STATISTICAL ANALYSIS OF TOMOGRAPHIC RECONSTRUCTION … · Submitted to Image Anal Stereol, 17 pages Original Research Paper STATISTICAL ANALYSIS OF TOMOGRAPHIC RECONSTRUCTION AL-GORITHMS

Submitted to Image Anal Stereol, 17 pagesOriginal Research Paper

STATISTICAL ANALYSIS OF TOMOGRAPHIC RECONSTRUCTION AL-GORITHMS BY MORPHOLOGICAL IMAGE CHARACTERISTICS

SEBASTIAN L UCK1, ANDREAS KUPSCH2, AXEL LANGE2, MANFRED P. HENTSCHEL2 ANDVOLKER SCHMIDT1

1Institute of Stochastics, Ulm University, D-89069 Ulm, Germany,2BAM Federal Institute for MaterialsResearch and Testing, D-12200 Berlin, Germanye-mail: [email protected](Submitted)

ABSTRACT

We suggest a procedure for quantitative quality control of tomographic reconstruction algorithms. Our task-oriented evaluation focuses on the correct reproduction ofphase boundary length and has thus a clear im-plication for morphological image analysis of tomographicdata. Indirectly the method monitors accuratereproduction of a variety of locally defined critical image features within tomograms such as interface posi-tions and microstructures, debonding, cracks and pores. Tomographic errors of such local nature are neglectedif only global integral characteristics such as mean squared deviation are considered for the evaluation of analgorithm. The significance of differences in reconstruction quality between algorithms is assessed using asample of independent random scenes to be reconstructed. These are generated by a Boolean model and thusexhibit a substantial stochastic variability with respectto image morphology. It is demonstrated that phaseboundaries in standard reconstructions by filtered backprojection exhibit substantial errors. In the setting ofour simulations, these could be significantly reduced by theuse of the innovative reconstruction algorithmDIRECTT.

Keywords: Tomography, Reconstruction Algorithm, Morphological Image Analysis, Phase Boundary, Metro-logy, Non-Destructive Testing.

INTRODUCTION

The principle of tomographic reconstruction of avolume from lower-dimensional projections as e.g. ap-plied in X-ray or electron tomography was mathe-matically discovered by Radon (1917). For a compre-hensive introduction and important aspects of applica-tions see e.g. Banhart (2007), Buzug (2008), and Frank(2005). Tomographic reconstructions can be computedby a variety of different techniques such as filteredbackprojection (FBP) (Feldkampet al., 1984; Kakand Slaney, 2001), algebraic reconstruction techniques(ART) (Carazoet al., 2005; Gilbert, 1972), geomet-ric tomography (Gardner, 1995) or discrete tomogra-phy (Batenburg, 2005; Herman and Kuba, 2007). Thecomparative evaluation of these algorithms is naturallydependent on the choice of quality criteria. These aremathematically formulated by a figure of merit (FOM)measuring the deviation of a phantom data set from itsreconstruction, which is computed from simulated pro-jections of the phantoms. Since ranking of the recon-struction quality provided by different algorithms isFOM-dependent (Herman and Odhner, 1991), FOMsneed to be chosen in a task-oriented way (Hanson,1990). That means, the FOM needs to detect differ-ences in reconstruction quality which are relevant to

the further analysis of the tomograms in a specific ap-plication e.g. from medicine or materials science. Typi-cal FOMs for medical applications are based on curvesof the receiver operator characteristics. They allow tostudy the detectability of different tissues and thus thereliability of diagnostics (Hanson, 1990; Herman andYeung, 1989). There is a large variety of FOMs fo-cusing on the correct reproduction of gray values. Themost common representative of gray value orientedFOMs is the mean squared deviation

d =

√

∑j(xrec

j −xphanj )2

/

√

∑j(xphan

j )2, (1.1)

which averages over the gray value differencesxrecj −

xphanj between all pixelsj in the reconstruction and the

phantom (Gilbert, 1972; Hansiset al., 2008). Alter-natively, mean absolute differences of gray values aswell as deviations of gray value means and variancesin the reconstruction from the respective values of thephantom have been considered (Sorzanoet al., 2001).Moreover, phase-specific mean gray values have beenused to assess the detectability of different componentswithin a material (Sorzanoet al., 2001).Especially if -as in real experiments- the density to re-construct is unknown, reconstruction residuals can be a

1

L UCK S et al.: Statistical analysis of tomographic reconstruction algorithms

valuable source of information for assessing the qual-ity of a tomogram (Langeet al., 2008). These residualsare obtained by subtracting a simulated projection ofthe reconstruction from the measured projection for allrotation angles.All approaches described above focus on global accu-racy of the reconstruction, whereas evaluation resultscan hardly be interpreted with respect to the preser-vation of locally defined image characteristics such aslittle cracks in the material or the exact shape of phaseboundaries. Correct reconstruction of locally definedcharacteristics is however crucial for unbiased compu-tation of morphological image characteristics such asconnectivity, boundary length or surface area. Quan-titative information on these characteristics is a valu-able source of information in a wide range of appli-cations such as metrology (Neuschaefer-Rubeet al.,2008), pathology (Mattfeldtet al., 2007), environmen-tal health (Stoegeret al., 2006) or design of materi-als (Frostet al., 2006). We therefore suggest to di-rectly incorporate measurements of morphological im-age characteristics into the FOMs used to evaluate to-mographic reconstruction algorithms. Apart from mea-surements of phase boundary length and surface area(see e.g. Parket al., 2000) many other morpholo-gical characteristics such as connectivity (Ohser andSchladitz, 2008) and fractal dimension (Baumannetal., 1993) are sensitive to the shape of phase boun-daries. We nevertheless decided to assess local recon-struction quality along phase boundaries by measur-ing deviations in boundary length between the original2D phantom images and their tomographic reconstruc-tions. Other approaches, which have been suggested tostudy reconstruction quality with focus on the vicin-ity of phase boundaries, are based on weighted avera-ging of gray value deviations between phantom and re-construction (Sorzanoet al., 2001), where weighting iswith respect to distance from phase boundaries. In con-trast to this FOM our method provides a direct assess-ment of reconstruction quality in terms of quantitativemorphological image analysis. Previous methods to lo-cally evaluate reconstruction quality consider averagegray value deviations within certain regions of inter-est (Furuieet al., 1994) and thus have the additionaldisadvantage that these regions need to be specifiedapriori .The two principle questions arising in comparison ofreconstruction algorithms ask for the relevance andfor the significance of differences in reconstructionquality, respectively. Relevant differences can possi-bly be detected by computation of an appropriatelychosen FOM for a single phantom such as the well-known section of a head introduced in Shepp and Lo-gan (1974). The use of such deterministic input offersthe advantage that specific features which typically

