IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 16, NO. 2, APRIL 1997 223

the P -matrix for the reconstruction of the real scanner data. Theoriginal listings forn� and -mask refer to theD = 384�80 tuberecordings in each sinogram and theB = 128� 128 pixels in eachimage, respectively. When applying focus-of-attention, these numbersdrop significantly. In the best case (head,z = 1), for example, about60% of the tubes and 90% of the pixels are not needed for thereconstruction. Even in the worst case (abdomen), 20% of the tubesand 70% of the pixels can be ignored. When it comes to the numberof nonzero elements in theP -matrix, which is the governing factorwith respect to memory consumption, we see reductions on the orderof 90% in the best case (head,z = 1) and 65% in the worst case(head,z = 2; abdomen,z = 1).

Table II lists the computational benefits translated into elapsed CPUtime (in seconds). Notice that the time it takes to compute then

�

and -masks is negligible. The fact that theP -matrix computes intime proportional to the size ofn�-mask and is less affected by thesize of the -mask is due to the implementation which is tube ratherthan pixel driven. The compensation for edge packing is not listedbut takes only about 0.15 s. The overall time spent on initializationis thus reduced in all cases. But perhaps more importantly, the timerequired for a single iteration of the EM-ML algorithm is reducedby 80% in the best case (head,z = 1) and 60–70% otherwise (head,z = 2; chest and abdomen;z = 1).

ACKNOWLEDGMENT

The authors wish to thank G. Smith, Director of Nuclear Medicineat the University of Tennessee Medical Center, Knoxville, for pro-viding the ECAT 921 PET scanner data.

REFERENCES

[1] G. T. Herman, A. Lent, and S. W. Rowland, “ART: Mathematics andapplication,”J Theor. Biol., vol. 42, pp. 1–32, 1973.

[2] L. A. Shepp and Y. Vardi, “Maximum likelihood reconstruction foremission tomography,”IEEE Trans. Med. Imag., vol. MI-1, pp. 113–122,1982.

[3] K. Lange and R. Carson, “EM reconstruction algorithms for emissionand transmission tomography,”J. Comput. Assist. Tomog., vol. 8, pp.306–316, 1984.

[4] T.-S. Pan and A. E. Yagle, “Numerical study of multigrid implementa-tions of some iterative image reconstruction algorithms,”IEEE Trans.Med. Imag., vol. 10, pp. 572–588, 1991.

[5] L. Kaufman, “Maximum likelihood, least squares, and penalized leastsquares for PET,”IEEE Trans. Med. Imag., vol. 12, pp. 200–214, 1993.

[6] , “Solving emission tomography problems on vector machines,”Ann. Opt. Res., vol. 22, pp. 325–353, 1990.

[7] T. Hebert, R. Leahy, and M. Singh, “Three-dimensional maximumlikelihood reconstruction for an electronically collimated single-photon-emission imaging system,”J. Opt. Soc. Amer., vol. 7, pp. 1305–1313,1990.

[8] H. M. Hudson and R. S. Larkin, “Accelerated image reconstructionusing ordered subsets of projection data,”IEEE Trans. Med. Imag., vol.13, pp. 601–609, 1994.

[9] C. M. Chen, C. Y. Lee, and Z. H. Cho, “Parallelization of the EMalgorithm for 3-D PET image reconstruction,”IEEE Trans. Med. Imag.,vol. 10, pp. 513–522, 1991.

[10] C. S. Butler, M. I. Miller, T. R. Miller, and J. W. Wallis, “Massively par-allel computers for 3-D single-photon emission computed tomography,”Med. Phys. Biol., vol. 39, pp. 575–582, 1994.

[11] Z.-W. Ju, E. C. Frey, and B. M. W. Tsui, “Distributed 3-D iterativereconstruction for quantitative SPECT,”IEEE Trans. Nucl. Sci., vol. 42,pp. 1301–1309, 1995.

[12] S. P. Olesen, J. Gregor, M. G. Thomason, and G. Smith, “EM-ML PETreconstruction on multiple processors with reduced communications,”Int. J. Imag. Syst. Technol., vol. 7, pp. 215–223, 1996.

