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A new simple iterative reconstruction algorithm for SPECT transmission measurementDoSik Hwang and Gengsheng L. Zeng Citation: Medical Physics 32, 2312 (2005); doi: 10.1118/1.1944288 View online: http://dx.doi.org/10.1118/1.1944288 View Table of Contents: http://scitation.aip.org/content/aapm/journal/medphys/32/7?ver=pdfcov Published by the American Association of Physicists in Medicine Articles you may be interested in An innovative iterative thresholding algorithm for tumour segmentation and volumetric quantification on SPECTimages: Monte Carlo-based methodology and validation Med. Phys. 38, 3050 (2011); 10.1118/1.3590359 Direction-dependent localization errors in SPECT images Med. Phys. 37, 4886 (2010); 10.1118/1.3481515 An iterative transmission algorithm incorporating cross-talk correction for SPECT Med. Phys. 29, 694 (2002); 10.1118/1.1472500 Fast algorithm for iterative image reconstruction with a small animal positron emission tomograph YAP-PET (inEnglish) Med. Phys. 28, 1141 (2001); 10.1118/1.1376642 Approximate 3D iterative reconstruction for SPECT Med. Phys. 24, 1421 (1997); 10.1118/1.598030
A new simple iterative reconstruction algorithm for SPECTtransmission measurement
DoSik Hwanga!
Department of Bioengineering and Department of Radiology, University of Utah, Salt Lake City,Utah 84108
Gengsheng L. ZengDepartment of Radiology, University of Utah, Salt Lake City, Utah 84108
sReceived 23 December 2004; revised 2 May 2005; accepted for publication 6 May 2005;published 21 June 2005d
This paper proposes a new iterative reconstruction algorithm for transmission tomography andcompares this algorithm with several other methods. The new algorithm is simple and resembles theemissionML-EM algorithm in form. Due to its simplicity, it is easy to implement and fast to computea new update at each iteration. The algorithm also always guarantees non-negative solutions. Evalu-ations are performed using simulation studies and real phantom data. Comparisons with otheralgorithms such as convex, gradient, and logMLEM show that the proposed algorithm is as good asothers and performs better in some cases. ©2005 American Association of Physicists in Medicine.fDOI: 10.1118/1.1944288g
I. INTRODUCTION
Transmission scans provide attenuation maps for SPECT im-age reconstruction with attenuation correction. Attenuationcorrection is important in SPECT, particularly when quanti-tative accuracy is desired. Some transmission measurementtechnologies require sequential emission and transmissiondata acquisition; others allow simultaneous acquisition ofemission and transmission data.1–8 If the transmission andemission projections are acquired simultaneously, the trans-mission source must emit photons with photon energies dif-ferent from the energies of the primary emissions of the ra-dioisotope or radioisotopes being imaged. Otherwise,emission and transmission data cannot be distinguished.High-quality transmission scans can also provide an ana-tomical representation to localize abnormal tracer uptake.State-of-the-art combined SPECT/CT systems provide excel-lent transmission images;9 however, these systems cost morethan conventional SPECT systems. For many applicationssuch as in cardiac imaging, SPECT-camera-based transmis-sion images are sufficient for attenuation correction pur-poses. These kinds of transmission data generally containlow-count photon measurements and the attenuation mapshave to be reconstructed from low-countsnoisyd transmis-sion data. Therefore several iterative algorithms have beendeveloped to handle the statistical noise.10–20 In this paper,we propose a new simple iterative reconstruction algorithm,which is very similar to the emissionML-EM algorithm inform. Due to its simplicity, it is easy to implement and fast tocompute each iteration. The algorithm also always guaran-tees non-negative solutions. Comparisons with other algo-rithms sconvex,13,21 gradient,21,22 and ML-EM with the loga-rithm of measurements,15,23 henceforward called logMLEMin this manuscriptd through computer simulation and realphantom studies show that the proposed algorithm is as goodas others, performs better in some cases, and handles thestatistical noise well.
II. RECONSTRUCTION ALGORITHM
Many iterative algorithms10–20 have been developed fortransmission tomography. Some of them are approximate andsome of them are Bayesian. Compared with the emissionEM
algorithm,10 they are more complicated to compute. In thispaper, we propose a new and simple reconstruction algorithmfor transmission data that is similar to the emissionEM algo-rithm.
