Int. J. Appl. Math. Comput. Sci., 2008, Vol. 18, No. 1, 63–73DOI: 10.2478/v10006-008-0006-y

INTERPOLATION-BASED RECONSTRUCTION METHODS FORTOMOGRAPHIC IMAGING IN 3D POSITRON EMISSION TOMOGRAPHY

YINGBO LI ∗ , ANTON KUMMERT ∗ , FRITZ BOSCHEN ∗ , HANS HERZOG ∗∗

∗ Faculty of Electrical, Information and Media EngineeringUniversity of Wuppertal, Rainer-Gruenter-Street 21

42119 Wuppertal, Germanye-mail: {yingbo, kummert, boschen@uni-wuppertal.de}

∗∗ Institute of Medicine, Research Centre Juelich52428 Juelich, Germany

e-mail: h.herzog@fz-juelich.de

Positron Emission Tomography (PET) is considered a key diagnostic tool in neuroscience, by means of which valuableinsight into the metabolism function in vivo may be gained. Due to the underlying physical nature of PET, 3D imagingtechniques in terms of a 3D measuring mode are intrinsically demanded to assure satisfying resolutions of the reconstruc-ted images. However, incorporating additional cross-plane measurements, which are specific for the 3D measuring mode,usually imposes an excessive amount of projection data and significantly complicates the reconstruction procedure. Forthis reason, interpolation-based reconstruction methods deserve a thorough investigation, whose crucial parts are the inter-polating processes in the 3D frequency domain. The benefit of such approaches is apparently short reconstruction duration,which can, however, only be achieved at the expense of accepting the inaccuracies associated with the interpolating process.In the present paper, two distinct approaches to the realization of the interpolating procedure are proposed and analyzed.The first one refers to a direct approach based on linear averaging (inverse distance weighting), and the second one refersto an indirect approach based on two-dimensional convolution (gridding method). In particular, attention is paid to twoaspects of the gridding method. The first aspect is the choice of the two-dimensional convolution function applied, and thesecond one is the correct discretization of the underlying continuous convolution. In this respect, the geometrical structurenamed the Voronoi diagram and its computational construction are considered. At the end, results of performed simulationstudies are presented and discussed.

Keywords: Tomographic reconstruction, three-dimensional positron emission tomography, Fourier slice theorem, frequ-ency sample distribution, two-dimensional interpolation, inverse distance weighting, gridding method.

1. Introduction

With the aid of Computerized Tomographic (CT) imagingmethods, the human ability to gain non-invasive insightinto the internal structure of living organisms can be si-gnificantly extended. Among the broad spectrum of va-rious imaging techniques, Positron Emission Tomogra-phy (PET) is regarded as a key diagnostic tool in neu-roscience for studying the metabolism function in vivo(Beutel et al., 2000). During the measuring procedure ofPET, tracer substances containing neutron-deficient radio-isotopes have to be injected into the human body at first,to spread in accordance with the metabolism in the body

and reach a quasi-steady distribution in the organ of inte-rest afterwards. Since the half-life period of utilized ra-dioisotopes is relatively short, the nuclear decay of theneutron-deficient nuclide in terms of emitting positronsmay occur shortly after the injection. However, each ofthe emitted positrons collides nearly immediately with anavailable electron in tissues so that the so-called annihi-lation process takes place. The two annihilation photons,which emerge during this process, have very high depar-ting velocities and may leave the human body in all like-lihood along opposed directions. In this respect, if twodistinct photons are detected by two different sensor unitsof the measuring system within a predefined time win-

64 Y. Li et al.

dow, this pair of photons can be assumed to belong tothe same annihilation process and can be hence registe-red as a single coincidence event. Its true occurring lo-cation, where the positron-emitting radioisotope actuallylocates, lies somewhere on the straight line between thesetwo detecting sensor units. Thereafter, in the absence ofeffects such as attenuation and scattering, the measurednumber of coincidence events along the individual stra-ight Line-Of-Response (LOR) between two sensor unitsapproximates the straight line integral of the underlyingtracer substance distribution. Since such substance distri-butions normally reflect the metabolic functions in vivo,malfunctions which are characteristic for diverse functio-nal disorders could be diagnosed by observing the propervisualization of the non-invasive distribution.

Due to the physical nature of the annihilation pro-cess, the two emerging photons are actually emitted intoall spatial directions, i.e., there is no preferential direc-tion in the three-dimensional (3D) spatial domain. Cor-respondingly, 3D imaging techniques are intrinsically de-manded for PET, since the more coincidence events aremeasured by the measuring system of the PET scanner,the better reconstruction quality may be achieved. Tofulfill this requirement, the measuring systems of mostPET scanners nowadays have cylindrical multi-ring struc-tures whose cylindrical surfaces are subdivided into uni-form sensor block units in both transaxial and axial direc-tions (Bendriem and Townsend, 1998). For convenience,the axial direction is conventionally designated as the z-direction, whereas the transverse ring plane correspondsto the x-y-plane, see Fig. 1. In order to enhance thesystem sensitivity, not only the sensor units on the samerings but also the ones on the different rings can be inter-connected electrically. Thereby the originated large axialfield of view permits measurements of the so-called cross-plane coincidence events and enables the achievement ofbetter reconstruction quality. To distinguish it from fromthe obsolete two-dimensional (2D) measuring mode, suchan imaging technique in PET is henceforth designated as3D-PET.