[13] J. Gregor and D. A. Huff, “A focus-of-attention EM-ML algorithm forPET reconstruction,” Dept. Comput. Sci., Univ., Knoxville, TN, Tech.Rep. CS-96-327, 1996.

[14] J. L. Prince and A. S. Willsky, “Reconstructing convex sets from supportline measurement,”IEEE Trans. Pattern Anal. Mach. Intell., vol. 12, pp.377–389, 1990.

[15] D. P. Bertsekas and J. N. Tsitsiklis,Parallel and Distributed Com-putation: Numerical Methods. Englewood Cliffs, NJ: Prentice-Hall,1989.

[16] G. T. Herman, A. R. De Pierro, and N. Gai, “On methods for maximum aposteriori image reconstruction with a normal prior,”J. Visual Commun.Image Repro., vol. 3, pp. 316–324, 1992.

[17] J. Llacer, E. Veklerov, K. J. Coakley, E. J. Hoffman, and J. Nunez,“Statistical analysis of maximum likelihood estimator images of humanbrain FDG PET studies,”IEEE Trans. Med. Imag., vol. 12, pp. 215–213,1993.

The Weighted-Distance Scheme: A Globally OptimizingProjection Ordering Method for ART

Klaus Mueller,* Roni Yagel, and J. Fredrick Cornhill

Abstract—The order in which the projections are applied in thealgebraic reconstruction technique (ART) has a great effect on speedof convergence, accuracy, and the amount of noise-like artifacts in thereconstructed image. In this paper, a new projection ordering schemefor ART is presented: the weighted-distance scheme (WDS). It heuris-tically optimizes the angular distance of a newly selected projectionwith respect to an extended sequence of previously applied projections.This sequence of influential projections may incorporate the completeset of all previously applied projections or any limited time intervalsubset thereof. The selection algorithm results in uniform sampling of theprojection access space, minimizing correlation in the projection sequence.This produces more accurate images with less noise-like artifacts thanpreviously suggested projection ordering schemes.

Index Terms—Algebraic reconstruction technique, computed tomogra-phy (CT), iterative reconstruction method, positron emission tomography(PET), projection ordering, single photon emission computed tomography(SPECT), 3-D reconstruction, 2-D reconstruction.

I. INTRODUCTION

The algebraic reconstruction technique (ART) is often preferredover filtered backprojection (FBP) methods for the two-dimensional(2-D)/three-dimensional (3-D) reconstruction from projections when

Manuscript received April 19, 1996; revised December 9, 1996. This workwas supported in part by the National Science Foundation under Grant CCR-9211288 and in part by the Defense Advanced Research Projects Agency(DARPA) under Grant BAA 92-36. The Associate Editor responsible forcoordinating the review of this paper and recommending its publication wasM. W. Vannier.Asterisk indicates corresponding author.

*K. Mueller is with the Biomedical Engineering Center, The Ohio StateUniversity, 270 Bevis Hall, 1080 Carmack Road, Columbus, OH 43210 USA(e-mail: klaus@chopin.bme.ohio-state.edu).

R. Yagel is with the the Biomedical Engineering Center, The Ohio StateUniversity, Columbus, OH 43210 USA. He is also with the Department ofComputer and Information Science, The Ohio State University, Columbus,OH 43210 USA.

J. F. Cornhill is with the Biomedical Engineering Center, The Ohio StateUniversity, Columbus, OH 43210 USA. He is also with the Department ofBiomedical Engineering, The Cleveland Clinic Foundation, Cleveland, OH43210 USA.

Publisher Item Identifier S 0278-0062(97)02404-X.

0278–0062/97$10.00 1997 IEEE

224 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 16, NO. 2, APRIL 1997

the projections are sparse, noisy, or nonuniformly distributed [1].Routine clinical computed tomography (CT) does not usually sufferfrom these conditions and, thus, the faster FBP methods are usedexclusively in this setting. However, these conditions do exist inother medical imaging applications. For instance, reconstruction fromnoisy positron emission tomography (PET) data has shown promisingresults [2]. In addition, ART and other iterative methods have enjoyedgreat interest in recent years due to their ability to incorporateinto the reconstruction procedure a proper model of the projectionprocess, as is needed in single photon emission computed tomography(SPECT) (see e.g., [3]). Certain constraints, such as object shapeand nonnegativity, can also be imposed. This proves helpful in someprototype CT applications such as 3-D angiography [4].