The well-known emissionML-EM algorithm10 is expressedin the following:
xinew=
xiold
o jaijo
j
aijpj
o kakjxkold
=xi
old
o jaijo
j
aijmeasured projection
modeled projection, s1d
wherepj is the measured emission projection,xi is the esti-mated radioactivity in image pixeli, and aij is the knowncoefficient that represents the contribution of image pixeli toprojection bin j . Hereokakjxk
old is the projector in the algo-rithm, and the summation overj is the backprojector.
Equations1d may be rewritten in the additive form:
xinew=
xiold
o jaijo
j
aijpj
o kakjxkold
=xi
old
o jaijo
j
aij
pj − o kakjxkold + o kakjxk
old
o kakjxkold
= xiold +
xiold
o jaijo
j
aij
pj − o kakjxkold
o kakjxkold
= xiold − seF ]F
]xiG
wj=1/okakjxkold,xi=xi
old, s2d
where the objective functionF is defined as
2312 2312Med. Phys. 32 „7…, July 2005 0094-2405/2005/32 „7…/2312/8/$22.50 © 2005 Am. Assoc. Phys. Med.
F = oj
wjSpj − ok
akjxkD2s3d
and the step sizese is
se =xi
old
2o jaij. s4d
The functionF in Eq. s3d becomes a conventional weightedleast-squares objective function if the weighting factorwj is1/s j
2 wheres j2 is the variance of the measured projectionpj.
Therefore, the emissionML-EM algorithm can be treated as analgorithm that minimizes the weighted least-squares objec-tive function s3d using okakjxk
old to approximate the Poissonprojection data variances j
2 at each iteration.For transmission data, let the projection model be
qj0 exps−oiaijmid, where qj
0 is the projection ray countsorfloodd before entering the object, andmi is the linear attenu-ation coefficient for pixeli. Let qj be the measured transmis-sion projection, andaij the length of the segment of rayjwithin pixel i.
We now propose a transmission algorithm similar to Eq.s1d:
minew=
miold
o jaijo
j
aij
qj0 exps− o kakjmk
olddqj
=mi
old
o jaijo
j
aijmodeled protection
measured protection. s5d
Hereqj0 exps−okakjmk
oldd is the projector in the algorithm, and
the summation overj is the backprojector. Algorithms5d canbe rewritten as an additive form:
minew= mi
old +mi
old
o jaijo
j
aij
qjSqj
0 expS− ok
akjmkoldD − qjD . s6d
Equations6d can be further expressed in a gradient type al-gorithm with the objective functionG:
minew= mi
old − stF ]G
]miG
wj=1
qjqj0 exps−okakjmk
oldd,mi=mi
old, s7d
where the objective functionG can be shown as
G = oj
wjSqj0 expS− o
k
akjmkD − qjD2, s8d
wherewj is
wj =1
qjqj0 exps− o
kakjmk
oldds9d
and the step sizest is
st =mi
old
2o jaij. s10d
For transmission Poisson data, the variance ofqj can be ap-proximated by eitherqj or qj
0 exps−okakjmkoldd. Therefore, the
objective functions8d is a form of weighted least-squareswith the weight of 1/s̃ j
4, which is approximated by the re-ciprocal of qjqj
0 exps−okakjmkoldd. To prevent the occurrence
FIG. 1. Attenuation phantoms. The numbers represent the attenuation coef-ficientsscm−1d. Rl, R2, and R3 represent the regions of interest for the meanand variance studies.
FIG. 2. Reconstruction of phantom 1 in the absence of statistical noise.Images were reconstructed by four different algorithms: TEMF, convex,logMLEM, and gradient from the first, second, third and fourth column,respectively. The top row is for the 6th iteration. The bottom row is for the20th iteration.
FIG. 3. Reconstruction of phantom 2 without any statistical noise.
FIG. 4. Reconstruction of phantom 3 without any statistical noise.
2313 D. Hwang and G. L. Zeng: A simple iterative reconstruction algorithm for transmission SPECT 2313
Medical Physics, Vol. 32, No. 7, July 2005
of zero in the denominator of Eq.s5d, a positive constantecan be added as in the following:
minew=
miold
o jaijo
j
aij
qj0 exps− o kakjmk
oldd + e
qj + e. s11d
The derivation of the transmission algorithms11d is adhoc, and we do not yet have a proof of the convergence ofEq. s11d. However, the additive forms7d of the algorithm isof gradient type. Therefore the algorithm is able to minimizethe objective functions8d if the iteration step-size is smallenough. To realize a smaller step size, the relaxed algorithmof Eq. s11d can be expressed as
minew= ami
old + s1 − ad
3mi
old
o jaijo
j
aij
qj0 exps− o kakjmk
oldd + e
qj + e, s12d
where a is a user-chosen relaxation parameter with 0øa,1.