On the other hand, the incorporation of additio-nal cross-plane events often results in an excessiveamount of projection data, which significantly com-plicates the reconstruction procedure. By using stan-dardized reconstruction algorithms such as analyticalFiltered-BackProjection (FBP) or the iterative Maximum-Likelihood-Expectation-Maximization (MLEM) method,clinically impractical reconstruction time durations be-come necessary (Moon, 1996). For this reason, recon-struction approaches based on the Fourier slice theorem,by means of which spectral values at particular frequencysampling points in the discrete object spectrum can be ob-tained, deserve a thorough investigation. Starting fromthose frequency samples, the original distribution functionin the spatial domain can be directly calculated by perfor-

ming the inverse Fourier transform. Because of the sim-plicity of such approaches, the reconstruction time can beconsiderably reduced. Since the available sampling pointsin the 3D frequency domain are naturally unevenly distri-buted, theoretically the time-consuming Discrete FourierTransform (DFT) should be applied to perform the inversetransform. In practice, an additional interpolating step forestimating the unknown spectral values on a predefinedCartesian grid is usually inserted to enable the utilizationof the Fast Fourier Transform (FFT) algorithm. Once theinterpolation is accomplished, the sought-after discrete di-stribution function can be calculated by performing the in-verse FFT. In this way, the reconstruction can be sped upthanks to the FFT’s speed advantage. Due to the usage ofthe interpolating step, such reconstruction approaches arereferred to as interpolation-based reconstruction methodsin this paper and will be closely addressed.

In the present paper, the principle of interpolation-based reconstruction approaches will be presented first.Apparently, their crucial parts are the interpolating stepsin the frequency domain which are always associated withinevitable inaccuracies (Thevenaz et al., 2000). In the 2Dmeasuring mode, the fact that the sampling points withknown spectral values are highly unevenly distributed le-ads to a very tricky problem. In the low frequency regions,the sample densities are still high enough to guaranteethe interpolation results, but in the high frequency regionsthe sample densities are simply too low to ensure achie-ving rational results. Compared with the 2D measuringmode, this problem still remains in 3D-PET but turns outto be less critical because there is a larger number of fre-quency samples available in the frequency domain due tothe additional cross-plane events. This is particularly be-neficial for minimizing the unavoidable artifacts introdu-ced by the interpolation procedure. Unfortunately, someanalysis reveals also the side-effect of this benefit (Li etal., 2005). The fluctuation of the sample density becomesalso much more unpredictable, which increases the com-plexity of a rational implementation. Based on this fact,two distinct realizations of the interpolation procedure arepursued. The first approach uses the straightforward li-near averaging method to estimate the unknown spectralvalues, whereas the second approach applies convolutionoperations to obtain the values at desired interpolation po-ints. Simulation results of these two approaches are pre-sented subsequently. Due to the achieved different perfor-mances, specific details and possible reasons are presentedand discussed at the end.

2. Principle of the interpolation-basedreconstruction approach

The fundamental concept of interpolation-based recon-struction methods in 3D-PET is given by the 3D Radontransform, which relates the measurable projection data to

Interpolation-based reconstruction methods for tomographic imaging . . . 65

Fig. 1. Schematic illustration of the sensor arrangement ofa PET scanner.

the underlying continuous tracer distribution via integralsalong straight lines (Kak and Slaney, 1988). With the 3Dtracer distribution defined as f(x, y, z), the mathematicaldefinition of the associated projection signal p(u, v, ϑ, ϕ)at a certain projection angle (ϑ, ϕ) can be formulated as

p(u, v, ϑ, ϕ) =∫ ∞

−∞f(x, y, z) dt, (1)

where the involved 3D coordinate transform is defined by⎡⎢⎣ x

y

z

⎤⎥⎦ =

⎡⎢⎣ cos ϕ cos ϑ − sin ϕ − cos ϕ sin ϑ

sin ϕ cos ϑ cos ϕ − sin ϕ sin ϑ

sin ϑ 0 cos ϑ

⎤⎥⎦

·

⎡⎢⎣ t

u

v

⎤⎥⎦ .

The spatial slope of an individual straight line is determi-ned by the azimuthal angle ϕ and the co-polar angle ϑwith respect to the transverse ring plane, while the para-meters u and v depict its displacement with respect to theorigin in the space domain.