present challenges for correct reconstruction (e.g. cer-tain gray value gradients) can be incorporated. How-ever, insights with respect to the statistical significanceof differences in reconstruction quality can only begained if instead of a small number of determinis-tic phantoms a sufficiently large sample of randomphantom data is investigated (Hanson, 1990; Herman,2009; Herman and Yeung, 1989; Matejet al., 1994and 1996). This way statistical tests can be applied tocompare reconstruction errors caused by different al-gorithms. Many types of artifacts in tomographic re-constructions such as smearing or stripes result fromthe relative positions of single objects within the sceneto be reconstructed (Hanson, 1990). Thus, a sample ofrandom phantoms resembling the experimentally oc-curring range of scenes ensures that statistically signif-icant errors of this type are taken into account, whereasthey may not even occur in deterministic phantomdata. To the best of our knowledge in previous stud-ies samples of phantoms have only been generatedunder moderate randomization with respect to objectpositioning. In particular, objects were clearly sepa-rated from each other (Hanson, 1990). For studies dis-cussing detectability of tumors in biological tissue po-tential tumor locations were even fixeda priori andrandom effects were limited to Bernoulli experimentsmarking the sites as occupied by a tumor or by reg-ular tissue (Hanson, 1990; Herman 1989). Comparedto these images the phantoms considered in our studyexhibit a substantially higher morphological variabil-ity. They are realizations of a model from stochasticgeometry, namely a 2D Boolean model consisting ofoverlapping discs (Molchanov, 1997; Schneider andWeil, 2008), which are placed at randomly chosen lo-cations within the image (for details see Section Phan-tom Data). Thus, our phantom data reflect propertiesof composite or porous materials with an irregular spa-tial structure, which nevertheless exhibit spatial homo-geneity in the sense of stochastic geometry. Repeatedsampling from the Boolean model sets us in a posi-tion to compare reconstruction errors of different al-gorithms by two-sample-goodness-of-fit tests. In thisway, differences in reconstruction quality of two algo-rithms can be assessed on a statistically sound basis.The approach we suggest occurs particularly naturalfor the selection of tomographic reconstruction algo-rithms in applications aiming at the quantitative anal-ysis of complex materials. For a specific experimentalsetting, the methodology suggested in this study canbe adapted to phantom data sets resembling structuralproperties of the experimentally investigated material.Additionally, artifacts related to the specific imagingtechnique or even to a specific experimental instrumentneed to be taken into account. Simulations of projec-tions would thus e.g. incorporate noise, beam harden-ing, limited rotation or alignment problems (Carazoet

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Image Anal Stereol ?? (Please use\volume):1-17

al., 2005; Haibel 2008).In the following the suggested methodology will be ap-plied to investigate the performance of standard FBPalgorithms in comparison to the innovative reconstruc-tion technique DIRECTT (Direct Iterative Reconstruc-tion of Computed Tomography Trajectories) (Langeetal., 2008). The phantom data will be two-dimensional.In many applications such as electron tomography,projections are acquired in parallel beam geometry. Asa consequence, tomograms of 3D volumes are a stackof 2D reconstructions from 1D projection data. Thus,in parallel beam geometry our results on 2D datasetsare also relevant for the 3D case. The exact conse-quences for 3D morphological image analysis remainan interesting subject for future applications of ourmethodology.We will see that the applied FBP algorithms alter thestructure of phase boundaries in such a way that boun-dary length measurements are substantially affectedwhereas the DIRECTT reconstructions preserve boun-dary structures in a much better way. In a first stepwe will demonstrate these effects for projections ofphantoms without noise (referred to as ‘ideal projec-tions’ below). Afterwards we will show that the supe-riority of the DIRECTT reconstruction remains validunder the addition of simulated noise to the projectiondata.The paper is organized as follows. After discussingphantom generation and simulation of projections wegive an introduction to the investigated reconstructiontechniques in order to illustrate their principle ideas toa non-expert reader. Then we briefly discuss the es-timation techniques for measuring boundary length,leaving additional details for the appendix. Further-more, we introduce the FOMs serving as basis for theevaluation and the statistical tests we applied. We con-clude by the presentation of the results and their dis-cussion.

METHODS

PHANTOM DATA

The phantoms projected and reconstructed forthis study were discretized versions of realizations ofBoolean models in 2D, which were sampled on an ob-servation windowW = [0,500]2 (Fig. 1(a)). Booleanmodels are a class of random closed sets whose con-struction is based on a homogeneous Poisson point pat-tern{Sn}n≥1. In our specific setting we used a Booleanmodel which is defined as the unionB=

⋃∞n=1B(Sn, r)

of circlesB(Sn, r) of radiusr = 10 centered atSn. Theintensity of the homogeneous Poisson process on IR2

determining the random locations{Sn}n≥1 was chosen

such that the mean number of points inW = [0,500]2

was 1200. Since reconstructions are pixel images,throughout our study we investigated discretized reali-zations of this Boolean model on a grid of 500×500pixels. The gray value of each pixel was chosen pro-portional to its area fraction covered by the realizationof the Boolean model. Notice that gray values are cho-sen independently of the number of circles coveringa location. All gray values were rounded to integersand the maximum gray value was set to 255. In or-der to allow for statistical analysis, 100 independentrealizationsb1, . . . ,b100 of B were generated and dis-cretized.

SIMULATION OF IDEAL AND NOISYPROJECTIONS

Projections were performed in parallel beam ge-ometry. The sample was rotated in steps of 0.5◦ upto a maximum angle of 180◦. For the ideal projec-tions model elements were projected individually andadded to the sinogram, i.e., we performed a projectionof mass (or density, resp.) instead of intensities. Thisapproach is equivalent to integration over strips of de-tector width, and thus reflects that detector elements aswell as the pixels of the phantom are not points but areaelements. The detector elements had exactly the samesize as the reconstruction pixels. The original observa-tion window was extended by a two pixel edge of zeroentries on all sides. The rotation axis was set at thewindows’ center. In order to ensure complete visibilityof the reconstruction pixels under all projection angles,the length of the line detector was chosen sufficientlylarge.In order to assess the impact of noise on the recon-struction results we simulated intensity sinograms un-der a noisy X-ray source. Other experimental artifactssuch as cross-talking, focal smearing, beam hardeningor limited dynamic range (non-linearities in detection)were not simulated. For the noisy intensity measured atdetector locationξ under rotation angleγ one obtainsthe approximative formula

Iγ(ξ ) = Xξ ,γexp(−pγ(ξ )), (2.2)

where the process{Xξ ,γ} denotes Gaussian white noisewith fixed expectationEXξ ,γ = µ > 0 and varianceVarXξ ,γ = σ2 > 0 andpγ(ξ ) denotes the correspond-ing ideal projection of density for a given object. Thisapproximates a Poisson-distributed number of X-rayquanta with high expectation (Buzug, 2008). Noticethat for each detector locationξ and rotation angleγ the recorded intensityIγ(ξ ) has a normal distribu-tion. However, expectation and variance differ fromthe respective values of the initial intensityXξ ,γ and

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(a) Phantom (b) DIRECTT

(c) FBP Inspect3D (d) FBP IMod

(e) fast FBP

Fig. 1.Sample phantom and its reconstructions from projection data. The right upper corner has been scaled byfactor 3.