The multiplicative form s11d is simple and as easy toimplement as the emission EM algorithm. Besides its sim-plicity, algorithm s11d or s12d has other desired properties.First, the results of the algorithm are always non-negative.Second, the objective function, hence the solution, is
weighted by the reciprocal of the approximate variance-square of the Poisson projection data. For the rest of thepaper, we will refer to the algorithm in Eq.s12d as TEMFstransmission algorithm in emission-MLEM formd.
III. COMPUTER SIMULATION STUDIES
Three different computer-generated phantoms were usedto characterize and compare TEMF with other algorithms.The transmission data were simulated with a parallel projec-tion configuration. Noiseless projection data were first gen-erated by an analytical formula, then Poisson noise wasadded into them. Three phantoms are shown in Fig. 1. Thenumbers on the phantoms represent the attenuation coeffi-cients scm−1d over the surrounding area. Using these phan-toms, four different situations were simulated: very low-countsI0=12d, relatively low-countsI0=30d, relatively high-countsI0=200d, and noiseless data.I0 is the mean count perdetector bin in the blank scan. A NCAT phantom was alsoused for generating both transmission and attenuated emis-sion data to study the effect on the resultant SPECT emissionreconstruction. The NCAT phantom is shown in Fig. 13.
From the simulated transmission projection data, the threephantoms were reconstructed using four different methods:TEMF fa was set to 0.5 ande was set to 2 in Eq.s12dg,convex,13,21 gradient,21,22 and logMLEM methodssthe emis-sion ML-EM algorithm applied to the logarithm of the trans-
FIG. 5. Line profile for the reconstruction of phantom 3.A dot represents the true value. Circle, triangle, square,and star marks represent for TEMF, convex, logMLEM,and gradient algorithms, respectively.
FIG. 6. Reconstruction of phantom 1 with very low countssI0=12d. FIG. 7. Reconstruction of phantom 2 with very low countssI0=12d.
2314 D. Hwang and G. L. Zeng: A simple iterative reconstruction algorithm for transmission SPECT 2314
Medical Physics, Vol. 32, No. 7, July 2005
mission datad.15,23The logMLEM algorithm and its ordered-subset version have been used by many groups even thoughit incorrectly treats the log-converted transmission sinogramsas Poisson-distributed data.19,23 The convex and gradient al-gorithms are shown in Eqs.s13d and s14d, respectively. Theconvex algorithm:
minew= mi
old +
mioldo
jaijsqj
0 exps− ok
akjmkoldd − qjd
ojaijo
kakjmk
oldqj0 exps− o
kakjmk
oldd. s13d
The gradient algorithm:
minew= mi
old +mi
old
ojaijqj
F ]
]miLsmdG
m=mold
= miold
ojaijqj
0 exps− ok
akjmkoldd
ojaijqj
, s14d
where Lsmd is the log-likelihood of the observed photoncounts and defined as
Lsmd = ojH− qj
0 exps− okakjmkd − qjo
k
akjmkJ + c,
s15d
wherec is a constant.
A. Studies with noiseless data
Figure 2 shows reconstructed images of phantom 1fFig.1sadg at the 6th and 20th iterations in the absence of statisti-cal noise. The images in the first column were reconstructedby TEMF, and the second, the third, and fourth by convex,logMLEM, and gradient methods, respectively. The imagesat the 20th iteration show that TEMF and logMLEM pro-duced good images clearly separating the three cold discs,whereas the convex and gradient algorithms could not sepa-rate them.sApproximately 100 iterations of the convex algo-rithm were required to separate themd. The processing timefor one update was 117.1, 212.8, 117.4 and 212.6 mssPen-tium 4 CPU 3.40 GHzd for the TEMF, convex,logMLEM, and gradient algorithms, respectively.