In ART, the object to be reconstructed is embedded in a regulardiscrete 2-D/3-D pixel grid. The reconstruction problem can then beformulated as a system of linear equations

N

j=1

wij � vj = ri 1 � i � R (1)

whereN is the number of grid pixels andR is the total numberof projection rays. WithM projections andRm rays per projection,R = Rm �M . The vj are the sought after values of the grid pixels,the ri are the magnitudes of the rays in the acquired projectionimages, and thewij are the weight factors that determine the amountof influence that thejth grid pixel has on theith ray. The weightfactors depend on the way the grid pixels are modeled. While someresearchers represent the pixel grid as a board of solid squares, othersmodel the grid as a field of overlapping interpolation kernels bearingsome functionh. In the former case, awij is simply given by thefractional overlap area of theith ray with thejth pixel. In the lattermethod, which is also the one we advocate, awij is computed as theline or strip integral that is produced by the intersection of theithray with the kernel placed at the location of thejth pixel.

In the case of sparse projection data, equation system (1) maybe underdetermined (i.e.,R < N ); but even ifR � N , which isusually the case in PET, SPECT, and CT, the solution by inversionmethods is hampered both by the size of the equation system andthe inconsistencies caused by the inherent noise in the acquiredprojection data and the approximate description of the weight factors.Thus, a method devised by Gordonet al. [5] is commonly used:Iteratively, for each projection image ray (selected in a certainorder), the grid is projected, the projection is compared with thecorresponding ray value in the acquired projection image, and acorrection term is computed and backprojected onto the grid. Ideally,each backprojection updates the grid to correspond more closely tothe acquired projection data. The iterative process is terminated whensome error or convergence-rate threshold is reached. It was shown byAndersen and Kak [6] that noise-like artifacts in the reconstructioncan be reduced if the grid is corrected only once per projectionimage and not for every projection ray. This discovery gave riseto Simultaneous ART (SART), an algorithm that we have adoptedfor our implementation as well. The SART procedure of updating apixel vj by the weighted contribution of all raysri that are part ofthe current projection imageP' can be captured by the followingequation:

v(k)

j = v(k�1)

j + �r 2P

r � w v

wwij

r 2Pwij

(2)

wherek is the number of the new state of the grid and� is therelax-ation factor typically chosen within the interval(0; 1]. A correctionstep can be made reasonably fast since efficient incremental, table-based methods are available that reduce the terms being summed in

(2) to simple table-lookups [7]–[9]. Note, that for the remainder ofthis article, an iteration constitutes a sequence of grid corrections inwhich all available projections are utilized exactly once.

Due to the problems associated with the linear equation system(1), many solutions may exist. It is thus the goal of ART to convergeto the solution that represents the closest approximation to the objectfunction from which the projection images were obtained. In thisrespect, it has been known for quite some time (see e.g., [10]) thatboth the quality of the approximation and the rate of convergenceof the iterative reconstruction procedure depends on, among otherfactors, the order in which the projections are selected for gridcorrection.

A number of researchers have pointed out [11], [12] that it isdesirable to order the projections in such a way that subsequentlyapplied projections are largely uncorrelated. This means that consec-utively applied projections must have significantly different angularorientations. Indeed, it is intuitive to recognize that if subsequentlyselected projections are chosen at similar orientations, one tends tooverly bias the reconstruction with respect to that viewing anglewithout adding much new information to the grid. Clearly, doing soprolongs the time for convergence and may also drive the approximatesolution away from the desired solution.

While all previously proposed ordering schemes take great careto space far apart consecutively chosen projections, they somewhatneglect the problem of optimizing the selection in a global sense.It is the argument of this paper that in the process of selectinga newly applied projection all, or at least an extended history of,previously applied projection orientations must be taken into accountand weighted by their time of application.