In accordance with the definition (1), the originally2D Fourier slice theorem may be generalized to

P (jωu, jωv, ϑ, ϕ)=F (jωx, jωy, jωz)∣∣∣∣∣∣∣

ωx = −ωu sin ϕ − ωv cos ϕ sin ϑ

ωy = ωu cos ϕ − ωv sin ϕ sin ϑ

ωz = ωv cos ϑ

,

(2)

which implies that the 2D Fourier transform of the pa-rallel projections p(u, v, ϑ, ϕ) with respect to the varia-

bles u and v at a certain projection angle (ϑ, ϕ) corre-sponds to a central plane crossing the 3D Fourier trans-form of f(x, y, z) at exactly the same projection angles,see Fig. 2. Theoretically, with the help of the extended3D Fourier Slice Theorem in (2), the 3D object spectrumF (jωx, jωy, jωz) could be entirely recovered by acqu-iring 2D projection spectra at all feasible projection an-gles. Despite possible data redundancy, the desired tracerdistribution could then be figured out by performing theinverse Fourier transform (Li et al., 2005). Unfortuna-tely, this procedure is merely applicable for the continu-ous case. In practice, the number of available projectionangles as well as the number of projections at each pro-jection angle are strictly limited. In this note, the recon-structed 3D object spectrum is no longer of continuousnature, but it is merely composed of frequency samples.Due to the additional cross-plane coincidence events inthe 3D measuring mode, the number of obtainable frequ-ency samples in the frequency domain is larger than thatin the 2D measuring mode. For an interpolation-basedreconstruction approach, the exact arrangement of thesefrequency samples in the 3D frequency domain has to beascertained for the subsequent interpolating procedure.

3. Distribution of frequency samples in the3D frequency domain

With the measurable projection data defined as straightline-integrals, projections having the same inclination canbe combined together as sets of parallel line-integrals. Al-though the sensor system of a modern PET scanner hasactually a cylindrical multi-ring structure, the measuredcoincidence events can be subsequently arc-corrected andrearranged in such a way as if they were consecutivelycaptured by two simultaneously rotating, parallel sensorpanels at various projection angles, see Fig. 3. It is worthemphasizing here that despite the slope of the two projec-tion panels with the angle ϑ in the axial direction, the rota-

Fig. 2. Schematic illustration of the 3D Fourier slice theorem.

66 Y. Li et al.

tional axis of the rotational movement remains the z-axis.The individual projection beam which is determined bythe two associated block-detectors is conventionally desi-gnated as the Tube Of Response (TOR). Along with thismodeling scheme, a related fictitious projection plane canbe introduced at each projection angle, which is parallel tothe two-sensor panels and additionally passes through theorigin of the spatial domain. If the inclination of a pro-jection plane is defined as the direction of its normal, theinclination of a projection plane is then consistent with theassociated projection angle.

Fig. 3. Modeling scheme of a cylindrical sensor system as twosimultaneously rotating sensor panels.

According to this definition, in the case of the 2Dmeasuring mode the projection plane has definitely no in-clination with respect to the z-axis, whereas in the caseof the 3D measuring mode the projection plane correla-ting with cross-plane coincidence events may be inclinedwith respect to the z-axis. In the latter case, the associatedco-polar angle ϑ has a nonzero value. Due to the geome-trical modeling scheme of the PET scanner, the measura-ble parallel projection samples in a projection plane areintrinsically equidistantly arranged along the z-axis. Onthe contrary, along the perpendicular transverse directionan arc-correction step has to be performed to ensure theequidistance between adjacent projection samples, whichis normally already integrated into the projection acquisi-tion procedure of hardware and hence causes no additionalcomputational overhead.

To this end, the projection samples in an arbitraryprojection plane are indeed ordered on a Cartesian gird.Thereupon, the 2D fast Fourier transform can be utilizedto compute the discrete projection spectrum economically,while the frequency samples are arranged on a Cartesiangrid as well. Considering the multi-ring block-detectorstructure of a PET scanner, the spatial distribution of fre-quency samples in the reconstructed 3D object spectrumcan be indeed constituted by consecutively gathering thefrequency samples with distinct inclinations, as if the 2D

Cartesian lattices of frequency samples were revolved inthe 3D frequency domain. In Fig. 4, such a constitutionprocedure is schematically illustrated. For the 2D measu-ring mode, the revolving Cartesian lattice in the frequencydomain has no slope with respect to the ωz-axis so thatthe resulted frequency samples are bounded within a cy-lindrical scope, see Fig. 4. In the 3D measuring mode,however, apart from the projections along the transversedirection, the revolving 2D Cartesian lattices correlatingwith cross-plane events definitely tilt to the ωz-axis. Inthis case, only part of the object spectrum can be recon-structed. The confined region in the 3D frequency domain,in which no frequency samples are obtainable, resemblesa head-to-head adhered truncated double-cone, see Fig. 4.The final distribution of frequency samples in the recon-structed 3D object spectrum is hence the accumulation ofall acquirable frequency samples, no matter whether fromthe inclined or the noninclined Cartesian lattices.