The results from phantom 2 experiments also showedsimilar effects sFig. 3d. TEMF clearly separated all ninediscs, whereas the others produced nine discs with dark shad-ows connecting them. In terms of separating those discs andimage contrast, TEMF produced the best image. Another er-roneous feature in the images reconstructed by the convexand gradient algorithms is the four dark areas in the areassurrounded by four cold discs. They were most obvious inthe images reconstructed by the convex and gradient meth-ods, while they were not noticeable in the images producedby TEMF and logMLEM.
In the experiments with phantom 3sFig. 4d, TEMF clearlyseparated the four discs with good contrast and clear edgeswhile convex and gradient methods produced a little blurringespecially around the edges of the discs. The convex andgradient methods yielded two artifactual bright areas at thecenter of the images that were not observed in images recon-structed by TEMF and logMLEM. Figure 5 shows the lineprofile of each reconstruction.
B. Studies with very low counts „I0=12…
We have also studied the cases with very low-count trans-mission data: each detector bin in the blank scan containedan average photon count of 12sI0=12d. In very low countexperiments with phantom 1sFig. 6d, the logMLEM algo-
FIG. 8. Reconstruction of phantom 3 with very low countssI0=12d.
FIG. 9. Line profile for the reconstruction from thenoisy data. A dot represents the true value. Circle, tri-angle, square, and star marks represent TEMF, convex,logMLEM, and gradient algorithms, respectively.
2315 D. Hwang and G. L. Zeng: A simple iterative reconstruction algorithm for transmission SPECT 2315
Medical Physics, Vol. 32, No. 7, July 2005
rithm distorted the three cold discs until hardly recognizableand apparently connected to each other. In images recon-structed by TEMF, convex, and gradient algorithms, the threediscs are recognizable as such. TEMF showed clear separa-tion of the three discs even at the sixth iteration. Similarresults were observed in the experiments with phantoms 2and 3sFigs. 7 and 8d.
In the experiments with phantom 3sFig. 8d, logMLEMproduced images in which four discs are hardly separable.They are blurred and mixed with each other. This result willbe clearly shown in the next sections where mean estimationvalues over several regions of interests will be presented.The other algorithms clearly separated four discs and kepttheir circular shapes. Figure 9 shows the line profile of eachreconstruction.
C. Mean estimation values and normalized standarddeviations
As an additional assessment of image quality, we com-pared mean estimation values and noise indexsa noise indexis defined as the normalized standard deviationd over three
regions of interestssROId of phantom 3 as shown in Fig.1scd. The first ROI is the inside circle marked as “R1” andthe true attenuation coefficient is 0.035/cm. The second ROIis the inside circle marked as “R2” and its true value is zero.The third ROI is the three small circles marked as “R3” andtheir attenuation coefficient is 0.07/cm. They were chosen toavoid partial volume effects at the edges. Mean value esti-mation and calculation of normalized standard deviationsover these ROIs were performed with different noise levels:very low-count, relatively low-count, relatively high-countmeasurements. Mean values and normalized standard devia-tions were calculated and averaged over 100 different noiserealizations.
1. Relatively high-count measurements „I0=200…
With high-count measurements, the estimations for allthree ROIs were very good. All four algorithms showed verysimilar performance in terms of mean estimation and nor-malized standard deviationssFig. 10d.
FIG. 10. Mean values and normalized standard deviations with relatively high counts.
FIG. 11. Mean values and noise indexes with relatively low counts.
2316 D. Hwang and G. L. Zeng: A simple iterative reconstruction algorithm for transmission SPECT 2316
Medical Physics, Vol. 32, No. 7, July 2005
2. Relatively low-count measurements „I0=30…
With relatively low-count measurements, all four algo-rithms made good estimations for ROI 1. At the earlier itera-tions, the noise in the reconstruction by TEMF was less thanthe noise in the reconstructions by the convex or gradientmethods. However, they reached the same level of noise atlater iterations. The logMLEM algorithm produced the leastnoise over all iterations; however, estimation of the attenua-tion coefficient of ROI 2 by the logMLEM method was theleast accurate. Conceivably, these erroneous estimationscould have produced less noise. Over ROI 3, TEMF slightlyoverestimated the mean values, whereas convex, gradient,and logMLEM slightly underestimated themsFig. 11d.