4


are given byµexp(−pγ(ξ )) andσ2exp(−pγ(ξ ))2, re-spectively. Thus, the stochastic counting rate at the de-tector depends not only on the input intensity but alsoon the projected object. The noise level is expressed bythe signal-to-noise-ratio (SNR)µσ . In an experimentalsetting under Gaussian approximation of Poisson noiseone hasσ =

√µ. However, for simulation purposeswe fixedµ and variedσ , since the parameter of inter-est was the SNR rather than the absolute value of theinitial intensity.

TOMOGRAPHIC RECONSTRUCTIONALGORITHMS

Filtered backprojection

A standard technique for the tomographic recon-struction of projection data is filtered backprojection(FBP) (Feldkampet al., 1984; Kak and Slaney, 2001).FBP is widely utilized in computed tomography usingX-rays (Buzug, 2008) as well as electrons (Carazoetal., 2005). In the following we will briefly summarizethe mathematical foundations of FBP.Let f : IR2 → [0,∞) denote a density distribution onIR2 with bounded support which is to be reconstructedfrom the set of projections{pγ : γ ∈ [0,2π)} whereγdenotes the rotation angle. A single projectionpγ(ξ )=∫

e⊥ξ ,γf (x)dx is an integral taken along the line

e⊥ξ ,γ = {ξ(

cos(γ)sin(γ)

)

+s

(−sin(γ)cos(γ)

)

: s∈ IR}, (2.3)

which is perpendicular to the first axis rotated byγand has distanceξ from the origin. The mathematicalkey ingredient of FBP is the Fourier slice theorem. Forthe Fourier transformPγ(q) =

∫ ∞−∞ pγ(ξ )e−2π iqξdξ of

a projectionpγ(ξ ) at a fixed rotation angleγ ∈ [0,2π)the Fourier slice theorem states the identity (cf. Buzug,2008)

Pγ(q) = F(qcos(γ),qsin(γ)). (2.4)

That is, the Fourier transform of the one-dimensionalprojection at rotation angleγ corresponds to the sliceof the two-dimensional Fourier transformed objectFpassing through the origin in directionγ . Substitutingpolar coordinates in the formula of the inverse Fouriertransform and a subsequent application of the Fourierslice theorem yields the identity (for details cf. Buzug,2008)

f (x) =∫ π

0

∫ ∞−∞ |q|Pγ(q)e2πqi(x1cos(γ)+x2 sin(γ))dqdγ ,

for all x∈ IR2.(2.5)

The most commonly used FBP algorithm is based onthis formula and organized as follows. In a first stepthe Fourier transformsPγ(q) of the projections arecomputed. These are subjected to an inverse Fouriertransform after radial weighting by the factor|q|,which yields filtered projection images. Backprojec-tions without weighting result in a convolution of theimage to be reconstructed with the point spread func-tion 1/‖x‖ (Buzug, 2008). In practice, the transformsare naturally done by discrete (inverse) Fourier trans-forms of the discretely sampled signal. In the backpro-jection step, for every pixel of the discrete output im-age the corresponding position on each filtered projec-tion image is determined and the corresponding valueis added to the sum which discretizes the outer inte-gral in (2.5). Since this position will in most cases be anon-integer pixel position, interpolation schemes needto be applied to neighboring pixels. By Shannon’s sam-pling theorem, at a given real space sampling distance∆ξ the signal can only be correctly reconstructed upto a frequencyQ = (2∆ξ )−1 in Fourier space. Thus,the radial weighting by the function|q| in (2.5) is onlyreasonable for|q| < Q. In other words, the samplingscheme imposes a band limitation which naturally de-termines the range of integration in the discrete ap-proximation of the inner integral in (2.5). In applica-tions high frequencies are often considered as noise.Therefore, in practice frequencies are not weighted ra-dially but the filter function is replaced by a modi-fied version|q|W(q), whereW(q) is a window func-tion that decreases the weight of the high frequencyband. Apart from a simple cutoff at frequencyqmax(i.e. settingW(q) = 1I[0,qmax](|q|)) (Ramachandran andLakshminarayanan, 1971), a variety of smoother ker-nel functions have been suggested, which suppress un-desired local extrema in the reconstructions (cf. Buzug,2008). A less commonly used alternative algorithm tothe real space FBP outlined above directly exploitsthe sampling on a polar grid, which is determinedby the Fourier slice theorem (Sandberget al., 2003).This Fourier space algorithm is more efficient than realspace FBP if the number of projections is sufficientlylarge. Moreover, it allows for higher order spline inter-polation of the data without additional cost. In order tomonitor the impact of the FBP algorithm, reconstruc-tions were computed by the real space and the Fourierspace approach. The latter will be referred to as ‘fastFBP’. For this study the implementation of the fastFBP provided by the IMod software (Kremeret al.,1996) was applied. Notice that the license for this com-ponent is not included in the standard version. In orderto assess the influence of specific implementations onthe reconstruction quality of the commonly used realspace FBP we computed real space FBPs by two dif-ferent software packages namely Inspect3D (version

5


3.0, FEI companyTM) and the IMod software (version3.11.2).An unmodified radial filter with simple cutoff was usedfor comparison of FBP to other reconstruction tech-niques. The cutoff frequency was set to the maximumspatial frequency that could occur in our image data.We will nevertheless demonstrate the effect on recon-structions which is caused by switching to a Gaussiandecay of the filter function at varying cutoff frequen-cies .Both software packages we applied are designed to re-construct 3D volumes from 2D projections. However,the software backprojects each 2D slice of the volumeseparately and thus each sample of our 2D phantomdata could be considered as a single slice of a 3D vol-ume. Sample reconstructions can be found in Fig. 1.