In the following section, we give a brief overview of previouswork on projection access ordering methods. Then, in Section III, wepresent a novel projection ordering method, the weighted distancescheme (WDS), which heuristically optimizes the angular distance ofa newly selected projection with respect to the complete sequenceof all previously applied projections (including those applied inthe previous iteration) or any continuous, time-wise adjacent subsetthereof. Finally, Section IV gives a numerical comparison of ourmethod with respect to previous methods.

II. PREVIOUS WORK

In order to minimize correlation in projection access it seemsadvantageous to arrange the projections such that (Postulate 1):

a) a series of subsequently applied projections is evenly distributedacross a wide angular range;

b) at no time is there an angular range that is covered more denselythan others.

All of the existing methods tend to be strong in one of the twoaspects, but weaker in the other. However, none of the previousmethods comments on how one should proceed with the projectionselection at iteration boundaries. It is clearly necessary to also includeprojections applied in previous iterations into the selection process.A smooth transition between iterations is warranted if the selectionscheme is continuous across iteration boundaries.

Denoting M as the total number of projections in the set and0 � j � M � 1 to be the number of projections already applied,Hamaker and Solmon [13] demonstrate in a fundamental treatmentthat a “good” permutation� of the ordering of theM projections isobtained when the minimum angle among(�(l)� �(k)) � 180�=M;

0 � l � j; 0 � k � j; l 6= k is maximized for eachj. Postulate 1could be regarded as a more heuristic interpretation of this finding.

Many implementations have used a fixed angle for projectionspacing: SART, for example, uses a constant angle of 73.8� when

IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 16, NO. 2, APRIL 1997 225

M = 100. For mostM there is, however, no fixed angle thatsatisfies the criterion of maximizing the minimum angle for all1 � j � M � 1.

An alternative method that uses variable angles between subsequentprojections was proposed by Herman and Meyer [11]. It is basedon the prime number decomposition (PND) ofM . This method,however, requires thatM be nonprime. The same authors also referto work performed by van Dijke [14] who concluded that, amongall schemes he tried, a random projection permutation gave the bestresults. However, we may prefer an ordering scheme that is morecontrollable and deterministic than a random number generator.

More recently, Guan and Gordon [12] presented, what they termed,the multilevel access scheme (MLS). This method works best whenthe number of projections is a power of two, but can also be used, withminor modifications, in the general case. The following descriptionis for the simple case ofM being a power of two: First, for levelone and two, the method chooses the projections at 0�, 90�, 45�, and135�. All subsequent levelsL = 3; � � � ; log

2M contain2L views.

The projection order at levelL is computed by simply going throughthe list of all applied projections at levelsl < L and addingM=2L totheir projection index. This method clearly covers all angular regionsevenly over time, but may not always maximize the angle betweensubsequent projections.

Fig. 1(a) shows, forM = 30, the obtained permutation of the first15 selected projection views when the scheme of PND is applied.Fig. 1(b) shows the permutation for the MLS method under the sameconditions. We observe that PND tends to cluster the projectionsaround certain viewing directions. This may not be advantageous inlight of our earlier comments with regards to an even spread of theapplied projections around the reconstruction cycle. As expected, theMLS method generates a permutation that conforms more closely tothis criterion.

III. T HE WEIGHTED-DISTANCE PROJECTIONORDERING METHOD

We now introduce a permutation scheme that seeks to enforceboth parts of Postulate 1. It is designed to maintain a large angulardistance among the whole set of used projections while preventingclustering of projections around a set of main view orientations.The method selects, from the pool of unused projections, thatprojection that optimizes both the angular spacing and the spreadwith respect to the complete set or a recent subset of all previouslyapplied projectional views. Hereby, it takes into account that morerecent applied projections should have a stronger influence in theselection process than projections that have been applied earlier inthe reconstruction procedure.

