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c163_par_iter_loc_rec

J Supercomput(2009)48:1–14

DOI10.1007/s11227-008-0198-9

Parallelism of iterative CT reconstruction based

on local reconstruction algorithm

Junjun Deng·Hengyong Yu·Jun Ni·Lihe Wang·

Ge Wang

Published online:29March2008

?Springer Science+Business Media,LLC2008

Abstract An iterative algorithm is suited to reconstruct CT images from noisy or truncated projection data.However,as a disadvantage,the algorithm requires signif-icant computational time.Although a parallel technique can be used to reduce the computational time,a large amount of communication overhead becomes an ob-stacle to its performance(Li et al.in J.X-Ray Sci.Technol.13:1–10,2005).To overcome this problem,we proposed an innovative parallel method based on the local iterative CT reconstruction algorithm(Wang et al.in Scanning18:582–588, 1996and IEEE Trans.Med.Imaging15(5):657–664,1996).The object to be recon-structed is partitioned into a number of subregions and assigned to different process-ing elements(PEs).Within each PE,local iterative reconstruction is performed to recover the subregion.Several numerical experiments were conducted on a high per-formance computing cluster.And the FORBILD head phantom(Lauritsch and Bruder http://www.imp.uni-erlangen.de/phantoms/head/head.html)was used as benchmark to measure the parallel performance.The experimental results showed that the pro-posed parallel algorithm signi?cantly reduces the reconstruction time,hence achiev-ing a high speedup and ef?ciency.

Keywords Computed Tomography(CT)·Image reconstruction·Iterative reconstruction·Local iterative CT reconstruction·Parallel computing·High performance computing·MPI

J.Deng·L.Wang

Department of Mathematics,University of Iowa,Iowa City,IA52242,USA

H.Yu·G.Wang

VT-WFU School of Biomedical Engineering and Sciences,Virginia Polytechnic Institute

and State University,Blacksburg,V A24061,USA

J.Ni( )

Department of Radiology,University of Iowa,Iowa City,IA52242,USA

e-mail:jun-ni@https://www.wendangku.net/doc/5211767082.html,

2J.Deng et al. 1Introduction

In the X-ray CT reconstruction,a cross-sectional or volumetric image of a patient is reconstructed from the projection data.There are two main approaches to perform im-age reconstruction,analytic,and iterative methods.Analytic methods,e.g.,the FDK and the Katsevich algorithms,utilize analytic formulas to reconstruct the image of the object.The iterative methods,e.g.,Algebraic Reconstruction Techniques(ART) and Expectation-Maximization(EM)[5–9],match the measured projection data with the calculated ones based on a currently approximated object density distribution, and subsequently make corrections according to the difference.This procedure is re-peated until some predetermined error level or maximum iteration number has been reached.

As well-known iterative methods are superior to the analytic ones,if the projection data contains high noise or is incomplete[10].A relatively high demand for compu-tational time is the main drawback to use iterative methods.For example,it may take numerous hours to accomplish a single iteration to reconstruct a3-D object with a moderate volume size from cone-beam projection data.Considering time constraints, analytic methods are favored in most tomography applications despite the limitations.

Several approaches have been developed to accelerate the computation of iterative methods.In the Ordered Subsets(OS)method,the projection data is divided into an ordered sequence of subsets(or blocks)and the image is updated after using only a subset,instead of compounding all of the projection data[11–13].This approach is reported to be able to substantially reduce computational time while maintaining image quality[11].In the parallel computing technology,a computational task is partitioned into multiple subtasks and the associated data is sent to different proces-sors connected through a network.After the subtasks are completed,the results are assembled by a master processor to obtain the?nal result.Efforts have been made to investigate the parallel implementation of the iterative algorithms in past years [14–17]Recently,Li et al.implemented the EM algorithm and ART algorithm using the data parallelism.For a reconstruction with a grid volume1283on a16-processors PC cluster,the obtained speedup was around9[1].

Although the data parallelism can be used to reduce the computational time,a suffering remains due to a heavy overhead.A collective communication is required among participating processors to update the estimation of the intermediate image during each iteration.As a result,a speedup is tremendously reduced as the number of processors increases.Moreover,the approach induces a valid problem.When a computation is conducted on a low-speed network or the processors are distributed geographically,the parallel schemes are not promising at all.To resolve this problem, we proposed a parallel scheme via the local CT reconstruction algorithm developed by Wang et al.[2,3].The parallel algorithm has the merit of reconstructing a local region of interest(ROI)without synchronizing to others processing elements(PEs). Thus,the heavy communication overhead is circumvented.

