Beam Hardening Correction in X-Ray ComputedTomography: A Comparison of Two Iterative

Model-Based Reconstruction Methods

Kilian DREMEL∗1, Theobald FUCHS2, Markus FIRSCHING2, and RandolfHANKE1

1X-ray Microscopy (LRM), Physics and Astronomy, University Wurzburg,Josef-Martin-Weg 63, D-97074 Wurzburg, Germany

2Fraunhofer Institute for Integrated Circuits IIS, Development Center forX-ray Technology (EZRT), Flugplatzstraße 75, D-90768 Furth, Germany

Abstract

While the expenses for computational power decrease, iterative reconstruction methodsfor x-ray computed tomography (CT) that allow the use of improved model assumptions,become more attractive. In applications for non-destructive testing one of the most com-mon degradations of image quality with standard x-ray CT reconstruction methods is beamhardening. Techniques for beam hardening correction using filtered back projection usuallyaccount for a single material only. In the case of multiple materials further improvement canbe achieved using iterative methods. We developed a framework for reconstruction basedon the principles of maximum-likelihood estimation that accounts for beam hardening byadapting the algorithms of L. Brabant (2012) and B. De Man (2001). These algorithmsare tested in simulation studies as well as in measurements with multiple materials. Wecompare the results and show that iterative model-based reconstruction methods turn out tobe a promising tool in order to reduce beam hardening.Keywords: computed tomography, CT, x-ray imaging, beam hardening, reconstruction

1 IntroductionToday, computed tomography (CT) has become a widely used method in non-destructive testing(NDT) applications in an industrial environment as well as in scientific applications. Especiallywhen searching for defects in cast parts or in dimensional metrology [1], CT has shown goodresults down to the µm-scale [2]. A major drawback in using CT in a laboratory environment onhighly absorbent metals like aluminium or iron remains beam hardening. The polychromaticnature of the laboratory sources causes cupping, shading, and streak artefacts in reconstruc-tions with standard filtered-back-projection (FBP), since FBP can only account for a singleenergy. The use of beam hardening reduction methods on the projections allows us to correct

∗kdremel@physik.uni-wuerzburg.de

11th European Conference on Non-Destructive Testing (ECNDT 2014), October 6-10, 2014, Prague, Czech Republic

the values of a known single material work piece from a known polychromatic spectrum to theirmonochromatic equivalent [3]. When objects with multiple materials are inspected the correc-tion becomes difficult, although multi-material iterative beam hardening correction methods inthe projection space have been developed ([4], [5]).Iterative reconstruction methods, like the simultaneous algebraic reconstruction technique (SART)[6] or maximum likelihood estimation methods have the potential to incorporate a model-basedforward-projection (see [7]) and, therefore, to handle the polychromatism during reconstruc-tion. Because iterative procedures require high computational power, they did not find theirway into standard applications so far. With decreasing expenses for highly parallel systems,these methods become more attractive and it is worth to have a look at their potential in model-based beam hardening reduction. In this paper, the authors will show a comparison between thestandard polynomial approach for beam hardening correction, the IMPACT method introducedby De Man in 2001 [8], which uses the decomposition in a photoelectric and in a Compton partand the method by Brabant from 2012 [9], which assumes the energy-dependency mainly in thephotoelectric part.Beginning with the theoretical part, this paper will give an overview over the beam hardeningproblem itself and the answers to this problem given by the mentioned authors. In the sectionon materials and methods the phantom and acquisition parameters for simulations and measure-ments will be shown. In the following two sections the reconstructed results will be presentedand discussed.