The algorithm implements the permutation of the projection or-dering as a circular queue�. The lengthS of � depends on howmuch of the projection access history is to influence the selectionprocess. Since due to the decaying weighting function the influenceof early projections diminishes to insignificance over time, earlyprojections can be replaced by later ones in� as time evolves. In ourimplementation, we choseS = M : A projection’s influential powerfades to zero afterM more projections have been selected. Note,however, that the number of influential projections represented byS

can be chosen arbitrary large.The projectionsPi; 0 � i � M � 1, are assumed to be equally

spaced by an angle' = 180�=M in the interval0 � i � ' < 180�.The first projection inserted into� (at position�[0]) and applied inthe reconstruction procedure is always the projection at orientationangle ' = 0�, i.e., P0. A list � is maintained that holds theLprojections not yet used in the current iteration. At the beginning ofeach iteration,� is filled with all M projections. When a projectionPi is selected from� by our selection criterion, it is removed from

(a)

(b)

Fig. 1. Projection order permutation� for M = 30 projections. Forillustrative purposes only the first 15 ordered views are shown. The height ofa bar corresponds to a view’s place in the ordered sequence� (For instance,in (a) projection 5 is applied as the third projection after view 0 and view15). The graphs shown are for: (a) PND and (b) MLS.

� and inserted into�. Insertion can occur within two phases: theinitial filling phase of� and the update phase of�. Let Q denotethe number of projections currently in�. In the filling phase,Q < S

and subsequently used projections are added to consecutive positionsin �. In the update phase,Q = S and� now functions as a circularqueue: The oldest projection in� is overwritten by the newly selectedprojection and ceases to influence the selection process. See Fig. 2for an illustration of the algorithm in pseudo-code.

We now describe the objective function used to select the nextprojection from � (i.e., the routine SelectProjection() in Fig. 2):First, for eachPl in �; 0 � l < L � 1, the weighted mean�1of the “repulsive forces” exerted ontoPl by the projectionsPq in�; 0 � q � Q � 1; �[q] 6= �[l], is computed. Therepulsiveforce is considered a measure of how close one projection is toanother, it decays linearly with increasing distance between twoprojections. A smaller repulsive force results from a larger spacingof two projections. The minimal distance of two projectionsPl andPq is given by

dlq = Min(jl � qj;M � jl � qj): (3)

226 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 16, NO. 2, APRIL 1997

Fig. 2. Pseudo-code to illustrate insertion/removal of projections into/from circular queue� and list� within the basic framework of ART.

The weighted mean of the repulsive forces acting on a projectionPl, denoted by�l, is given by

�l =

Q�1

q=0

wq � (M=2� dlq)

Q�1

q=0

wq (4)

where a weight factorwq = (q + 1)=Q. The weighting ensures thatprojections applied more recently have a stronger repulsive effectthan projections applied earlier in the reconstruction procedure.

However, using the distance criterion alone does not achieve thegoals of Postulate 1. It was observed that a newly selected projectioncould minimize the distance criterion by being very distant to someprojections in�, but at the expense of being very close to others. Thiscircumstance lead to a situation where projections were selected fromone of several clusters in a cyclic fashion, a condition we strived toavoid. In order to eliminate the large distance fluctuations that gaverise to this behavior we added a second measure to be minimized:the weighted standard deviation of the distancesdlq; �l

�l =

Q�1

q=0

wq � (dlq � �dl)2Q�1

q=0

wq: (5)

Here �dl = dlq=Q is the average distance ofPl to thePq. Main-taining a small�l of the projection distances prevents projectionsfrom clustering into groups of angular viewing ranges.

We then normalize the�l to a range of [0, 1]

~�l =�l �Min0�k<L�1(�k)

Max0�k<L�1(�k)�Min0�k<L�1(�k): (6)

The normalized standard deviations~�l are computed from the�l ina similar fashion.

Finally, we select that projectionPl 2 � to be applied next thatminimizes the weightedL2-norm

Dl = ~�2

l + 0:5 � ~�2

l : (7)

Experiments indicated that a factor of 0.5 to weigh~�2l seemed toyield the best results for a wide range ofM .

IV. RESULTS

Table I gives the projection access orders for all six orderingschemes discussed in the previous sections(M = 30): sequentialaccess scheme (SAS), fixed angle scheme (FAS) at 66.0�, PND [11],random access scheme (RAS) [14], MLS [12], and WDS.