DIRECTT

The algorithm DIRECTT (Langeet al., 2008) rep-resents a promising alternative to conventional re-construction algorithms such as FBP or ART. Fig. 2schematically displays the algorithm’s iterative phi-losophy. The 2D algorithm is applicable to parallel aswell as fan beam geometry of projection. In the follow-ing we study parallel beam projections as illustrated inFig. 2, which are computed in strips for each detectorelement. Subfigure 2(a) (top left) indicates a modelvolume at the example of a 14 pixel object. Subfig-ure 2(b) (bottom left) represents the respective densitysinogram (Radon transform (Radon, 1917)) which iseither achieved by computed projection of model den-sities (Fig. 2(a)) or is the initial experimental inten-sity data converted according to Lambert-Beer’s law(cf. Buzug, 2008). Demanding that each element ofthe reconstruction array corresponds to exactly onesinusoidal trajectory of the sinogram (Fig. 2(b)), theDIRECTT algorithm selects pixels, i.e. area elements,corresponding to trajectories of dominant weight foran update of the reconstruction. The given densitysinogram can optionally be filtered along the detectordirection. This is helpful to avoid artifact formationequivalent to the effects of an unfiltered backprojec-tion. However, an intriguing feature of DIRECTT isthat adaptations of the filter function can be used toevaluate trajectories with focus on specific aspectsof interest such as mass or contrast. Switching filtersbetween subsequent iteration steps can be used to in-corporate a variety of different aspects of the imageinto the reconstructions step by step. Let the discretesinogram be given by the matrixS= (si j ), such thati = 1, . . . ,N and j = −`, . . . , `, whereN denotes thenumber of projection angles and 2`+ 1 is the num-ber of detector elements. Then the filtered sinogramis given by the matrixS∗, where theith row s∗i of S∗

is obtained by a convolution of theith row si of S

(extended by zeros at both ends) with the discretefilter function c : {−2`, . . . ,2`} → IR, more preciselys∗i j = ∑`

k=−`sikc(k− j).Computation of the DIRECTT reconstructions in thepresent study involved switching between two differ-ent filter functions. The first six iteration steps wereperformed after application of the mass filter

cm(k) =

{

− 1k2 for k∈ {−2`, . . . ,2`}\{0},

2∑2`k=1

1k2 for k= 0.

(2.6)The subsequent steps of iteration (fourteen for recon-structions of ideal projections) were based on contrast-filtered sinograms, where the contrast filtercc is givenby

cc(k) =

−1 for k∈ {−1,1},2 for k= 0,

0 else.(2.7)

The weights of all possible trajectories (correspond-ing to integer reconstruction positions) are computedby averaging along the respective traces within the op-tionally filtered sinogram. This involves interpolationbetween neighboring sinogram pixels corresponding todifferent detector elements. This is a substantial dif-ference to FBP, where interpolation is performed inFourier space (fast FBP) or in the backprojection stepafter the sinogram has been (inversely) Fourier trans-formed and filtered (real space FBP).In the update step of DIRECTT, a fraction of the tra-jectory weight is added to the respective area elementin the reconstruction array if the weight ranks withina predefined top percentage of all trajectory weights.This is illustrated in Fig. 2(c), where 11 out of the 14original elements in the example have been added. Theprojection (Radon transform) of the reconstruction ar-ray (i.e. a computed sinogram) is then subtracted fromthe original data set. The obtained residual sinogram(Fig. 2(d), containing trajectories of 3 remaining ele-ments in the example) is subject to the same procedurein the subsequent iteration steps until a pre-selectedcriterion of convergence is reached. This procedure canbe described as an iterative Radon and inverse Radontransform. In contrast to FBP there is no integral com-putation along the line detector (including limited sam-pling due to its element size) but an optional over-sampling along the numerous projection angles. Oneof DIRECTT’s unique characteristics is its very pre-cise projection of reconstruction elements taking intoaccount their actual size and shape which is essentialfor enhanced spatial resolution. That is, reconstruc-tion pixels are considered as a set of densely packedelements instead of being (circularly smeared) pointfunctions only. Hence, all previously described calcu-lations are performed based on squared area elements

6


in a Cartesian matrix. DIRECTT is of particular inter-est when the focus is on reconstruction of finely struc-tured details or on precise location of reconstructedelements rather than on computing time. In contrast toFBP, DIRECTT does not treat each (detector) projec-tion individually, i.e. it is not deconvolved globally or(Fourier) filtered, but the entire trajectory of a recon-struction element is considered over all projections. Incontrast to ART, DIRECTT does not modify the entityof all reconstruction elements simultaneously.

Fig. 2.Reconstruction principle of the iterative proce-dure applied by DIRECTT.(a) Model volume of a14pixel object.(b) Density sinogram of the model. Eachtrajectory corresponds to one of the pixels.(c) Inter-mediate reconstruction array, where11 out of the14pixels have been added.(d) Residual sinogram aftersubtraction of the sinogram generated by the interme-diate reconstruction in (c) from the sinogram in (b).

ESTIMATION OF AREA AND BOUN-DARY LENGTH

For the evaluation of reconstruction algorithms wecompared the area as well as the boundary length asmeasured by two different computational methods inthe discretized input data to values found for the vari-ous reconstructed images. Notice that we are interestedin the boundary length and area of the discretized ver-sion of the entire Boolean modelB =

⋃∞n=1B(Sn, r)

rather than in the cumulated morphological characte-ristics of the single circlesB(Sn, r), n≥ 1. Comparisonof morphological image characteristics requires a bi-narization of the images, which was done by simplethresholding, where the threshold parameter was set to50% of the maximum greyvalue. In order to excludebias by pure scaling differences, thresholding was doneafter normalizing the gray values of each image in sucha way that the average gray values occurring in thebackground and the foreground phase were set to 0 and

255, respectively.Any attempt to measure morphological characteristicsof a discretized set faces the problem that the shapeof the set before discretization cannot be reconstructedfrom the pixel image. In order to estimate the fore-ground area we applied the natural approach of count-ing the number of foreground pixels. For estimating theoriginal boundary length, different formulae from inte-gral geometry and stereology can be exploited. Never-theless, estimation results and consequently approxi-mation errors are more than likely to differ with respectto the estimation method chosen. In order to ensurereliability of our statistical results on reconstructionquality, we therefore implemented two different meth-ods for measuring the boundary length of the fore-ground phase.The first method we applied has been introduced inKlenk et al.(2006) and further discussed in Guderleietal. (2007). It will be referred to as the Steiner methodsince it is based on a discretized version of a Steiner-type formula known from the geometry of polyconvexsets. Details of this method are given in the appendix.As input parameter the algorithm needs a sequence ofso-called dilation radiir1 . . . , rn. Measurement resultsare dependent on the choice ofr1 . . . , rn. Therefore, forour investigations we used two different sets of dilationradii. The first choicer i = 4.2+1.3i, i = 1, . . . ,1000,was suggested in Guderleiet al. (2007), whereas thesecond choicer i = 0.4+0.09i, i = 1, . . . ,158, was op-timized to obtain results whose mean coincides withthe theoretical mean boundary length of the Booleanmodel used for phantom generation. The correspond-ing mean value formulae of Boolean models can befound in Schneider and Weil (2008).The second method we applied in order to measurethe boundary length of the input images and the re-constructed data is discussed in Ohser and Mucklich(2000) and will be referred to as the Cauchy method. Itapproximates the boundary length of a discretized setby a discrete analog of Cauchy’s surface area formula,which expresses the boundary lengthL(K) of the setKas an integral of the total projection length ofK overall directions (see appendix). The algorithms discussedin this section were implemented in the Geostoch soft-ware library (Mayeret al., 2004).