2 TheoryCT measurements in NDT are usually done in cone-beam geometry with an object rotatingon a manipulator system between an x-ray source and a flat-panel detector. The total countdetected from the polychromatic spectrum of the source can be described by the integral overthe intensities I of all energies E of the source, multiplied with the corresponding detectorsensitivity S :

I0 =

∫I(E)S (E)EdE (1)

The incident beam gets attenuated by Beer’s law on its path s through the object:

I =

∫ Emax

Emin

I(E)S (E)E exp(−

∫µ(s, E)ds

)dE (2)

Since there is an energy-dependency of µ, the attenuation of the rays does not depend linearlyon the path length. In the energy range of typical X-ray sources the attenuation coefficient µbasically consists of two different parts, namely the attenuation by the photoelectric effect inthe lower energy range and the Compton effect at higher energies [10]. When this knowledge isused, µ can be modelled by

µ(E) = µphoto(E) + µCompton(E). (3)

Thereby the energy-dependency of the photoelectric and Compton part are

µphoto ∝1

E3 , µCompton ∝ KN( E511keV

)(4)

with KN representing the Klein-Nishina function.

Figure 1: Relative attenuation coefficient ( µ(s)µ(0) ) of the materials used in the phantom with the

source spectrum used in the measurements as function of their path length through the object.

2.1 Polynomial correctionAs the beam hardening artefacts in CT reconstruction are the result of a non-linearity of theattenuation coefficient when radiographing different path lengths in a material, it is possible tocalculate a polynomial correction function based on the known properties of the material. For arange of path lengths, the known spectrum and the known material-specific attenuation is usedto calculate the corresponding intensity at each point. With a polynomial function fitted to thedata points, a correction function for a single material can be obtained. A sample is shown infig. 1. In our case, we used a 5th degree polynomial function and applied the correction to theprojections. The projections are then reconstructed using FBP.

2.2 IMPACT algorithmDe Man (see [8]) uses eq. 3 to design a discrete acquisition model based on a finite number ofsample source energies. From eq. 2 follows:

I =∑

k

I0,k exp

−∑j

ai jµ j

. (5)

In this I0,k is the combined source intensity and detector sensitivity, and ai j represents thepath length of ray i in a discrete set of voxels at voxel j. In a maximum-likelihood algo-rithm the log-likelihood function (with forward projected intensity Iforw., calculated as in eq. 5)L =

∑i I ln (Iforw.) − Iforw. gets maximized by Newtons method (see [8]):

µn+1j = µn

j +

∂L∂ j∑

h∂2L∂ j∂h

(6)

The energy-dependency of the attenuation is modelled by

µ(E) = µphoto(E) + µCompton(E) = µp,const µp(E) + µc,const µc(E). (7)

When a reference energy E0 is used, the energy parts in eq. 7 become:

µp(E) =E3

0

E3 , µc(E) =KN(E)KN(E0)

(8)

By fitting eq.7 to known material-specific data, the energy-constant parts can be obtained. Formultiple materials a set of constant Compton and photoelectric parts has to be calculated beforereconstruction. Then, each voxel in the back projected volume is at each iteration assignedto one of the predefined materials and the constant parts are modelled as a piecewise linearfunction of µ. By using the reference energy E0 for the fit, the attenuation coefficients arereconstructed at this specific energy.

2.3 Algorithm by BrabantFor every polychromatic beam there is an effective energy, depending on the pathlength of theray through the material, that would, in a monochromatic case, show the same attenuation asthe polychromatic beam. This effective energy is modelled at voxel n in a linear approximationby Brabant as

Eeff,n = E0 + η

n∑i=0

µi(E0) (9)

Following eq. 4 and neglecting the energy-dependency of the Compton part one can obtain anattenuation coefficient depending on the previous ray path:

µn(Eeff) =µn(E0)

(1 + α∑n−1

j µ(E0))β(10)

In this context β = 3 is the photoelectric energy-dependency and α is the strength of the cor-rection that depends on the spectrum and the inspected material. As Brabant put this relationtogether with the SART (see [6]) algorithm, we put it into the context of maximum likelihoodestimation using eq. 6.

3 Materials and methodsIn order to test the described algorithms we developed a software-framework that can incorpo-rate different methods. The spectrum and detector response were calculated by Monte Carlosimulations.