In order to compare all presented methods with regards to Postulate1, we define a discrete 2-D space that is spanned by the projectionindex number and the projection access time instance. This space,called the projection access space (PAS), is “sampled” by the projec-tion ordering methods in aN -rooks fashion, i.e., no line and column

TABLE IPROJECTIONACCESSORDERS FORALL SIX ORDERING SCHEMES(M = 30)

can be sampled twice since the tuple (projection index, access time)is one-to-one and onto. Fig. 3 shows the sampling patterns for thesix projection ordering schemes forM = 30.

By visual inspection, FAS, PND, MLS, and WDS all seem to havea fairly uniform sample distribution. However, to capture the qualityof the distributions in light of Postulate 1 in a more quantitativeway, a descriptive metric is needed. Part a) of this postulate calls fora uniform distribution of the sample points in PAS. We measurethis property by sliding a square box across the PAS, countingthe number of sample points inside the box at each box position,and computing the standard deviation of all counts. If all areas areequally sampled then the standard deviation should be small. Weperformed this analysis forM = 30; 80, and 100, respectively.The sliding square box was dimensioned to capture about 10% ofthe samples on the average. Thus, the box sizes were 10�10 forM = 30, 25� 25 for M = 80, and 30�30 for M = 100.Part b) of the postulate was designed to prevent the clustering of

IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 16, NO. 2, APRIL 1997 227

Fig. 3. Sampling patterns in projection access space for the six projection ordering schemes (M = 30). In all plots, the time coordinate runs from leftto right, while the projection index runs from bottom to top.

projections around a few angular ranges. We evaluate this propertyafter half of the projections (i.e.,M=2) have been applied. Forthis purpose, we slide aM=2 � 4–sized box along the verticaldirection of the PAS, aligned to the left PAS border. Ideally, allboxes should have an equal number of applied projections in them(i.e., two). Again, we count the number of incidences within eachbox and compute the standard deviation of all counts. A largerstandard deviation is evidence for an uneven distribution caused byclustering.

The results of this analysis are listed in Table II. We see while PNDperforms well with respect to projection access uniformity, it tends tocluster projections into angular groups. On the other hand, MLS tendsless to clustering (except forM = 80), but exhibits inferior projectionaccess uniformity. Table II also shows that WDS behaves equally wellin both categories, access uniformity and cluster-freeness, where it isbetter or at least as good as any other method investigated.

Finally, we tested all projection access schemes on the low-contrastShepp–Logan phantom as described in [15] and shown in Fig. 5.Eighty projections of 128 rays each were computed analytically fromthe mathematical description of the ellipses that make up the phantom.In the reconstruction procedure� was set to a fixed value of 0.30and an interpolation kernel based on the Bessel–Kaiser function wasused. Multidimensional Bessel–Kaiser functions have many desirableproperties, such as fast decay for higher frequencies, tunability, andradial symmetry, as is described in [16] and [17]. (Referring tothe nomenclature of [17], our function kernel has a radius of twopixel lengths, an orderm = 2 to ensure a continuous derivativeat the function borders, and a taper parameter� = 10:80 to forcethe kernel’s frequency spectrum to a minimum at the samplingfrequency.)

TABLE IISTANDARD DEVIATIONS OF BOX COUNTS FORTHREE PROJECTION-SET

MAGNITUDES (M = 30, 80, AND 100) TO MEASUREPROJECTIONACCESS

UNIFORMITY AND CLUSTERING. (THE FIXED ANGLE USED IN FAS FOR

M = 30, 80, AND 100 WAS 66.0�, 69.75�, AND 73.8�, RESPECTIVELY)

For estimation of the reconstruction error we use the normalizedroot-mean-squared error measure[10]

error=N

i=1

(oi � vi)2

N

i=1

(oi � �o)2

: (8)

Here,oi is the value of pixelvi in the original Shepp–Logan phantom.This error is plotted in Fig. 4 for all six permutation schemes(M = 80) for a) the entire head and b) the region around the threesmall tumors in the bottom half of the phantom (see Fig. 5). We wouldalso like to compare the various methods in terms of the level of thenoise-like reconstruction artifacts. For this purpose, we compute thevariance within the union of two circular regions to the right and

228 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 16, NO. 2, APRIL 1997

(a)

(b)

(c)

Fig. 4. Reconstruction errors for Shepp–Logan phantom (80 projections of128 pixels each, 128�128 grid): (a) entire head segment, (b) area with thethree small tumors only, and (c) reconstruction noise.

left, respectively, of the two brain ventricles (i.e., the dark ellipticalstructures in the head center, see Fig. 5). These circular regions arehomogeneous in the original phantom. The variance is plotted inFig. 4(c) for all ordering methods.

