60 80

1090807060504030201

00 50 100

07

06

05

04

03

02

01

00 50 100

Imag

e cou

nts

Imag

e cou

nts


(c)







5 Conclusion



Acknowledgments


References

















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










10

20

30

40

50

60

70

80

90

20 40 60 80

1090807060504030201

00 50 100

0908

07

06

05

04

03

02

010

0 50 100

Imag

e cou

nts


Imag

e cou

nts

(a)

10

20

30

40

50

60

70

80

90

20 40 60 80Im

age c

ount

s09

08

07

06

05

04

03

02

01

00 50 100

Horizontal pixels

07

06

05

04

03

02

01


Imag

e cou

nts

(b)

10

20

30

40

50

60

70

80

90

20 40 60 80

Imag

e cou

nts

090807060504030201

00 50 100


Imag

e cou

nts

0 50 100

1 08

07

06

05

04

03

02

01

0

(c)






Ideal

Evaluation point

06

03

00 03 06


Hot

cont

rast

reco

very

coeffi

cien

t









5


4 Discussion



Ideal

Evaluation point

06

03

00 03 06


Col

d co

ntra

st re

cove

ry co

effici

ent







True102

3061

5102

918

100


0

02

04

06

0 02 04 06


1

2

3

Hot

cont

rast

reco

very

coeffi

cien

t




True102

3061

5102

918

100


08

07

06

05

04

03

02

01

00 02 04 06


1

2

3

Col

d co

ntra

st re

cove

ry co

effici

ent





10

20

30

40

50

60

70

80

90

20 40 60 80

1090807060504030201

00 50 100

07

06

05

04

03

02

01

00 50 100

Imag

e cou

nts

Imag

e cou

nts


(a)

10

20

30

40

50

60

70

80

90

20 40 60 80

09

08

07

06

05

04

03

02

01

00 50 100

08

07

06

05

04

03

02

01

00 50 100

Imag

e cou

nts

Imag

e cou

nts


(b)

10

20

30

40

50

60

70

80

90

20 40 60 80

1090807060504030201

00 50 100

07

06

05

04

03

02

01

00 50 100

Imag

e cou

nts

Imag

e cou

nts


(c)







5 Conclusion



Acknowledgments


References

















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Ideal

Evaluation point

06

03

00 03 06


Hot

cont

rast

reco

very

coeffi

cien

t









5


4 Discussion



Ideal

Evaluation point

06

03

00 03 06


Col

d co

ntra

st re

cove

ry co

effici

ent







True102

3061

5102

918

100


0

02

04

06

0 02 04 06


1

2

3

Hot

cont

rast

reco

very

coeffi

cien

t




True102

3061

5102

918

100


08

07

06

05

04

03

02

01

00 02 04 06


1

2

3

Col

d co

ntra

st re

cove

ry co

effici

ent





10

20

30

40

50

60

70

80

90

20 40 60 80

1090807060504030201

00 50 100

07

06

05

04

03

02

01

00 50 100

Imag

e cou

nts

Imag

e cou

nts


(a)

10

20

30

40

50

60

70

80

90

20 40 60 80

09

08

07

06

05

04

03

02

01

00 50 100

08

07

06

05

04

03

02

01

00 50 100

Imag

e cou

nts

Imag

e cou

nts


(b)

10

20

30

40

50

60

70

80

90

20 40 60 80

1090807060504030201

00 50 100

07

06

05

04

03

02

01

00 50 100

Imag

e cou

nts

Imag

e cou

nts


(c)







5 Conclusion



Acknowledgments


References

















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










True102

3061

5102

918

100


0

02

04

06

0 02 04 06


1

2

3

Hot

cont

rast

reco

very

coeffi

cien

t




True102

3061

5102

918

100


08

07

06

05

04

03

02

01

00 02 04 06


1

2

3

Col

d co

ntra

st re

cove

ry co

effici

ent





10

20

30

40

50

60

70

80

90

20 40 60 80

1090807060504030201

00 50 100

07

06

05

04

03

02

01

00 50 100

Imag

e cou

nts

Imag

e cou

nts


(a)

10

20

30

40

50

60

70

80

90

20 40 60 80

09

08

07

06

05

04

03

02

01

00 50 100

08

07

06

05

04

03

02

01

00 50 100

Imag

e cou

nts

Imag

e cou

nts


(b)

10

20

30

40

50

60

70

80

90

20 40 60 80

1090807060504030201

00 50 100

07

06

05

04

03

02

01

00 50 100

Imag

e cou

nts

Imag

e cou

nts


(c)







5 Conclusion



Acknowledgments


References

















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










10

20

30

40

50

60

70

80

90

20 40 60 80

1090807060504030201

00 50 100

07

06

05

04

03

02

01

00 50 100

Imag

e cou

nts

Imag

e cou

nts


(a)

10

20

30

40

50

60

70

80

90

20 40 60 80

09

08

07

06

05

04

03

02

01

00 50 100

08

07

06

05

04

03

02

01

00 50 100

Imag

e cou

nts

Imag

e cou

nts


(b)

10

20

30

40

50

60

70

80

90

20 40 60 80

1090807060504030201

00 50 100

07

06

05

04

03

02

01

00 50 100

Imag

e cou

nts

Imag

e cou

nts


(c)







5 Conclusion



Acknowledgments


References

















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Research Article Evaluation of the Feasibility and ...

Documents

Transcript of Research Article Evaluation of the Feasibility and ...