3. Very low-count measurements „I0=12…
For very low-count measurementssI0=12d, all four algo-rithms underestimated the attenuation coefficient for ROI 1and overestimated it for ROI 2. The logMLEM algorithmyielded the worst estimations for the mean values of ROI 2and ROI 3. The normalized standard deviations as a noiseindex show that convex and gradient algorithms followedalmost the same pattern. TEMF produced a low noise indexat early iterations, which then approached the noise valuesproduced by the convex and gradient algorithms at later it-erations. The logMLEM gave a lower noise index over alliterations; however, this does not mean that logMLEM per-forms better than others since logMLEM made the worst
estimation for the mean values of ROI 2 and ROI 3. This badestimation of logMLEM with low-count measurements wasalso revealed in the reconstructed images in Fig. 8. For ROI3, logMLEM underestimated the mean value, while the oth-ers made a good estimationsFig. 12d.
D. Effects on SPECT emission reconstruction: NCATphantom
A NCAT phantom was used to generate both transmissionand attenuated emission data to study the effect that the dif-ferent algorithms might have on the resultant SPECT emis-sion reconstruction. The NCAT phantom is shown in Fig. 13.The transmission and emission data were generated in a 256-bin detector based on a 2563256 NCAT phantom and thenrebinned into a 64-bin detector in order to introduce a mod-eling error in our simulation studies. The reconstruction wasperformed on a 64364 image. The reconstructed attenuationmaps and emission activity maps are presented in Fig. 13.The results indicate that there are some differences betweenthe reconstructed attenuation maps, but these differences hadvery little effect when they were applied to noisy emissiondata.
IV. APPLICATION TO REAL TRANSMISSION DATA
An x-ray torso phantomfsee Fig. 14sadg was used in atransmission scan on a Philip’s IRIX SPECT systemsPhilipsMedical Systems, Cleveland, OHd at the University of Utah
FIG. 12. Mean values and noise indexes with very low counts.
FIG. 13. Reconstructions of the NCAT phantom. Toprow: attenuation map. Bottom row: activity map.
2317 D. Hwang and G. L. Zeng: A simple iterative reconstruction algorithm for transmission SPECT 2317
Medical Physics, Vol. 32, No. 7, July 2005
Hospital. A 60mCi Ba-133 point source was used as thetransmission source. It was located perpendicular to the cen-ter of the detector and 98 cm away. The distance from theBa-133 point source to the center-of-rotation was 74.5 cm.The detector rotated 360° with 120 stops. Due to the weaksource strength, the count rate was only 0.4 kcounts/s. Theacquisition time was 10 s/view. The attenuation maps werereconstructed using four different methodssFig. 14d. TEMFproduced as good an attenuation map in this physical phan-tom experiment as the other methods.
V. CONCLUSIONS
We proposed a new, simple iterative reconstruction algo-rithm sTEMFd for transmission tomography and comparedthis algorithm with several other methods. Using computersimulation studies with different noise levels and differentphantoms, we showed that TEMF is as good as other algo-rithms, performs better in some cases, and handles the statis-tical noise well. Each algorithm seems to have its own ad-vantage over others depending on situations such as the typeof objects to be reconstructed and the level of photon counts.The logMLEM algorithm showed good performance in termsof separation of two adjacent discs when the transmissiondata contained high-count or noiseless measurements. How-ever, it produced biased estimations when data containedvery low photon counts. TEMF showed good performancewith both high-count and low-count measurements, while theconvex and gradient methods produced some artifacts withhigh-count measurements and needed a lot more iterations toresolve them. Another advantage of TEMF is its simplicity ofimplementation. TEMF is more efficient than the other threealgorithms considered in this paper. The processing time forone update in TEMF was 117.1 mssPentium 4 CPU 3.40GHzd, which is almost half that for the convex and gradientalgorithms, which were 212.8 and 212.6 ms, respectively.The logMLEM method required 117.4 ms. TEMF also al-ways guarantees non-negative solutions and takes care ofPoisson statistics through weights in the objective functionas described in the Sec. II. More studies are needed to decidethe optimal parameterssa anded in TEMF. These parametershave been introduced in order to make sure that the algo-rithm converges and to prevent the occurrence of zero in thedenominator. In many cases,a of 0.5 ande of 2 workedwell; in some cases, a largere produced a better attenuationmap. The convergence of the algorithm also needs furtherinvestigation.
ACKNOWLEDGMENTS
This work was supported by the National Institutes ofBiomedical Imaging and Bioengineering, and National Can-cer Institute of the National Institutes of Health under grantsCA 100181 and EB00121. We thank Dr. Roy Rowley of theUtah Center for Advanced Imaging Research for editing ourmanuscript.
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