STATISTICAL TESTS FOR COMPARI-SON OF RECONSTRUCTION ERRORS

Reconstruction algorithms were statistically com-pared via the empirical probability distribution of thereconstruction error. The error was defined as relativedeviation of the morphological characteristics on thereconstructions from the phantom data. The morpho-logical characteristics measured were boundary length

7


and area of the foreground. The following definition ofreconstruction error is given for the example of a mea-surement method for the boundary length which leadsto an estimatorL for this morphological characteristic.Effects of different reconstruction algorithms on areameasurements were compared in an analogous way.Given two reconstruction algorithmsA1 andA2 and anestimatorL for the boundary length, for the phantomsbphan

1 , . . . ,bphan100 and their reconstructionsbAk

1 , . . .bAk100,

k= 1,2, the relative reconstruction errors

eA1i =

∣

∣

∣

∣

L(bphani )−L(b

A1i )

L(bphani )

∣

∣

∣

∣

and

eA2i =

∣

∣

∣

∣

L(bphani+50)−L(b

A2i+50)

L(bphani+50)

∣

∣

∣

∣

(2.8)

were computed fori = 1, . . . ,50. Since the phan-tomsbphan

1 , . . . ,bphan100 were discretized versions of in-

dependently sampled realizations of a Boolean model,the entire collection of errors from the two sampleseA1

1 , . . . ,eA150,e

A21 , . . . ,eA2

50 inherited stochastic indepen-dence. Consequently, two-sample-goodness-of-fit testscould be applied in order to compare the two distri-butions the error sampleseA1

1 , . . . ,eA150 andeA2

1 , . . . ,eA250

were drawn from. Given a pair of reconstruction al-gorithms A1 and A2, Kolmogorov-Smirnov (KST),Wilcoxon rank (WRT) and Ansari-Bradley (ABT) testswere performed. For KST and WRT two different nullhypotheses were considered, firstly that the cumulativedistribution functions (CDF)FA1 andFA2 of the errordistributions are equal, and secondly, that the error ofalgorithmA1 tends to be smaller than the one producedby A2 in the stochastic sense defined below.

– TheKolmogorov-Smirnov testchecks the null hy-pothesisH0 : FA1(x) = FA2(x) for all x∈ IR againstthe two-sided-alternative that the values of the twoCDFs differ for somex ∈ IR. Any differences be-tween the two samples will lead to the rejectionof H0 if they are too large in the statistical sense.In the one-sided version of the KST the null hy-pothesisH0 : FA1(x)≥FA2(x) for all x∈ IR is tested,which would imply that the first sample ofA1 sta-tistically consists ofsmaller values than the sec-ond one. For details see Conover (1971, p 309) andGibbons (1985, p 127).

– TheWilcoxon rank testis equivalent to the Mann-Whitney U-test. It is especially sensitive to devi-ations in the location parameters ofFA1 and FA2,i.e., it is used to determine whether one of the dis-tribution functions is shifted relative to the other.If the random variablesX1 and X2 have CDFsFA1 and FA2, respectively, the two-sided versionof the WRT testsH0 : P(X1 > X2) =

12 against

H1 : P(X1 > X2) 6= 12. Thus, differences in variabil-

ities of reconstruction errors within the two sam-ples will not lead to the rejection ofH0 as easilyas differences in the means or medians of the twosamples. The one-sided version testsH0 : P(X1 <X2)≥ 1

2 againstH1 : P(X1 < X2)<12, i.e. whether

the sample ofA1 is statistically smaller than thesecond sample. For mathematical details, see e.g.Gibbons (1985, p 164) and Lehmann and Romano(2005, p 243).

– TheAnsari-Bradley testassumes thatFA1(x−m) =FA2(θ (x−m)) for all x∈ IR, an unknown nuisanceparameterm used to normalize the location of thesample and some scaling ratioθ > 0. The test fo-cuses on the question if the distributions differ indispersion rather than in location. Thus, the ABTtestsH0 : θ = 1 againstH1 : θ 6= 1. One-sided al-ternatives are possible but not considered in thisstudy. Details can be found in Gibbons (1985, p179).

For all tests in this study version 2.8.1 of theRprogramming language (R Development Core Team,2007) was applied. Test results are given in terms ofa p-value, which is the largest level of significance atwhich the null hypothesis is not rejected.

RESULTS

Reconstruction errors are visualized by boxplots(Fig. 3 – 6). The box depicts the median and the (pos-sibly approximated) quartiles of the data. The centeredvertical lines show the smallest and largest observa-tions if their distance from the box does not exceed1.5 times the box size. More extreme values within thesample are plotted as circles. Note that in the boxplotswe consider signed relative errors, where these quanti-ties are defined as in (2.8) but without taking absolutevalues. However, all statistical tests are based on un-signed relative errors.

RECONSTRUCTIONS FROM IDEALPROJECTIONS

For DIRECTT the classic FOM of MSD definedin (1.1) had a value of 0.0139, whereas the corre-sponding values for the FBP algorithms were in theinterval between 0.069 and 0.074, with best resultsfor the Inspect3D software and the highest error mea-sured for the standard FBP in IMOD (Fig. 3). Althoughvery small, the differences in MSD between the FBPtechniques were found to be statistically significant byKST and WRT. This is plausible since there was hardlystochastic variability in the single MSD samples.

8


The area measurements of the foreground phase be-haved rather stable under all reconstruction algorithmsand relative errors were only at the level of few permille (Fig. 4). Errors fluctuated around 0 for DIRECTTand were only around 0.002 for the fast FBP and theInspect3D software. The standard FBP implemented inIMod showed a slightly increased error level of around0.006. KST and WRT classified the differences be-tween the algorithms as statistically significant, thoughthe absolute level was very small.Boxplots in Fig. 5 indicate that the signed relative er-rors for measurements of boundary length differed be-tween reconstruction algorithms. Stochastic variabilityof the errors within the sample was similar for all fourreconstruction algorithms as indicated by the highp-values of the Ansari-Bradley test (Tab. 1). In the DI-RECTT reconstructions the Steiner method measureda decrease in boundary length of around 1.5% and theCauchy method a decrease of around 0.5% in compar-ison to the original phantoms. However, the FBP al-gorithms produced significantly higher relative errorsthan DIRECTT as indicated by thep-values of thetests in their one-sided versions in Tab. 1. The errorproduced by the fast FBP algorithm was significantlysmaller than the error produced by the standard FBPimplemented in IMod, which in turn was slightly butsignificantly smaller than the error found in the FBPreconstruction done with Inspect3D.For all reconstruction algorithms error levels dependedon the method used to measure boundary length. Ave-raging over all three FBP implementations, the Steinermethod yielded a slightly stronger decrease in boun-dary lengths of around 13%. The choice of radii forthe Steiner method had only a hardly noticeable im-pact on the measurements of relative errors (Fig. 5(a)vs. 5(b)). In summary, it should be emphasized that onthe FBP-reconstructions all methods of measurementconsistently indicated a decrease of boundary lengthin comparison to the original phantoms, whereas thedeviation on the DIRECTT reconstructions was signif-icantly smaller. Allp-values of tests for equality of er-ror distributions were so small that the KST as well asthe WRT detected differences when the level of signi-ficance was set toα = 0.001/2. This in particular im-plies that the hypothesis of equal error distributionswas rejected in a Bonferroni-corrected setting for mul-tiple testing at levelα = 0.01. The latter defines thata hypothesisH0 is rejected at a levelα once a singleone ofn tests performed on the same data rejectsH0 atlevelα/n. As a consequence, the probability of a falserejection is bounded byα , which in most cases is quitea conservative estimate for the type 1 error of the test.The cutoff frequency, at which FBP algorithms switchfrom highpass to Gaussian filtering of the Fouriertransformed projections before backprojecting them,