The resulting frequency samples are obviously irre-gularly distributed in the frequency domain. For the sakeof rapid reconstruction, interpolating steps have to be con-ducted to enable the usability of the FFT algorithm. The-oretically, a true 3D interpolating procedure in the 3D fre-quency domain seems to be inevitable. But by observingit closely, it turns out that only consecutive 2D interpo-lations in a set of transverse planes are really required,because the obtainable frequency samples fall exactly inthese equidistantly displaced planes which are perpendi-cular to the ωz-axis (Li et al., 2006). This characteristicsimplifies the computational complexity of the interpola-ting procedure significantly and speeds up the reconstruc-tion again. Generally speaking, incorporating cross-planemeasurements leads to an increased density of frequencysamples in all transverse planes, which induces to scaledown the interpolation error effectively. However, depen-ding on the axial location of the transverse plane, the den-sity increase fluctuates considerably in various regions of

(a) (b)

Fig. 4. Schematic illustration of the frequency sample structurein the reconstructed 3D object spectrum: (a) Fouriertransformed projection plane with no inclination , (b) Fo-urier transformed projection plane with inclination.

Interpolation-based reconstruction methods for tomographic imaging . . . 67

transverse planes. For a transverse plane lying closely tothe origin of the 3D frequency domain, the additional fre-quency samples concentrate predominantly in the centralregion, where the density is already high enough for ratio-nal interpolation, see Fig. 5. Otherwise, for a transverseplane lying comparatively far from the spectral origin, thepredominant concentration of additional frequency sam-ples shifts outwards, see Fig. 5. From the experience ga-ined during the simulation, the density increase of frequ-ency samples in the border area is actually more beneficialfor improving the interpolation results than the densityincrease in the central region of an individual transverseplane, because the density of only obtainable frequencysamples in the 2D measuring mode is too low for a ratio-nal interpolation in the border region.

(a) (b)

Fig. 5. Schematic illustration of frequency sample distributionsin distinct transverse planes in the reconstructed 3D ob-ject spectrum: (a) distribution of frequency samples ina transverse plane lying close to the spectral origin, (b)distribution of frequency samples in a transverse planelying far from the spectral origin.

4. Consecutive 2D interpolation proceduresin the 3D frequency domain

For the purpose of speeding up the reconstruction pro-cedure, interpolation operations have to be performed ineach transverse plane of the reconstructed 3D object spec-trum to enable the usability of the FFT algorithm. Thisprocedure is illustrated schematically in Fig. 6. Althoughthe intrinsic error associated with the interpolating proce-dure cannot be completely avoided, distinct interpolatingschemes do provide varying performances. In the presentpaper, two distinct interpolating approaches are pursuedand analyzed. The first approach presents a straightfor-ward solution, for which the unknown spectral values areestimated by the weighted linear averaging of the spec-tral values of the relevant neighboring frequency samples.On the contrary, the second approach uses the 2D convo-lution operations in each transverse plane to acquire theunknown values at the interpolation sites. Due to the 2Dconvolution operation in the frequency domain, an addi-

Fig. 6. Illustration of the interpolating procedure in alltransverse planes.

tional division step in the spatial domain has to be accom-plished subsequently to acquire the correct results.

4.1. Inverse distance weighting interpolation. TheInverse Distance Weighting (IDW) interpolation method,which was originally utilized in geoinformatics, can beeasily adopted for the case of interpolating from 2D une-venly distributed frequency samples onto frequency sam-ples arranged on a Cartesian grid in the transverse plane.The IDW method is based on the assumption that the fre-quency sample which is to be interpolated should be moreinfluenced by the closely located than by the remotely lo-cated neighboring samples (Fisher and Embleton, 1987).In this respect, the value at the desired location is then aweighted linear average of the neighboring values, whe-reas the associated weight decreases if the distance be-tween the interpolation site and the scattered neighbor in-creases. The corresponding mathematical definition ofIDW interpolation can be therefore formulated as

Fregular =N∑

i=1

wi · fi, with wi =h−p

iN∑

j=1

h−pj

, (3)

where fi represents the spectral value of the i-th scatteredfrequency sample and wi the associated weight. In the for-mula for calculating individual wi, hj depicts the distancebetween the j-th neighboring sample and the interpolationsite, while N is the total number of all existing neighborswithin a pre-specified neighborhood. In Fig. 7, the selec-tion of a relevant neighborhood and the related neighbo-ring samples are exemplified, where the pre-defined rele-vant neighborhood is illustrated as a gray-shaded square.