3.1 PhantomAs a phantom for our measurements and simulations we used an AlCuMgPb (consisting ofabout 95 % Al) disc with drill holes that can be filled with pins of iron or acrylic glass. For asingle material measurement we used the disc without pins, for the multi material case the con-figuration consisted of 4 steel pins in the outer holes and one pin of acrylic glass in the centerhole (see fig. 2). In this way a multiple material phantom with known dimensions and attenua-tion coefficients was realized. The exact material composition of alloys like the AlCuMgPb isonly known within limits. Since in our case especially the amount of materials with high atomicnumber make a difference in the attenuation, the composition was additionally determined byan EDX (energy-dispersive X-ray spectroscopy) analysis.

3.2 Reconstruction settings and material propertiesFor the algorithm by Brabant, the constants α and β had to be determined manually. We foundα = 0.06 for the multi material phantom and α = 0.073 for the single material phantom to fit

Figure 2: The phantom that was used in the measurements. The holes are filled with pins madeof steel or acrylic glas. Dimensions are given in mm.

both measurements and simulations the best. The value β = 3 was left unchanged. For thealgorithm by De Man we used the XCom database [11] to obtain reference values for the usedmaterials. These data were fitted using eq. 7 to get the photoelectric and Compton parts of theattenuation at 70 keV as listed in tab. 1. In all cases only a single middle slice was reconstructed.

material µCompton,c,70keV/1

cm µPhoto,c,70keV/1

cm µ70keV/1

cm

acrylic glas 0.21 0.01 0.22

AlCuMgPb 0.48 0.39 0.87

steel 1.2 5.2 6.4

Table 1: The materials used in the phantom, their attenuation coefficients at 70 keV and thefitted values for Compton part and photoelectric part.

3.3 Simulations and measurementsAll simulations were created using Scorpius XLab c©, a raytracing based simulation softwareby Fraunhofer EZRT with settings that were designed to be consistent with the measurements.The measurements were made with a Comet MXR-225 HP/11 source at 220 kV (multi materialphantom) or 120kV (single material phantom) and a current of 2.5 mA (multi material phantom)or 1.0 mA (single material phantom). For the multiple material phantom a 2.0 mm Cu prefilterand for the single material phantom a 0.25 mm Ti prefilter was used to minimize the lowestenergies in the spectrum. As detector a Perkin Elmer XRD820 CN14 with 1024x1024 pixelswith a pixelsize of 200 µm was used. The beam was collimated vertically to avoid scatteredradiation and each projection was integrated for 199 ms. The sample was put on a rotary plateand a full 360◦ scan with 400 projections was performed. The distances and reconstructiondetails can be found in tab. 2. Fig. 3 shows the measurement setup.

4 ResultsIn fig. 4 the reconstructions of the single material case are shown. The reconstructed valuesin the FBP image vary a lot between the outer to the inner parts of the disc and the unfilled

parameter valuedetector pixels used for reconstruction 288

distance source-detector 105.30 cm

distance source-object 71.19 cm

number of projections 400

reconstructed voxels 288x288

voxel size 135 µm

iterations 100

projection subsets 16

Table 2: Measurement and reconstruction details.

Figure 3: Measurement setup with a Comet MXR-225 HP/11 source and a Perkin ElmerXRD820 CN14 detector. Photo: Fraunhofer EZRT.

holes cannot be identified as empty. Using the polynomial correction on the projections, thereduction of artefacts results in an improved contrast. By using the algorithm by Brabant theshading artefacts in the measurements are better corrected. With the IMPACT algorithm, notonly the artefacts are reduced but also the attenuation coefficients can be connected with thecorresponding material, because the reference energy is fixed at 70keV (see lineplots in fig. 4).