had a noticeable impact on the error in boundarylength measurement. Suppression of high frequenciesincreased the relative deviation in boundary lengthmeasurements on the reconstructions from the originalimages independently of the method of measurement(Fig. 6).

DIRECTT FBP IMod FBP Inspect3D fast FBP

0.00

0.02

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Fig. 3. Mean squared deviations of reconstructionsfrom original phantoms.


−0.

002

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006

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Fig. 4.Relative deviation of area measurements on re-constructions from original phantoms.

RECONSTRUCTIONS FROM NOISYPROJECTIONS

The sensitivity of the relative error in boundarylength measurements to noise in the projection datawas investigated for the DIRECTT algorithm and FBP(Fig. 7). Since the errors of the different FBP algo-rithms were of similar order, we chose the standardFBP implementation of IMod for the comparison. Em-pirical 96% confidence intervals were computed fromthe two error samples under ideal projections. SNRswere considered between 50 and 400 in steps of 50. Foreach SNR a phantom was picked at random and noisewas added to its projections as described in Sec. 2.2.These were then used as input data for DIRECTT andFBP.For noisy projections the quality of the DIRECTT re-constructions improved in terms of MSD over the first

9


iterations but from a certain point on decreased again.This deterioration of the reconstruction quality occurswhen the residual sinograms are dominated by noiseand thus, further iterations introduce erroneous infor-mation into the reconstructions. The number of iter-ations for the DIRECTT reconstructions under noiseevaluated in Fig. 7 was chosen such that MSD wasminimized. Fig. 7 shows that the error of the DI-RECTT reconstructions under noise was not containedin the confidence interval of the error under ideal pro-jections. For SNRs higher than 150 the FBP recon-structions from noisy input data stayed within or closeto the range of error under ideal projections. Never-theless, the relative loss in boundary length caused byDIRECTT was in all cases less than in the FBP recon-structions. SNRs of less than 20 did not yield reason-able reconstruction results.

DISCUSSION

Our results demonstrate that standard FBP recon-struction algorithms for projection data may alter theboundary structure of two-phase phantom images insuch a way that measurements of boundary length aresubstantially affected. As a standard gray value ori-ented global measure of reconstruction quality, MSDalready indicated errors in the FBP reconstructions.Locally defined image characteristics such as phaseboundaries and quantitative image characteristics canhowever hardly be related to global integral FOMssuch as MSD in a direct way. Thus, for monitor-ing reconstruction artifacts distorting fine details andtheir consequences for quantitative image analysis it isimportant to consider alternative FOMs. In this con-text boundary length measurements can be a valu-able source of information since they are sensitive tochanges of local pixel configurations. CorrespondingFOMs can thus provide a more comprehensive viewon reconstruction quality.Apart from boundary length also other characteristicsfrequently considered in quantitative morphologicalimage analysis and spatial statistics such as connec-tivity (Ohser and Schladitz, 2008; Thiedmannet al.,2009), spherical contact distribution function (Mayer,2004; Thiedmannet al., 2008) and fractal dimen-sion are dependent on adequate reproduction of phaseboundaries. Since we have seen that FBP algorithmsalter the structure of phase boundaries, estimation ofthese image characteristics from FBP reconstructionsoccurs to be problematic, even if comparative studiesof different materials or scenarios may still be possible.On the other hand, foreground area showed very lim-ited sensitivity to the reconstruction artifacts producedby FBP algorithms. Thus, measurements of foreground

area can be regarded as stable with respect to standardFBP techniques.Our evaluation was based on a set of phantom im-ages consisting of discretized realizations of a Booleanmodel, which were independently sampled. Therefore,classical two-sample tests could be applied to com-pare the errors of different algorithms. Since differ-ences between error distributions of boundary lengthmeasurements for the considered reconstruction algo-rithms were rather pronounced (see the boxplots inFig. 5) the unambiguousp-values of the Kolmogorov-Smirnov and the Wilcoxon rank test in Tab. 1 wereto be expected. The Ansari-Bradley test indicated thatdispersion of the error samples was not significantlydifferent between algorithms and was probably mainlycontrolled by the stochastic variability of the phantomdata.The testing methodology we suggested can be trans-ferred to any other FOM and set of randomly sam-pled phantom data. This way also subtle differencesin reconstruction quality can be evaluated with statisti-cal rigor. It should again be emphasized that a statisti-cal approach to the evaluation of reconstruction qualityessentially relies on randomly sampled phantoms. Re-construction of deterministic phantom data is a valu-able tool to investigate the capability of reconstructionalgorithms to reproduce certain predefined image fea-tures. An approach based on randomly sampled phan-toms is complementary since it can be used to monitorthe statistical significance of errors produced by recon-struction algorithms. This information is especially im-portant for the quantitative investigation of irregularlystructured materials.Throughout this study the relative error in boundarylength was measured by two different techniques and– for the Steiner method – two choices of parame-ters. This way bias introduced by boundary measure-ment techniques could be excluded. Since the meth-ods are based on discrete approximations of differentformulae for the boundary length of a polyconvex set(see Appendix), deviations in measurement results nat-urally occurred. Although relative errors measured bythe Cauchy method were found to be slightly smallerthan the errors measured by the Steiner method, quali-tative as well as statistical findings agreed for all meth-ods applied. Thus, our findings were not tied to a spe-cific measurement approach.In order to relate the errors caused by the general tech-nique of FBP to the effect caused by different algorith-mic approaches, FBP reconstructions were conductedby the standard real space FBP and the fast FBP al-gorithm suggested by Sandberget al., 2003. More-over, for the real space FBP two different implementa-tions from the IMod software and the Inspect3D pack-age were compared. The reconstruction errors as as-sessed by the relative boundary length was found to

10



−0.