In fact, the larger the relevant neighborhood is selec-ted, the larger the number of scattered frequency samplesmay be incorporated into the interpolating procedure, andin turn the more accurate interpolation results could beexpected. However, a large number of sample neighborsmeans also a higher computational expense, which cor-respondingly slows down the reconstruction speed. The-refore, a trade-off between the reconstruction time andthe interpolation accuracy has to be made. In the pre-

68 Y. Li et al.

sent paper, two different solutions are proposed in thisrespect. On the one hand, the neighborhood’s extent isuniformly defined independently of the related frequencysample densities. The drawback of this easily realizableapproach is that the number of incorporated frequencysamples is permanently variable. Particularly in the bor-der region of the traverse plane, the number of samplescould be very low because neighboring samples for a spe-cified interpolation site are often barely available. On theother hand, instead of a uniform extent of neighborhoodan equal number of neighboring samples can be used asthe criterion to determine the unequally bounded neigh-borhood. In the border region with a low sample den-sity, the seeking area has to be expanded gradually untilenough neighboring samples are encountered. Compa-red with the former case, this seeking procedure is muchmore time consuming. Fortunately, the exact locations ofunevenly distributed frequency samples in each transverseplane can be determined in advance, because they are onlyaffected by the geometry of the PET scanner. Consequ-ently, the unevenly distributed frequency samples in eachplane can be sorted according to their locations by usingan efficient data structure such as the balanced binary tree.Based on the specified neighboring samples, the associa-ted weights wi can be directly calculated according to (3).Since the locations of the frequency samples as well astheir associated weights can be calculated without know-ledge about their spectral values, the seeking and the suc-cessive calculating procedure can be completed prior tothe actual reconstruction. In this way, the reconstructionduration can be kept short.

4.2. Gridding method. The second approach to es-timate the spectral values on a specified 3D Cartesianlattice is based on separate 2D convolution operationsin each transverse plane. This approach was conventio-nally used in the 2D reconstruction scenario and embra-ced by the term “gridding method” (Schomberg and Tim-mer, 1995). Since the irregularly distributed frequency

Fig. 7. Selection of the relevant neighborhood for a frequ-ency node in the transverse plane to be interpolated.

samples in the 3D frequency domain merely locate in cer-tain transverse planes, the gridding method known fromthe 2D scenario can be easily extended to the 3D recon-struction scenario, whose functional principle is outlinedin the following way: Consider a discrete 3D object spec-trum Firregular, whose frequency samples are irregularlydistributed in a set of equidistantly displaced transverseplanes in the frequency domain. The equivalent discretespectrum FCartesian, but with sampling points arranged ona 3D Cartesian grid, can be obtained by consecutively per-forming 2D convolutions with a predefined 2D windowfunction Wwindow in all the transverse planes:

{FCartesian}layer = {Firregular}layer ∗ Wwindow, (4)

where ∗ denotes the 2D convolution operation and thesubscript { · }layer indicates that the convolution operationmerely takes place layer-wise. Due to the consecutive 2Dconvolutions, the obtained FCartesian actually is not thesampled object spectrum of the desired distribution func-tion fCartesian, but rather an intermediate outcome. Inorder to compensate the effect of convolution in the fre-quency domain, an additional division step in the spatialdomain has to be introduced in the inversion step, prior toa straightforward 3D inverse Fourier transform. With F−1

xy

defined as the 2D inverse Fourier transform in each trans-verse plane with respect to the variables x and y, and F−1

z

defined as the 1D inverse Fourier transform with respectto the variable z, respectively, the correct inversion pro-cedure to acquire the desired fCartesian can be ultimatelyexpressed by the formula

fCartesian = F−1z

{F−1

xy {FCartesian}layer

wwindow

}, (5)

where wwindow denotes the 2D inverse Fourier transformof the 2D convolution window Wwindow used in the fre-quency domain.

Despite the seemingly uncomplicated expressionin (5), two significant aspects of the gridding method haveto be carefully considered. The first one is the selection ofthe 2D window function Wwindow. Theoretically, the opti-mal convolution function in the frequency domain shouldbe of infinite extent, which is, however, impractical withrespect to the computational cost. For this reason, the co-nvolution function has to be truncated and windowed, sothat a trade-off between the reduced computational effortand the accompanied accuracy deficit has to be taken intoaccount. Several authors have been engaged in this aspectfor finding an optimally appropriate solution and sugge-sted the Modified Kaiser-Bessel (MKB) window functionas the most promising candidate (Schomberg and Tim-mer, 1995; Jackson et al., 1991; Matej and Lewitt, 2001).Although the conducted works deal mainly with the 2Dreconstruction scenario, the proposed MKB window func-tion can be adopted here without significant modification,

Interpolation-based reconstruction methods for tomographic imaging . . . 69

since for 3D-PET merely successive 2D gridding proce-dures are required. With properly selected parameters, thebell-shaped MKB function and its inverse Fourier trans-form may be free from discontinuities. Actually, the 2Dconvolution window used is the multiplicative product oftwo 1D MKB functions.