Fig. 5. Original Shepp–Logan phantom with circled areas indicating theregions in which the error measurements were performed: Regions A wereused to evaluate the level of noise artifacts, B is the region in which thereconstruction error for the three small tumors was measured.

The behavior of the selection process at iteration boundaries wasas follows: While SAS, FAS, PND, and MLS apply their previousaccess order anew (which is probably the strategy used by therespective authors), RAS clears its list of used projections and choosesthe next projection at random, thus generating a different accessorder for each iteration. WDS by design generates a different accesssequence in every iteration as well, however, with the additionalconstraint of optimal fit with regards to the previous sequence. Atthis point it should also be mentioned that some authors (such as[18]) linearly increase the relaxation factor� from a small valueat initial projections to a larger fixed value for use in later gridcorrections. To eliminate the effects of yet another variable in thecomparison process, we chose to use a fixed value of� throughoutthe reconstruction procedure. Hereby� = 0:3 was found to have thebest convergence properties for all access ordering schemes.

For the full head section, all five nonsequential ordering schemesreach their minimum error at about the same time, (i.e., at the endof the fourth iteration). However, this error is smallest with WDS.Hence, even though WDS does not provide a faster convergence to itsminimum-error solution, the solution is more accurate compared tothe solutions obtained with the competing methods at all iterations (atleast until the overall error increases again). Although the differencein error is rather small for the full head section, it is considerablymore significant for the isolated tumor area. This could be viewedas evidence that by using WDS small object detail can be betterrecovered. It is also interesting to note that all ordering schemes reachtheir error minimum for the tumor area about one iteration later thanfor the full head section. We further observe that images producedby WDS have the least amount of noise-like reconstruction artifacts,with RAS being the closest competitor.

Fig. 6 shows the reconstruction results obtained after three iter-ations using the various projection ordering schemes. Even thoughthe numerical results are to a great extent visible in the images, wewould like to note that, for the human vision system, larger numericalerror does not always translate to a more visible artifact. For instance,larger random noise is much more gracefully tolerated by the humaneye than a small but periodic noise pattern such as ringing. FromFig. 6 we observe that, apart from SAS, which after three iterations isfar from reaching its minimal error, FAS and PND have considerablymore reconstruction artifacts than MLS, RAS, and WDS. The artifactsare least noticeable with RAS and WDS, while the contrast for thesmall tumors is best with MLS and WDS.

IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 16, NO. 2, APRIL 1997 229

Fig. 6. Reconstruction of the Shepp–Logan phantom after three iterations on a 128�128 grid using 80 projections of 128 rays each and� = 0:3

for the six projection schemes.

V. FUTURE PLANS

Even though the results for the simulated sparse datasets arepromising, further research is needed to investigate if the methodand its principles offer the same advantages also for real-life datasets,such as noisy PET or SPECT data, projection sets that are distributednonuniformly in angle (including the limited-angle problem [18]),and very sparse datasets with 40 projections and less.

It would also be interesting to investigate the effect of incorporatingknowledge about the shape and dimensions of the object into theselection process. Our test object was fairly circular in shape whichyielded a uniform correlation among the projections. Had the objectbeen more elongated in shape, this may not have been the case.Although in medical application most objects, such as the heart orbrain, are fairly spherical, in industrial settings this is far less the case.

VI. CONCLUSION

We have presented a new projection ordering scheme for ART: theWDS. It heuristically optimizes the angular distance of a newly se-lected projection with respect to an arbitrary-length sequence of previ-ously applied projections. WDS exhibits more uniform projection ac-cess space sampling than existing methods and delivers more accuratereconstructions. Especially, fine detail is more faithfully recoveredand the degree of noise-like reconstruction artifacts is smaller. Both ofthese features are important as they reduce the chance of ambiguitiesfor both the clinician and computerized image analysis systems.