In the following sections,the local iterative CT reconstruction algorithm is?rst outlined.Then the corresponding parallel computing scheme is presented.Next,the performances in terms of computational times,overall speed-up,and parallel ef?-ciency are measured and presented.Finally,we discuss some relevant issues and con-clude the paper.

Parallelism of iterative CT reconstruction based on local3

Fig.1Geometrical illustration of the cone-beam CT

system

2Parallel iterative reconstruction for local CT

2.1Local iterative CT reconstruction algorithm

As illustrated in Fig.1,the local iterative algorithm is initially proposed to address the CT reconstruction problem when projection data is incomplete[2].Assuming a region of interest(ROI)is contained in a convex set C in the2-D parallel beam case, the characteristic function M(x,y)in C can be expressed as

M(x,y)=

1,(x,y)∈C, 0,otherwise,

where x and y are the Cartesian coordinates.Denoting the projection pro?le of M(x,y)as

P M(θ,t)= R

M(x,y)δ(x cosθ+y sinθ?t)dx dy,

whereδ(t)is Dirac’s delta function,one can de?ne a parameter set

(θ,t):θ∈[0,π],t∈[?R,R]and P M(θ,t)>0

The projection pro?le along a localized parallel beam can be written as:

P(θ,t)= R

f(x,y)δ(x cosθ+y sinθ?t)dx dy,(θ,t)∈Z.

4J.Deng et al. According to Wang et al.[2,3],the local iterative reconstruction formula is given by:

f k+1(x,y)=f k(x,y)

n(x,y)

δ(x cosθ+y sinθ?t)

P(θ,t)

P k(θ,t)

dθdt

=f k(x,y)

n(x,y)

P M(θ,x cosθ+y sinθ)>0

P(θ,x cosθ+y sinθ)

P k(θ,x cosθ+y sinθ)

dθ

=f k(x,y)g k(x,y), where

g k(x,y)=

n(x,y)

P M(θ,x cosθ+y sinθ)>0

P(θ,x cosθ+y sinθ)

P k(θ,x cosθ+y sinθ)

dθ,

n(x,y)=

P M(θ,x cosθ+y sinθ)>0

δ(x cosθ+y sinθ?t)dt dθ, and

P k(θ,t)= R

f k(x,y)δ(x cosθ+y sinθ?t)dx dy,(θ,t)∈Z

is the reprojected data based on current image estimate f k(x,y).

In the cone-beam geometry,the projection is considered as a blurred three dimen-

sional function:

P( α)=

X p

X p, α

d X p,

where α=(β,p,ξ) ,βdenotes the X-ray source rotation angle,(p,ξ)speci?es the detector position,

X p, α

X p, X s, X d

=δ

x p?x s

x d?x s

?y p

?y s

y d?y s

x p?x s

x d?x s

?z p

?z s

z d?z s

p≡(x p,y p,z p) , X

s≡(x s,y s,z s) ,

d≡(x d,y d,z d) ,are vectors for specimen,

source,and detector coordinates,respectively.The associated iterative formula is:

f k+1

X p

=f k(

H0( X p)

X p, α

P( α)

P k( α)

d α=f k

X p

g k

X p

where

g k

X p

H0( X p)

X p, α

P( x d)

P k( x d)

d α,

and

X p

X p, α

d α

x p?x s

x d?x s

?y p

?y s

y d?y s

x p?x s

x d?x s

?z p

?z s

z d?z s

dp dξdβ.

Parallelism of iterative CT reconstruction based on local5 Geometrically,P k( X p)is a synthesized cone beam projection based on the current estimate f k( X p),g k( X p)is the overall correction factor computed by backproject-ing the ratios of measured and synthesized projections,and H0( X p)is the weight

compensating for cone beam divergence.

The local iterative algorithm can be considered as a generalized EM-type algo-rithm.If the set C further represents the whole object scanned,the algorithm is virtu-ally identical to the conventional EM algorithm.In general,the set C is a nontrivial part of the object,and the algorithm can accurately recover high-frequency infor-mation in the set C,while faithfully providing low-frequency information outside of it.