The second important aspect of the gridding methodis the correct discretization of the underlying 2D continu-ous convolution depicted in (4), since in Firregular onlyfrequency samples are available. In this case, the continu-ous integral operation must be approximated and replacedby a double summation, where each involved frequencysample receives a weighting factor. Generally speaking,the related weighting factor for a certain frequency sam-ple is inversely proportional to the local density of its ne-ighboring frequency samples. However, since the frequ-ency samples are highly nonuniformly distributed in eachtransverse plane, accurate calculation of an individual we-ighting factor poses a computational challenge. In this re-spect, a new modeling scheme for the assigned weightingfactors of scattered frequency samples are proposed, na-mely, accurately modeled as the normalized area of a po-lygon, which belongs to the specified frequency sample inthe transverse plane. According to this modeling scheme,the higher the local density frequency sample, the smal-ler the assigned polygon and correspondingly the smallerthe related weighting factor. In computational geometry,the subdivision induced by this scheme is called the “Vo-ronoi diagram.” Compared with the modeling schemesproposed in the literature so far, the resulting weightingfactors are also much more precise in terms of discretiza-tion. In contrast to some proposed time-consuming itera-tive approaches, this modeling scheme is straightforwardand time efficient.

4.3. Computational construction of the Voronoi dia-gram. The Voronoi diagram is a versatile geometricstructure. For a set of distinct frequency samples in atransverse plane, the associated Voronoi diagram is defi-ned as the subdivision of the plane into various polygons,one for each sample, with the property that an arbitrarypoint lies within a specified polygon if and only if the di-stance from this point to the sample of the associated po-lygon is shorter than all other distances between this pointand the remaining samples (De Berg et al., 1997). Suchpolygons are often called Voronoi cells, whose normali-zed area contents are of interest for the gridding method.In Fig. 8, an exemplary Voronoi diagram for some givenfrequency samples is schematically illustrated. Due to thefinite extent of Wwindow, a bounding neighborhood foreach interpolation location must be selected first, withinwhich the Voronoi diagram is to be generated.

For computing the Voronoi diagram, Fortune’s algo-rithm commonly known and named after its inventor, ispreferred and implemented in this paper (Fortune, 1987).

Fig. 8. Schematic illustration of an exemplary Voronoi diagram.

The strategy of this efficient, event-driven algorithm is tosweep a straight line over the concerned transverse plane,during which action two different sorts of events are to becorrespondingly handled, the so-called site-event and thearc-event. These two different events can be distinguishedsubject to the intersection of the already constructed partdiagram structure with the sweeping line. In case of enco-untering a site-event, a new Voronoi cell with associatedpending edges has to be added to the existing structure,whereas for encountering an arc-event, the pending edgesof an open Voronoi cell are terminated by attaching inter-section vertices. The affected Voronoi cell is thus closed.However, after all detected events are handled, there arestill some open Voronoi cells. To close them, a boundingbox which is equal to the predefined relevant neighbor-hood has to be added to the existing structure. In thismanner the Voronoi diagram, which is only composed ofclose Voronoi cells, can be constructed step by step tillcompleteness. Once the complete Voronoi diagram is ge-nerated, the areal content of each single Voronoi cell canbe easily calculated based on its determined vertices andaccordingly the weighting factor of the associated frequ-ency sample can be figured out as well.

4.4. Remaining difficulty. Despite this clear construc-tion scheme, there is a remaining difficulty in constructingthe Voronoi diagram. Since the encountered events do nothave to be handled immediately, they need to be sorted ina queue temporarily subject to their associated coordina-tes. The coordinate of a site-event is simply the coordinateof the associated frequency sample itself, whereas the co-ordinate of an arc-event is actually the coordinate of theassociated circle’s center. Since the associated circle of anarc-event is normally determined by three points duringthe construction, numerical calculations based on their co-ordinates are necessary. In this case, numerical inaccura-cies cannot be completely excluded. Particularly, sincethe frequency samples in the transverse plane may be so-metimes very densely distributed, numerical inaccuracies

70 Y. Li et al.

turn out to be a really serious problem so that in the worstcase the affected events may be sorted into a queue witha wrong order. Due to the progressive nature of Fortune’salgorithm, even a simply inverted order may lead to a to-tally false outcome for the Voronoi diagram. Despite greatefforts such as iteratively solving the underlying quadraticequation, this problem cannot be optimally and comple-tely avoided so that within the scope of the present paperan additional verifying step has to be accomplished afterthe constructing procedure. If degeneracy is spotted, theaffecting frequency sample has to be regrouped with a ne-arby neighboring frequency sample and the constructingprocedure has to to be repeated under the modified con-dition. Once the Voronoi diagram is generated, the regro-uped frequency samples receive a weighting factor whichis equal to the normalized areal content of the Voronoicell divided by the number of regrouped samples. Thismeans effectively that the erroneously calculated weigh-ting factor has to be accepted to by-pass the numericalproblem. However, as the degeneracy occurs seldom andthe number of neighboring samples is relatively large, theinfluence of such approximations could be reasonably ne-glected.