In fig. 5 the FBP reconstructions of the measured and simulated multi material object are shown.Although in this case a 2mm Cu filter was used, cupping and shading artefacts are clearly vis-ible in the corresponding line plots. The acrylic glass in the center pin is impossible to beidentified, because the cupping artefacts cause a massive gradient in the reconstructed attenu-ation. The polynomial correction method cannot be used in this context, because this methodcan only account for a single material - the attenuation of other included materials would beover- or underestimated. Fig. 1 shows the polynomial functions used for the three materials inthe phantom. Using the algorithm proposed by Brabant, the artefacts caused by beam hardeningare clearly reduced. The lineplots (see fig. 5) show a much more constant attenuation inside thethree materials than the FBP and the areas, especially the center pin, can clearly be assigned totheir material.The IMPACT reconstruction shows higher contrast because the attenuation is calculated at70 keV. Cupping and shading artefacts are clearly reduced in measured and simulated data,although the good results of the simulations could not be achieved in the measurements.

Figure 4: Single material reconstructions and corresponding lineplots by using (from left toright) FBP, polynomial corrected FBP, algorithm by Brabant and IMPACT algorithm. Top rowshows simulations, bottom row shows measurements. The red line in the FBP reconstructionsindicates the plotted line. The grey values of the reconstructions vary as the effective energygets changed by the algorithm.

Figure 5: Multi material reconstructions and corresponding lineplots by using (from left toright) FBP, algorithm by Brabant and IMPACT algorithm. Top row shows simulations, bottomrow shows measurements. The red line in the FBP reconstructions indicates the plotted line.For the measurements an additional 45 degree lineplot, indicated by the dotted line in the FBPreconstrction, is shown. The grey values of the reconstructions vary as the effective energy getschanged by the algorithm.

5 DiscussionThe polynomial correction method shows acceptable results when only one single material isfound in the workpiece. It is easily implemented and, by using FBP, results are created nearlyas fast as without a correction method.If there are more materials to be reconstructed, an iterative reconstruction approach is a chancenot only to reduce beam hardening artefacts but also to provide possible improvements in imagequality when using iterative model-based reconstruction. Both introduced methods show im-provements in the single and multi material case. When using the method described by Brabantone has to try different parameters to get an improved result. The reconstructed attenuationsare based on the effective energy and are therefore defined by the spectrum. Although thereare improvements in handling the beam hardening effects, there has to be a trade-off betweenthe different materials, because they are only separated by their attenuation and not by theirmaterial-specific behavior. Also, the only considered dependency is the photoelectric effect.Therefore, the different behaviour of the steel and Al parts in the measurements results in aclear reduction of cupping in the steel, but in less significant improvements in the Al parts (seethe corresponding lineplots in fig. 5).The discrimination between the photoelectric effect and the Compton effect allows the bestclassification of the materials. The IMPACT algorithm is a little more costly in terms of time,because for every sampling point in the spectrum the energy dependency of the material hasto be calculated and the forward projection has to be performed for both the photoelectric andthe Compton part. But, considering the lineplots in fig. 5, especially those of the simulateddata and the 45 degree plot of the measured data, it shows the best results. Although it is ableto to determine the correct attenuation coefficients at a given energy, it is dependent on the fitthat has to be obtained from a known database, because in most cases the material compositionis not exactly known. Moreover, the spectrum with detector efficiency has to be known. Thedifference between reconstructed values and the effectiveness of the beam hardening correctionin simulation and measurement shows how sensitivly the algorithm reacts to the knowledge ofthe exact composition of the material and the spectrum, especially when it comes to absolutevalues. Moreover, in the simulations no scattering was included, so the measured data cannotbe perfectly corrected since the algorithms only account for beam hardening. Building a model-based reconstruction method accounting for both photoeffect and Compton effect is a physicallyaccurate but also a little more time-consuming way to handle polychromatic CT which requiresknowledge about object, source and detector behavior.A comparison with a dual energy approach to reduce beam hardening artefacts will be part offuture work.

6 AcknowledgementsWe like to thank Stephan Braxmeier from ZAE Bavaria in Wurzburg for the possibility to per-form EDX reference scans.

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