2.2Parallel reconstruction using iterative local CT algorithm

This section presents the strategy for reconstructing a3-D object in parallel,by us-ing the local iterative CT reconstruction algorithm.Conventionally,to parallelize the computation of an iterative algorithm,the projection data is?rst partitioned into several groups and sent to different PEs.Then each PE uses the projection data to complete the reconstruction.After each iteration,the PEs exchange the current estimation with the other PEs and continue the next iteration,until either a pre-determined error is tolerated or maximum iteration is reached.The approach re-constructs images identically to the associated sequential algorithm.However,as mentioned in the introduction section,it suffers from a heavy communication over-head.Hence,the performance is compromised if a large number of processors are used.

The proposed local reconstruction algorithm allows boosting the performance by reducing heavy communication overhead.And the algorithm can reconstruct the lo-cal region of interest C with high accuracy.Therefore,one can partition a3-D object data into multiple sub-ROIs,and assign each sub-ROI to a single PE.Within each PE,the local iterative CT reconstruction algorithm is deployed to perform concrete reconstruction by regarding the assigned sub-ROI as the set C in Wang’s algorithm. Once all the PEs have accomplished their tasks,a master node assemble the sub-ROIs images collected from all PEs into the?nal reconstruction.In practice,to ensem-ble the?nal image,a collective communication operation“MPI_Gatherv”is used to gather the results from slave nodes in order.Finally,the master node could ei-ther save the result to disk or send it to a remote user.Figure2illustrates the whole ?owchart.

It can be observed that unlike the conventional approaches,there is no commu-nication among the PEs during each iteration.The reason is the reconstruction of a sub-ROI on the assigned PE is fully independent of others.The only communica-tion time used is to collect images of all the sub-ROIs from all the PEs once.In this way,the communication overhead among processing elements(PEs)is eliminated and the parallel performance signi?cantly increases.The approach maximizes the ef-?ciency because a single sub-ROI result can be achieved independently while other sub-ROIs results are under computation.This property is favorable in a distributed environment.

6J.Deng et al.

Fig.2The?owchart of the parallel algorithm

Since both the size and the position of a sub-ROI in the whole ROI in?uence the reconstruction time,load imbalance is a more sensitive issue than that in the con-ventional approaches.Without careful consideration,the parallel performance would be compromised by the load imbalance.Due to the symmetry of the scanning locus in X and Y direction,one can partition data evenly in both directions.In the primary study,we partitioned the ROI into2by2equal grids in the X–Y plane.In the Z di-rection,the partition is more complicated.It can be veri?ed that the computational load on each PE is roughly proportional to the number of X-rays that intersect with the associated sub-ROI.Since the X-ray source emits a cone beam consisting of equal number X-rays at all positions on the circular scanning locus in our simulation model, partitioning evenly along the Z-direction seems to be a good choice.However,when the X-ray source starts from the bottom and ends at the top of the phantom,such as a spiral scanning locus,some of the X-rays emitted from these positions do not inter-sect with the phantom.Such X-rays have little contribution to the whole computation and are removed before the iteration begins.Consequently,the computation for the X-rays from these positions is less than that for other positions.Figure3gives an illustration of this situation.It’s impossible to give a universal partitioning criterion so that the parallel computing is synchronized perfectly.Nevertheless,we could man-ually adjust the partitioning ratio for a much smaller group of projection data.Since the CT scanning geometry is not changed,if the load imbalance is resolved for the smaller case,then the larger cases are also settled.

3Experiments

To demonstrate the feasibility of the parallel iterative local CT algorithm,numerical experiments were designed and conducted using the FORBILD head phantom[4]. The parallel algorithm was implemented on a PC cluster at Medical Imaging High Performance Computing Lab(MIHPC Lab)at the University of Iowa.The cluster has16nodes,each consisting of two64-bit AMD Opteron processors with4GB memory.Message Passing Interface(MPI),a parallel library,was used to perform message passing(process of data communication)among the PEs.The program was written in C,and compiled by the Porland Group’s c compiler.

Parallelism of iterative CT reconstruction based on local7 Fig.3Illustration where only

part of the cone beam X-ray

intersect with the object

Table1Parameters of the

spiral cone beam geometry Case I Case II

Scanning radius(cm)6464

Source to detector distance(cm)128128

Helical pitch(cm)12.8 6.4

Object radius(cm)12.812.8

Detector size(width,height)28.41×22.5328.41×22.53

Number of projections per turn96192

Number of detector cells128×64512×256

Reconstruction matrix12832563 As an example,we chose the practical spiral cone beam scanning geometry in our simulation.The geometrical parameters were summarized in the Table1.A planar detector was used to collect the projection data.Two cases with different projection data and reconstruction matrix size were used.