5. Simulation results

For the purpose of evaluation, the two interpolation-basedreconstruction methods were implemented using the pro-gramming language C++. Although the two proposed me-thods are generally applicable to truly measured projec-tion data, in the scope of this paper only simulation stu-dies on the basis of simulated projection data are perfor-med. This is due to the fact that the simulated projec-tion data are free from the acquisition noise, which is ty-pical for the truly measured projection data. In the case ofusing simulated projection data, the reconstruction resultsare merely influenced by the utilized reconstruction me-thods and reasonable evaluation with respect to the recon-struction quality may be achieved, whereas in the case ofusing truly measured projection data, the cause of possi-ble reconstruction deviation cannot be clearly ascertained.For this reason, using noise-free projection data is signifi-cant for a correct evaluation of the discussed interpolation-based reconstruction methods.

Consequently, a 3D phantom object consisting layer-wise of the so-called 2D Shepp-Logan phantoms is defi-ned as the original distribution function, which is of the di-mension 128 × 128 × 32, i.e., 32 layers in the z-directionand 128 × 128 pixels on each transverse layer. The 2DShepp-Logan phantom contains several ellipses with dif-ferent sizes and absorption properties to resemble the fe-atures in the human brain. In order to imitate the 3D brainstructure, the extents of the 2D Shepp-Logan phantoms onvarious layers are intentionally differently specified andthe change of extents between layers basically occurs gra-

Fig. 9. 3D phantom object consisting layer-wise of 2DShepp-Logan phantoms.

dually, e.g., the 2D phantoms on the topmost and lower-most layers are much smaller than that on the central layer,see Fig. 9. Based on this defined 3D phantom, the corre-sponding projection data can be then computed by meansof the accurately modeled system matrix, which is prima-rily based on the geometry of the individual TOR and self-implemented before (Li et al., 2006).

By using interpolation-based reconstruction me-thods, i.e., the direct approach (IDW) and the convolution-based indirect approach (gridding method), distinct recon-struction results of the original 3D phantom can be achie-ved. In Fig. 10, the layer-wise depiction of the reconstruc-ted 3D phantom object by using the IDW method is gi-ven. The observation of this reconstruction result confirmsclearly the correctness of the implemented IDW method,in which 2D Shepp-Logan phantoms of various extentsin individual transverse layers are properly reconstructed.This conclusion is also applicable to the reconstructionresults achieved by using the gridding method. Its com-plete layer-wise depiction is omitted here, because, due tothe restricted depiction extents, differences between theresults of the two discussed methods can be barely di-stinguished optically. Instead, for the purpose of a sub-jective evaluation, the middle layers of differently recon-structed 3D phantoms as well as the original Shepp-Loganphantom of the same extent are exemplarily selected anddepicted side by side in Fig. 11. Although both of theinterpolation-based reconstruction methods yield seemin-gly correct results, optical observation shows anyhow thatthe reconstruction result achieved by using the IDW me-thod exhibits more distortions than the result achieved byusing the gridding method. On the contrary, the differencebetween the reconstruction results based on the 2D filteredbackprojection and the gridding method is minor. Never-theless, one can say that the contrast in the reconstructedphantom by using the gridding method is slightly higherso that the contour is correspondingly clearer.

As for the simulation time, less than 10 minutes on aPC are really needed to complete the reconstruction for a

Interpolation-based reconstruction methods for tomographic imaging . . . 71

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72 Y. Li et al.

Fig. 12. RMSD values of various reconstruction results.

given dimension, which may appear at first glance not tobe very efficient. However, most of the simulation timeactually has to be spent to load the pre-calculated we-ighting factors from the hard disc into the main memory.In this respect, there is no significant time difference be-tween the two reconstruction methods, since no matterhow the weighting factors are calculated, they are bothpre-calculated and stored on the hard disc as files. Apartfrom this reading time, the real reconstruction time is infact much shorter, e.g., a couple of minutes. Neverthe-less, further analysis regarding the data arrangement andefficient data input/output is inevitable.

In order to quantify the differences between thediverse reconstructions, the values of the Root-Mean-Square-Deviation (RMSD) for individual reconstructionresults can be calculated, with the mathematical definitiongiven by

RMSD =

√√√√ 1MN

M∑i=1

N∑j=1

(fori [i, j] − frec [i, j])2 ,

(6)where fori [i, j] and frec [i, j] denote respectively the ori-ginal and reconstructed pixel values in the i-th row andj-th column of the same layer with M rows and N co-lumns. In Fig. 12, the calculated RMSD values dependingon layer numbers are depicted for various reconstructionresults. Besides IDW and the gridding method, the resultsof the 2D filtered-backprojection serving as a referenceare also evaluated. For the IDW method, both the uniform