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(a) Steiner method; dilation radiir i = 4.2+1.3i, i = 1, . . . ,1000


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(b) Steiner method; dilation radiir i = 0.4+0.09i, i = 1, . . . ,158


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(c) Cauchy method

Fig. 5.Relative deviation of boundary length measurements on reconstructions from original phantoms.

0.1 0.2 0.3 0.4 0.5

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(b) Cauchy method

Fig. 6.Relative deviation of boundary length measurements on FBP reconstructions from original phantoms fordifferent cutoff frequencies. These reconstructions wereperformed by the real space FBP algorithm implementedin the IMod software. The highest spatial frequency that could occur in the image data was0.5.

11


SNR

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(b) Cauchy method

Fig. 7. Sensitivity of relative errors of boundary length measurements to noise in the projections. The red linesmark a96%confidence interval of the error under ideal, i.e. noiselessprojections found for FBP, the blue linesmark a corresponding confidence interval for DIRECTT. Red points are reconstruction errors of FBP found forsingle randomly picked phantoms under the noise level depicted on the x-axis. The blue points are the corre-sponding errors using DIRECTT.

be qualitatively similar for all three FBP implemen-tations, even if the fast FBP yielded significantly bet-ter results than the two real space algorithms. The sta-tistical significance of the differences in FBP recon-struction errors produced by the real space FBP inIMod and the Inspect3D software suggest that the per-formance of FBP techniques depends on their imple-mentation. It should however again be pointed out thaterror levels are very similar and our qualitative find-ings are implementation-independent. It should alsobe emphasized that rankings of implementations canonly be given with respect to a specific FOM. Thisis clearly illustrated by the rankings of the Inspect3Dreconstructions which exhibit a higher boundary er-ror than the other FBP reconstructions but performedbest within the FBP group with respect to MSD. Thisrather good representation of gray values is possiblythe consequence of the 16 bit image representationused in Inspect3D, whereas IMod computes recon-structions based on 8 bit images. Principle sources oferrors in real space FBP algorithms are the interpola-tion schemes applied in the backprojection step. Bothreal space FBP implementations used a computation-ally fast linear interpolation. The slight superiority ofthe fast FBP is possibly the result of the Fourier spaceapproach, which may reduce interpolation errors alongboundaries.Cutoff frequencies applied in FBPs can hamper cor-rect reconstructions of phase boundaries in a substan-tial way (Fig. 6). This effect is plausible since edgesare represented by high frequencies in Fourier space.It can also be expected to occur for other types of win-

dow functions which reduce the impact of high fre-quencies in comparison to simple radial weighting.For ideal projection data DIRECTT presented itselfas a powerful reconstruction algorithm, which repro-duced phase boundaries in an almost perfect way. Thishas been achieved under the simple but essential as-sumptions of homogeneous density within the pixels,regarded as area elements, and identical pixel sizeswithin the model, the detector and the reconstruction.First tests with DIRECTT however indicated that thereconstruction quality is still remarkable when pixelsof smaller size than the detector elements are recon-structed (Lange and Hentschel, 2007).In order to challenge the results obtained for noise-less projections, noise of different level was added tothe projection data of single randomly chosen phan-toms and the reconstruction results of DIRECTT andFBP were compared. The FBP reconstructions exhib-ited a high noise tolerance, since the relative error inboundary length measurement did hardly leave an em-pirical 96% confidence interval that had been com-puted for the ideal projections (Fig. 7). This showsthat under noisy projections the reconstruction er-ror in the phase boundaries is not dominated by thenoise but by properties of the FBP technique. TheDIRECTT reconstructions reacted more sensitively tothe noise, since the errors increased and were outsidethe 96% confidence interval for the DIRECTT recon-structions under ideal projections. Nevertheless, in allcases the DIRECTT reconstructions exhibited a sub-stantially smaller error than the FBP tomograms. Thus,the improvement in reconstruction quality that can be

12


achieved by DIRECTT appears to not to be limited toideal sets of input data but can also be expected in ex-perimental settings. The simulated projections of pixeldata serving as input for the reconstruction algorithmsdo not exactly describe a tomographic experiment witha spatially continuous material. However, our methodschosen for the transformation of the spatially continu-ous realizations of the Boolean model into pixel phan-toms and the computation of the projections by stripintegrals yield an adequate approximation of an exper-imental setting.It should be mentioned that it is difficult to determinethe optimal number of iterations for the reconstructionof noisy experimental input data. It is a challengingproblem to judge whether a residual sinogram is dom-inated by erroneous information from projection noiseand hence, further steps of iteration will only result inartifacts. This is an important subject for further stud-ies.The merits of the DIRECTT reconstructions come atthe cost of increased computation time, which totaled22 minutes on an Intel Xeon 5130 processor (2.0 GHz,one core) for a single input image. However, stan-dard algebraic reconstruction algorithms such as SIRT(Gilbert, 1972), which is commonly used in electrontomography (Balset al., 2007), are also computation-ally more demanding than FBP. In contrast to DI-RECTT, in addition to the number of iterations, theyusually require optimization of other parameters in or-der to yield satisfactory results (Carazoet al., 2005).Since for our phantom data a SIRT reconstructioncomputed by the Inspect3D software with 20 iterationsexhibited substantially increased blurring at the phaseboundaries in comparison to the FBP results, we didnot include SIRT in our comparative analysis.It is possible that other backprojection techniques thanstandard FBP are capable of an improved represen-tation of phase boundaries. These algorithms com-prise λ -tomography, where local inversion formulaeensure that space-continuously defined functions andtheir theoretical reconstructions have the same jumps.One should however point out thatλ -tomography doesnot reconstruct the density distributionf itself but thefunction Λ f , whereΛ =

√−4 denotes the Calderon

operator which does not preserve gray values (for de-tails see Faridaniet al., 1997; Kuchmentet al., 1995;Louis and Maass, 1993). An innovative and compu-tationally efficient reconstruction technique has beenproposed by Louis (2008). This approach combines re-construction and edge detection and could also enablesuperior reconstructions of phase boundaries in com-parison to standard FBP techniques. Furthermore, forsamples consisting of few different materials such asour phantom data, algorithms from discrete tomogra-phy have been reported to be very promising tools for

tomographic reconstruction (Balset al., 2007; Baten-burg, 2005, Herman and Kuba, 2007). Discrete tomo-graphy exploits a-priori information on the object tobe reconstructed, namely the number of materials it iscomposed of. Therefore, the algorithms can partiallycompensate for missing information, e.g. caused bylimited rotation of the sample (Langeet al., 2008).Furthermore, they return an image which does not re-quire segmentation of the different materials. By con-struction, DIRECTT also offers the option to computediscrete tomograms and thereby to reconstruct detailswhich cannot be extracted from the projections in thestandard setting. Nevertheless, our results demonstratethat even without the a-priori information needed fordiscrete tomography mode, DIRECTT reconstructionsexhibit a level of contrast and detail preservation whichis not achieved by conventional FBP reconstruction.We have illustrated that under ideal and noisy projec-tions reconstruction algorithms can cause statisticallysignificant changes of image morphology. For prac-tical applications the error caused by the reconstruc-tion algorithms needs to be carefully related to the ef-fect of experimental imperfections in the projectiondata. These may comprise alignment deviations, thespecific noise level or limited rotation. Expert knowl-edge about these experimental conditions is importantto identify appropriate reconstruction algorithms andtheir parameters, that meet the specific needs of an ap-plication. Nevertheless, whenever realistic projectionscan be simulated and a FOM capturing the aspects ofinterest has been defined, randomly generated phantomdata reflecting the structural properties of the investi-gated object and statistical analysis provide a powerfulsetting to compare different reconstruction techniques.