Table2gives the results of the measured computational time with respect to the number of PEs.In both cases,the computational time is signi?cantly decreased as the number of PEs increases.

8J.Deng et al. Table2Reconstruction time with different number of processors(NP)

NP148121620242832

Case I1310771418302229194168147138 Case II1574489473250500351252787823294203081819016113

Note:The unit of time is second

Table3Speedup and ef?ciency with different number of processors(NP)

NP148121620242832

Speedup(Case I)1.001.703.134.345.726.757.808.919.49 Speedup(Case II)1.001.663.124.485.656.767.758.669.77 Ef?ciency(Case I)1.000.420.390.360.360.340.330.320.30 Ef?ciency(Case II)1.000.420.390.370.350.340.320.310.31

To examine the performance of the proposed parallel algorithm,we computed the two standard benchmarks,speedup S p and ef?ciencyη,which are de?ned as

S p=T s

T np

,andη=

S p

n p

Here n p is the number of processors,T s is the total execution time when one proces-sor is used,and T np is the total parallel execution time when n processors are used.

The speedup and ef?ciency were computed from the Table1.And the results were presented in Table3and plotted in Fig.4.In Fig.4(a),the speedup linearly increases with the increase of the number of processors.This is a considerable advantage over the conventional parallel iterative algorithms,where the speedup increases initially and then decreases due to an inevitably large amount of communication overhead[1]. This behavior is very promising to achieve high performance in a large-scale system with more computer processors.Another interesting observation is that the speedup curves for the two cases are close to each other,regardless of the difference of data size(projection data and the reconstruction matrix).

To further accelerate the parallel computing,a special strategy,reconstructing the region outside of the ROI with lower resolution,can be applied by taking advantage of the local iterative CT reconstruction algorithm.Since the algorithm only recovers the low frequency information for the regions outside the sub-ROIs and we are only interested in the reconstruction inside the sub-ROIs,we can tolerate lower resolution outside of the sub-ROI while reconstructing high-resolution inside the sub-ROI.This is feasible since the iterative CT algorithm is implemented in a ray-tracing manner. Along the ray,we use larger step size to trace forward and backward when the ray is outside of the sub-ROI,and keep the step size when the ray is inside of the sub-ROI. Therefore,the computational time for the outside region can be reduced and the total

Parallelism of iterative CT reconstruction based on local9 Fig.4Comparison of the(a)

speedup,and(b)ef?ciency of

the parallel iterative algorithm

(a)

(b)

computational time can be decreased as well.Upon this idea,we conducted several experiments.Table4gives the computational time when using different resolutions for the inner and outside sub-ROI regions.The computational time is further reduced comparing with the previous homogeneous resolution approach.The speedup and ef?ciency in Fig.4clearly verify this point.

In order to show the applicability of the parallel algorithm to preserve the recon-struction quality,some typical reconstructed image slices were presented in Fig.5 and representative pro?les were plotted in https://www.wendangku.net/doc/5211767082.html,paring these results with the one using sequential algorithm,a congruency can be seen from the Figs.5and6,indi-cating the image quality was quite stable regarding the different ways of partitioning the ROI.This was consistent with what Wang et al.mentioned that the local iterative approach will naturally reconstruct an actual image when suf?cient projection data is available,or will produce an optimal image[2].It could also be observed that the image quality was well maintained when moderately lower resolution for the region outside of the sub-ROI was used.

10J.Deng et al. Table4Computational time with different NP when using heterogeneous resolution

NP148121620242832

Case I,131054530420616113612010696 double step size

Case I,1310467248169134112958070

4times step size

Case II,1574486844436066255231991316299142111239811376 double step size

Case II,15744862444287162273816765127441071795448685 4times step size

Note:The unit of time is second.Here,we compare the results when double step size and4times step size are used for the region outside the sub-ROIs when local reconstruction is carried out on the PEs

Table5Speedup and ef?ciency with different number of processors(NP),inhomogeneous cases

NP148121620242832

Speedup(Case I,1.002.404.316.368.149.6310.9212.3613.65 double step size)

Speedup(Case I,1.002.675.027.379.2911.1213.1115.5617.79 4times step size)

Speedup(Case II,1.002.304.376.177.919.6611.0812.7013.97 double step size)

Speedup(Case II,1.002.525.486.929.3912.3514.6916.5018.13 4times step size)

Ef?ciency(Case I,1.000.600.540.530.510.480.450.440.43 double step size)