(a) (b)

(c) (d)

Fig. 11. Middle layers in the reconstructed 3D phantoms byusing the Fourier-based inversion methods and the 2Dfiltered backprojection: (a) Shepp-Logan phantom, (b)2D filtered backprojection , (c) inverse distance weigh-ting , (d) gridding method.

neighborhood and the uniform number of neighbors aretaken into account. The diagram in Fig. 12 reveals thatboth of the IDW implementations provide equivalent out-comes, which are, however, worse than that of the FBP

Interpolation-based reconstruction methods for tomographic imaging . . . 73

method, whereas the result achieved by using the grid-ding method has the best reconstruction quality among allthe methods. The reason for the unsatisfying result of theIDW method traces back to the fact that the straightfor-ward interpolation in the frequency domain corresponds toan equivalent low-pass filtering in the space domain, thro-ugh which artefacts are introduced into the reconstructionresults. On the contrary, the gridding method uses convo-lution instead of direct interpolation and therefore balan-ces the effect of low-pass filtering to some degree. Whilefor the IDW method the weighting factors are exclusivelydetermined by distances between frequency samples andthe interpolation location, the gridding method has a muchhigher degree of freedom due to the versatile choices ofconvolution functions. In particular, by choosing the para-meters of the MKB function properly, a good compromisethat both the MKB function in the frequency domain andits inverse transform in the space domain decay relativelyrapidly and smoothly can be reached so that the low-passeffect can be compensated more effectively.

6. Conclusions

In this work interpolation-based reconstruction methodsfor 3D-PET were proposed and analyzed. Since merelyconsecutive 2D interpolation operations in the 3D frequ-ency domain are necessary, reconstruction procedures canbe sped up significantly. However, simulation studies re-veal that classically implemented interpolation does notprovide satisfying results, while interpolation performedbased upon convolution operations in the frequency do-main leads indeed to a better reconstruction quality. Al-though additional compensating steps in the spatial do-main are demanded, the required weighting factors can becalculated in advance and fetched during the convolutionprocedure so that the thereby caused computational over-head may be even neglected.

Acknowledgments

This work was supported by the German Research Foun-dation (DFG) under the grant KU 678/10–2.

ReferencesBeutel J., Kundel H. L. and Van Metter R. L. (2000). Handbook

of Medical Imaging, SPIE Press, Bellingham, Washington.

Bendriem B. and Townsend D. W. (1998). The Theory and Prac-tice of 3D PET, Kluwer Academic Publishers, Dordrecht.

De Berg M., Van Kreveld M., Overmars M. and Schwarzkopf O.(1997). Computational Geometry, Springer-Verlag, Berlin.

Fisher N. I. and Embleton B.J. (1987). Statistical Analysis ofSpherical Data, Cambridge University Press, Cambridge.

Fortune S. (1987). A SWEEPLINE algorithm for Voronoi dia-grams, Algorithmica 2(2): 153–174.

Jackson J. I., Meyer C. H., Nishimura D.G. and Macovski A.(1991). Selection of a convolution function for Fourierinversion using gridding, IEEE Transactions on MedicalImaging 10(3): 473–478.

Kak A. C. and Slaney M. (1988). Principles of ComputerizedTomographic Imaging, New York, IEEE Press.

Li Y., Kummert A., Boschen F. and H. Herzog (2005). Investiga-tion on projection signals in 3D PET systems, Proceedingsof the 12th International Conference on Biomedical Engi-neering, Singapore.

Li Y., Kummert A., Boschen F. and H. Herzog (2005). Spectralproperties of projection signals in 3-D tomography, Proce-edings of the 16th Triennial World Congress of the Inter-national Federation of Automatic Control, Prague, CzechRepublic.

Li Y., Kummert A. and Herzog H. (2006). Direct Fourier methodin 3D PET using accurately determined frequency sampledistribution, Proceedings of the 28th Annual InternationalConference of the IEEE Engineering in Medicine and Bio-logy Society, New York, USA.

Li Y., Kummert A., Li H. and Herzog H. (2006). Evaluation ofthe direct Fourier method for 3D-PET in the case of accu-rately determined projection data, Proceedings of the 11thIASTED International Conference on Signal and ImageProcessing, Honolulu, USA.

Matej S. and Lewitt R. M. (2001). 3D-FRP: Direct Fourier re-construction with Fourier reprojection for fully 3-D PET,IEEE Transactions on Medical Imaging 48(4): 1378–1385.

Moon T.K. (1996). The expectation-maximization algorithm,IEEE Signal Processing Magazine 13(6): 47–60.

Schomberg H. and Timmer J. (1995). The gridding methodfor image reconstruction by Fourier transformation, IEEETransactions on Medical Imaging 14(3): 596–607.

Thevenaz P., Blu T. and Unser M. (2000). Interpolation revisited,IEEE Transactions on Medical Imaging 19(7): 739–758.