ACKNOWLEDGMENT

The authors would like to thank the anonymous reviewers for theirconstructive comments.

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[3] D. Ros, C. Falcon, I. Juvells, and J. Pavia, “The influence of a relaxationparameter on SPECT iterative reconstruction algorithms,”Phys. Med.Biol., no. 41, pp. 925–937, 1996.

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Reliability of an Image Analysis System for Quantifyingthe Radiographic Trabecular Pattern

Clara M. Korstjens,* Rob J. Spruijt, Wil G. M. Geraets,Lis Mosekilde, and Paul F. van der Stelt

Abstract—A reliability evaluation technique was used to examine thereliability of an image analysis system of the trabecular pattern and todetermine the contribution of three possible sources of error variance.Two series of radiographs were taken of 14 lumbar vertebral slices(28 radiographs). Every radiograph was placed on a viewing box fordigitization four times by a single operator (112 positions of radiographs)and from every position of a radiograph an area of 15 mm�15 mm wasdigitized twice (224 samples for analysis). Ten geometrical characteristicsof the trabecular pattern were studied and its orientation was analyzedin 12 directions. Reliability was determined by calculating Cronbach’s�.This design enabled dividing the measurement error(1��) into fractionsassociated with the X-ray procedure, the operator and the system. Usingthis reliability evaluation technique, it was found that the orientationvariables are more reliable than the geometric variables. It was foundthat effort to increase the reliability should be directed toward improvingthe technical procedure of this image analysis system. Also, repeatedmeasurements will increase the reliability. The number of repeatedmeasurements based on a desired reliability can be calculated. Thisprocedure of evaluation gives the opportunity to select a source of errorvariance which have to be reduced to increase reliability most effectively.

Index Terms—Image processing, radiography, reliability, trabecularpattern.

I. INTRODUCTION

Digital imaging and digital image processing techniques are newdevelopments which may increase the diagnostic value of radiographs[1]–[5]. In earlier publications, a new image analysis system, intendedto increase the diagnostic yield of radiographs with respect to bonearchitecture, was introduced. Bone architecture is an important factorin relation to bone strength. During aging the quality and strength ofbone changes leading to a reduction of the load-bearing capacity andto the occurrence of fragility fractures [6]–[7]. New geometrical andtopological features were developed to characterize bone architecture.These features were preferred above complex statistical measureswhich are less visible and more difficult to interpret [8]–[10]. Withthis noninvasive image analysis system, the structure of and changesin the architecture of trabecular bone can be described [8]–[11].Another regularly applied texture analysis technique measures fractaldimension, which is the degree of “roughness” of an image, and quan-tifies how the fractal dimension varies with the size of the structures

Manuscript received January 3, 1995; revised September 9, 1996. TheAssociate Editor responsible for coordinating the review of this paper andrecommending its publication was S. M. Pizer.Asterisk indicates correspond-ing author.

*C. M. Korstjens is with the Department of Orthodontics, Academic Centrefor Dentistry Amsterdam (ACTA), Amsterdam 1066 EA, the Netherlands. Sheis also with the Department of Oral Radiology, Academic Centre for DentistryAmsterdam (ACTA), Louwesweg 1, Amsterdam 1066 EA, the Netherlands (e-mail: c.korstjens@acta.nl).

R. J. Spruijt is with the Department of Social Dentistry and Dental HealthEducation, Academic Centre for Dentistry Amsterdam (ACTA), Amsterdam1066 EA, the Netherlands.

W. G. M. Geraets and P. F. van der Stelt are with the Department of OralRadiology, Academic Centre for Dentistry Amsterdam (ACTA), Amsterdam1066 EA, the Netherlands.

L. Mosekilde is with the Department of Cell Biology, Institute of Anatomy,University of Arhus, DK-8000Arhus-C, Denmark.

Publisher Item Identifier S 0278-0062(97)02403-8.

0278–0062/97$10.00 1997 IEEE