APPENDIX

MEASURING BOUNDARY LENGTH BYTHE STEINER METHOD

The algorithm for boundary length measurementwe refer to as the Steiner method exploits the follow-ing Steiner-type formula: LetK ⊂ IR2 be polyconvexset, i.e. a finite union of convex sets, then forr > 0 theso-called weighted volumeρr(K) of the setK can bewritten as

ρr(K) = r2πV0(K)+ rV1(K), (5.9)

where 2V1(K) is the boundary lengthL(K) of K andV0(K) denotes the Euler-Poincare characteristics ofK. In the 2D setting the latter counts the number ofconnectivity components in the foreground minus thenumber of holes. For a convex setK the weighted vol-umeρr(K) coincides with the volume of the parallel

13


set(K ⊕B(r,o)) \K, which consists of all points notcontained inK but whose distance toK is at mostr (Fig. 8). In case the boundary ofK has concavitypoints,ρr(K) is obtained by partitioning(K⊕B(r,o))\K in a specific way and counting the volumes of the ob-tained components with certain multiplicities (Klenketal., 2006). Ifρr(K) is known for two different dilationradii r0 andr1, by (5.9) we obtain the linear equationsystem

ρr0(K) = r20πV0(K)+ r0V1(K),

ρr1(K) = r21πV1(K)+ r1V1(K),

(5.10)

which can be uniquely solved forV0(K) andV1(K).This equation system can also be exploited to obtain anestimator for the boundary length of a discretized ver-sion of a setK on a square lattice. For details on howto compute the left-hand sides in (5.10) for discretizedsets we refer to Klenket al. (2006). Nevertheless, itshould be pointed out that a central aspect of the algo-rithm is a polyhedral approximation of the set whichis used to determine its boundary. For approximat-ing ρr(K) the occurrences of certain 8-neighborhoodconfigurations around boundary pixels are counted. Insimulation studies, estimation results for the boundarylength were shown to significantly improve if insteadof approximatingρr(K) for only two dilation radiia higher number of dilation radiir1, . . . , rn was used(Klenk et al., 2006). This usually yields an overdeter-mined system of equations, and thus, a solution can beobtained by the standard least-squares method. Esti-mation results are dependent on the choice of the radiir1 . . . , rn.

Fig. 8. Parallel set (K ⊕ B(r,o)) \ K of a rectangleK (gray). The dilation K⊕B(r,o) of K by the circleB(r,o) around the origin with radius r consists of allpoints whose distance to K is at most r.

MEASURING BOUNDARY LENGTH BYTHE CAUCHY METHOD

The algorithm for boundary length measurementwe refer to as the Cauchy method relies on Cauchy’s

surface area formula, which expresses the boundarylengthL(K) of the polyconvex setK as an integral ofthe total projection lengthπθ (K) of the setK in direc-tion θ :

L(K) =12

∫ 2π

0πθ (K)dθ . (5.11)

The total projection length is defined asπθ (K) =∫ ∞−∞V0(K ∩er,θ )dr, i.e., as the integral of the number

of connected components of the one-dimensional in-tersection ofK and the lineser,θ , whereer,θ is theline with directionθ ∈ [0,π) and the directed distancer ∈ IR from the origin (Fig. 9). Integration is with re-spect to the distancesr from the origin. On discretizedbinary images, the total projection length can be ap-proximated by computing relative frequencies of cer-tain pixel configurations at the phase boundary (Ohserand Mucklich, 2000). Based on these approximationsof the total projection length in the 8 canonical direc-tions in a 2D square lattice, the integral (5.11) canbe numerically evaluated by means of a quadraturescheme.

Fig. 9.A polyconvex set K intersected by the line er,θ ,which has directionθ ∈ [0,π) and the directed dis-tance r∈ IR from the origin. In the example depicted,the number of connected components V0(K∩er,θ ) is 2.

ACKNOWLEDGMENTS

This work was supported by a grant from DeutscheForschungsgemeinschaft to VS (SFB 518, projectB21). The authors thank Thomas Steinle for his help inperforming parts of the tomographic reconstructions.We are also grateful to Gregory Beylkin and DavidMastronarde who kindly provided the license for thefast FBP algorithm. Valuable comments by the refer-ees and Yannick Anguy as the handling editor helpedto substantially improve the manuscript.

14


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p-values KST KST WRT WRT ABTtwo-sided one-sided two-sided one-sided two-sided

DIRECTT vs. FBP IMod < 0.0001 > 0.9999 < 0.0001 > 0.9999 > 0.9999DIRECTT vs. FBP Inspect3D < 0.0001 > 0.9999 < 0.0001 > 0.9999 > 0.9999DIRECTT vs. fast FBP < 0.0001 > 0.9999 < 0.0001 > 0.9999 > 0.9999fast FBP vs. IMod < 0.0001 > 0.9999 < 0.0001 > 0.9999 > 0.9999fast FBP vs. Inspect3D < 0.0001 > 0.9999 < 0.0001 > 0.9999 > 0.9999FBP IMod vs. Inspect3D < 0.01 > 0.9999 < 0.001 > 0.9999 > 0.5

Table 1.Bounds for the p-values of the tests conducted for comparison of the relative errors of boundary lengthproduced by the different reconstruction algorithms undernoiseless projections. Small p-values of two-sided WRand KS tests indicate that error distributions are significantly different. Large p-values of the one-sided KST andWRT mean that the first of the algorithms in column 1 produces asignificantly smaller error than the other one.Large p-values of the two-sided ABT suggest that the variability of the errors is similar for the two reconstructionalgorithms considered.

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STATISTICAL ANALYSIS OF TOMOGRAPHIC RECONSTRUCTION … · Submitted to Image Anal Stereol, 17 pages Original Research Paper STATISTICAL ANALYSIS OF TOMOGRAPHIC RECONSTRUCTION AL-GORITHMS