Ef?ciency(Case I,1.000.670.630.610.580.560.550.560.56 4times step size)

Ef?ciency(Case II,1.000.580.550.510.490.480.460.450.44 double step size)

Ef?ciency(Case II,1.000.630.690.580.590.620.610.590.57 4times step size)

As we could observe,although the speedup consistently increases,it continues to show a gap between the ideal speedup—the straight line with a unit slope.The reason is that although the PE reconstructs the sub-ROI independently,it also recovers the low-frequency information outside of this sub-ROI,which introduces the redundant computation in the parallel scheme.Therefore,theoretically,the parallel reconstruc-tion using the local iterative CT algorithm won’t obtain the linear speedup or unit ef?ciency.

Parallelism of iterative CT reconstruction based on local11

Fig.5Representative slices of reconstructed2563volume.(a)Original phantom,(b)sequential EM algorithm,(c)homogenous step size,(d)double step size.Displaying window for call cases is[0.95,1.15], where the value in the range is linearly rescaled to[0,255]

4Discussion and conclusion

Although a good parallel performance was achieved,the load imbalance caused by the ROI partitioning was not solved thoroughly.As we mentioned in Sect.2,the par-titioning criterion in Z direction was based on the test for smaller data set,which was more or less imprecise.Besides,it is not convenient to adjust the partitioning ratio when the number of processors is large.More handy methods need to be exploited to solve this problem thoroughly.

Another concern is about the quality of the reconstructed image.It is clear that the parallel algorithm was not identical to its sequential prototype.Therefore,there was a bright spot in the center of the slice,which was the boundary of the sub-ROIs.To

12J.Deng et al.

Fig.6Representative pro?les of reconstructed slices.(a)The pro?les of the original phantom,reconstruc-tion result of EM algorithm,and reconstruction results of the parallel algorithm,respectively.(b)The pro-?les of the reconstruction results when using homogeneous step size,double step size for outside sub-ROIs region,and4times step size for the outside sub-ROIs region,respectively

remove this,one could append more layers to the boundary of the sub-ROIs when reconstructing and retrieving only the central parts to resemble the?nal result.

In addition to the ROI partitioning,the heterogeneous resolution is also a key factor that determines the load of each PE,and thus affects the speedup and load im-balance potentially.Generally speaking,the lower resolution for the region outside ROI,the higher speedup could be expected.However,the increase in speed for each PE might not be identical since the partition itself is not homogeneous.Furthermore, there should be a tradeoff between it and the image quality,since the coarser resolu-tion in the outside region still has impact on the ROI.As a result,a balanced point needs to be carefully chosen so as to achieve an optimal result.

In conclusion,a parallel computing strategy based on local iterative CT recon-struction algorithm was investigated in this paper.To perform the parallel computing, a ROI was partitioned to sub-ROIs and each sub-ROI was assigned to a PE.On each PE,the local iterative CT reconstruction algorithm was used to conduct the recon-struction.Then the master node collected all the sub-ROIs from the worker nodes to assemble the image.As a result,the computational time was greatly reduced and high speedup was achieved.A special strategy using inhomogeneous resolution was taken to further speedup the computation while the image quality was preserved.Future research should include investigating the impact of different partition methods on the performance of the parallel algorithm,a more detailed investigation into the effect of inhomogeneous resolution on the speedup and image quality and the study on how to removing the bright spot on the boundary of the sub-ROIs while preserving the parallel performance.

Acknowledgements The project was supported by National Health Institute(NIH/NIBIB)grants EB001685,EB002667,and EB006412-01.The authors would like to thank Mark Fleckenstein,Univer-sity of Michigan,for editorial help in preparing the manuscript.Thanks also go to Research Services of Information Technology Services,the University of Iowa,for their administrative and computing supports.

Parallelism of iterative CT reconstruction based on local13

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Junjun Deng received his B.S.(2000)and M.S.(2003)in mathematics from

Peking University,Beijing.Currently he is a Ph.D.candidate in the program in

mathematics and computational sciences,the University of Iowa,USA.His inter-

ests include computed tomography,medical image processing and parallel com-

puting.

Hengyong Yu earned his B.S.degrees in information science&technology

(1998),computational mathematics(1998),and Ph.D.in information&telecom-

munication engineering(2003)from Xi’an Jiaotong University,China.He was

Instructor and Associate Professor with the College of Telecommunication Engi-

neering,Hangzhou Dianzi University,from July2003to Sept.2004.During Sept.

2004and Nov.2006,I was postdoctoral fellow and Associate Research Scientist

with the Department of Radiology,University of Iowa,Iowa City,IA.During Sept.

2004and Nov.2006,I was postdoctoral fellow and Associate Research Scientist

with the Department of Radiology,University of Iowa,Iowa City,IA.During Sept.

2004and Nov.2006,he was postdoctoral fellow and Associate Research Scientist

with the Department of Radiology,University of Iowa,Iowa City,IA.Currently,

14J.Deng et al. he is a Research Scientist and the Associate Director of the CT Laboratory,VT-WFU School of Biomed-ical Engineering and Science,Virginia Tech,Blacksburg,V A.His interests include computed tomography and medical image processing.He has authored or coauthored more than50peer-reviewed journal papers. He serves as the Editorial Board Member of Signal Processing,Guest Editor of the International Journal of Biomedical Imaging and Guest Associate Editor for Medical Physics.He is a senior member of the IEEE and member of the Chinese Institute of Electronics.In2005,he was honored for an outstanding doctorial dissertation by Xi’an Jiaotong University,and received the?rst prize for the best natural science paper from the Association of Science&Technology of Zhejiang Province,China.

Jun Ni received his B.S.from Harbin Engineering University,China,in1982,

M.S.from Shanghai Jiao Tong University(SJTU),China,in1984,and his

Ph.D.from the University of Iowa(UI),USA,in1991.He had a research as-

sociate/lecturer position in SJTU.He worked as postdoctoral associate at UI and

Purdue University from1992to1994.Since1994,he worked at UI as a senior

computing consultant,associate research scientist,research scientist,and the di-

rector of Scienti?c Computing(high performance computing and grid comput-

ing).He has been an adjunct assistant and associate professor in the Department

of Computer Science and the Department of Mechanical&Industrial Engineering

at UI.Currently,he is an Associate Professor and director in the Department of

Radiology,College of Medicine,Associate Professor of Biomedical Engineering and Mechanical Engineering at UI,and Adjunct Associate professor in Computer Science.He is the direc-tor of Medical Imaging HPC Lab(MiHi)and HPC Nanotechnology Lab(HPCNano)at UI.He has over 100peer-reviewed conference and journal papers,30edited books.He is an editor-in-chief,associate and guest editor of more than20journals.He is a member of IEEE,ASME,SMII,AASA,FAS,and RSNA.

Lihe Wang received his B.S.from Peking University,China,in1983,M.S.from

the University of Chicago,in1986,and his Ph.D.from New York University1989.

He worked as instructor and assistant professor at Princeton University from1989

to1993.Since1993,he worked at University of Iowa as Associate Professor and

full professor.He has been as associate professor in UCLA from1995to1997and

member of IAS in2002.He was also a Sloan Fellow in1994.

Ge Wang(S’90–M’92–SM’00–F’03)received B.E.in electrical engineering from

Xidian University,Xian,China,in1982,M.S.in remote sensing from Graduate

School of Academia Sinica,Beijing,China,in1985,and M.S.and Ph.D.in electri-

cal and computer engineering from State University of New York,Buffalo,in1991

and1992.He was Instructor and Assistant Professor with Department of Electrical

Engineering,Graduate School of Academia Sinica in1984–1988,Instructor and

Assistant Professor with Mallinckrodt Institute of Radiology,Washington Univer-

sity,St.Louis,MO,in1992–1996.He was Associate Professor with University of

Iowa from1997–2002,and then Professor with Departments of Radiology,Bio-

medical Engineering,Mathematics,Civil Engineering,Electrical and Computer

Engineering,and Director of the Center for X-Ray and Optical Tomography,Uni-versity of Iowa.Currently,he is Director of the Biomedical Imaging Division and Samuel Reynolds Pritchard professor,WFU-VT School of Biomedical Engineering and Sciences at Virginia Polytechnic Institute and State University,Blacksburg,V A.His interests include computed tomography,biolumines-cence tomography,and systems medicine.He and his coauthors have published over400journal articles and conference papers,including the?rst paper on spiral/helical cone-beam CT,the?rst paper on biolu-minescence tomography,and the?rst paper on interior tomography.He is the founding Editor-in-Chief for International Journal of Biomedical Imaging,and Associate Editors for IEEE Trans.Medical Imaging and Medical Physics.He is an IEEE Fellow,SPIE Fellow,and an AIMBE Fellow.He is also recognized by a number of awards for academic achievements.