an analytical method for reducing metal artifacts in x-ray...

Research ArticleAn Analytical Method for Reducing Metal Artifacts inX-Ray CT Images

Ming Chen 1 Dimeng Xia1 DanWang1 Jingqi Han2 and Zhu Liu2

1College of Mathematics and Systems Science Shandong University of Science and Technology Qingdao 266590 China2Department of Radiology Traditional Chinese Medicine Hospital of Huangdao District of Qingdao City Qingdao 266500 China

Correspondence should be addressed to Ming Chen mingchen gang163com

Received 7 November 2018 Accepted 20 December 2018 Published 13 January 2019

Academic Editor Haipeng Peng

Copyright copy 2019 Ming Chen et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Medical CT imaging often encounters metallic implants or some metal interventional therapy apparatus These metallic objectscan produce metal artifacts in reconstruction images which severely degrade image quality In this paper we analyze the differencebetween polychromatic projection data and Radon transform data and develop an analytical method to reduce metal artifactsApproximate features of metal artifacts can be obtained by a simplified energy spectrum function of x-ray beam The developedmethod can reduce most artifacts and preserve more original details It does not require prior knowledge of x-ray energy spectrumand original projection data avoiding iterative calculation and saving reconstruction time Simulation experimental results showthat the method can greatly remove metal artifacts

1 Introduction

X-ray CT imaging plays an indispensable role in clinicaldiagnosis and interventional therapy However due to beam-hardening caused by some metallic implants in patientsor some metallic interventional therapy apparatus such asdental fillings orthopaedic implants or microwave ablationneedle there are strong streak or star-shape artifacts inreconstruction images [1ndash3] which are usually called metalartifacts They seriously degrade image quality and bringtrouble to clinical applications Although many metal artifactreduction (MAR)methodswere developed their applicationsin clinical settings are not totally successful because ofthe complexity of their forming cause and characteristicsCurrently there is no standard solution [4 5]Therefore howto reduce metal artifacts still remain a challenging problemin x-ray medical CT imaging

The effects of x-ray beam-hardening the photon starva-tion and the partial volume can all result in metal artifacts[6] In the past few decades a large number of MARalgorithms have been proposed to correct or reduce metalartifacts Interpolation was widely used for data completion[7ndash10] where missing projection data are approximatedby an interpolation technique But due to the inaccuracy

of data interpolation additional fringe artifacts and otherdeformations were introduced in a new reconstruction image[10 11] The missing data became more accurate by use offorward projections of a prior image [12 13] A combinationmethod of normalization and interpolation was proposed toremove most of the artifacts [11] However since artifacts arevery strong for some cases some pixels are often classifiedinto wrong types leading to unsatisfactory results In orderto correct beam-hardening some iterative algorithms wereproposed which reconstructed images from some processedprojections [14ndash16] They can suppress some artifacts butthere is still no satisfactory result for all images Recentlythere were some researches about the deep learning strategyto reduce metal artifacts [17 18] One of their drawbacksis that there is no common CT image database for modeltraining and another is that somemild artifacts typically stillremain

The forming cause of metal artifacts is mainly the highattenuation of metallic objects which leads to x-ray beam-hardening and aggravates the scattering phenomenon for thepolychromatic x-ray beam spectrum For low-atom numbermetals satisfactory results were achieved by correcting beam-hardening [19ndash25] Some dual-energy correction [21] andstatistical iterative correction [23 24] were proposed to

HindawiMathematical Problems in EngineeringVolume 2019 Article ID 2351878 7 pageshttpsdoiorg10115520192351878

2 Mathematical Problems in Engineering

reduce beam-hardening effects The former required longerpostprocessing and higher doses of radiation while the latterneeded more prior information about the energy spectrumof the incident x-ray and the energy-dependent attenuationcoefficient of the materials Park et al [25] put forwarda MAR method by giving the approximate expression ofmetal artifacts where choosing the approximate alternativeof energy spectrum is very critical to obtain more accurategeometrical characterizations of artifacts

For a polychromatic x-ray spectrum CT imaging thispaper analyzes the geometrical features of metal artifacts ina two-dimensional fan-beam system constructs an approx-imate energy spectrum function and gives an approximateexpression ofmetal artifacts by using an excess photon energyrelation The artifacts can be analytically expressed as theapproximate mathematical equation in this method

The rest of this paper is organized as follows In Section 2we summarize some background knowledge about Radontransform and the projection expression for a polychromaticx-ray spectrum In Section 3 an analytical MAR method isdeveloped by looking for geometrical characterizations ofmetal artifacts In Section 4 we give numerical simulationsto verify the effectiveness of the developed method Finallyconclusions will be made and some related issues will bediscussed

2 Background Knowledge

In this section we give Radon transform of a two-dimensional (2D) function and projection expressions for apolychromatic x-ray spectrum This section will give theo-retical basis for finding the relations between single-energyprojection data and polychromatic projection data in nextsection

Mathematically 2D Radon transforms can be seen as aline integral process from one 2D function to another Let119877 denote Radon transform For 119891(119883) and a line 119871 119877 can beexpressed as follows

119877119891 (119871) = int119871119891 (119883) 119889119897 (1)

or

119877119891 (119903 120593) = int119871(119903120593)

119891 (119909 119910) 119889119897 (2)

where 119883 = (119909 119910) is a reconstructed point 119871 = 119871(119903 120593) 119883 sdot 0 = 119903 0 = (cos120593 sin 120593) 119889119897 is a arc length on 119871 120593 is aprojection angle and 119903 is a sampling variable along a detectordirection In fact (2) is also called ray sum line integral orthe projection under single energy [26]

When 120593 is fixed and 119903 is (minusinfin+infin) in (2) 119877119891(119903 120593) isa set of parallel projections shown in Figure 1 For all 120593 isin[0 120587) 119877119891(119903 120593) is usually called parallel-beam projections inCT imaging The expression of a fan-beam projection forsingle energy is similar to (2) by converting the parametersof parallel-beam scan mode to those of fan-beam scan mode

For CT system with a polychromatic x-ray spectrum theprojection expression above will become difficult Let 1198680119864 and

x

y

r

O

rdetector

f(x y)

Rf(r )

Figure 1 Radon transform of 119891(119909 119910) under 120593

0 20 40 60 80 100 120minus0005

0

0005

001

0015

002

0025

003

0035

Figure 2 An example of x-ray energy spectrum when x-ray tubeoperates at 120 KV where a horizontal axis is x-ray energy and alongitudinal axis is a normalized output

119868119864 denote the incident x-ray intensity and the transmittedintensity at an energy 119864 respectively Let 119891(119883 119864) denotethe linear attenuation coefficient of the detected object at119864 To emphasize an energy-dependent nature of materialattenuation Lambert-Beer law [27 28] can be written asfollows

119868119864 = 1198680119864 exp minusint119871119891 (119883 119864) 119889119897 (3)

X-ray beam produced by x-ray tube operating at 120 KVgenerally covers a broad spectrum which is with two sharppeaks in an example shown in Figure 2 by a solid line Thespectrum distribution shows that the range of the output x-ray photon energy is [20 keV 120 keV] where energy photonsless than 20 keV are absorbed by somematerials in x-ray tube[29]

Mathematical Problems in Engineering 3

Projection data for a polychromatic x-ray source can beexpressed as follows

119901 (119903 120593) = minusln int 119868119864dEint 1198680119864dE

= minus ln (int 120578 (119864) exp minusint119871119891 (119883 119864) 119889119897 119889119864)

(4)

where 120578(119864) = 1198680119864 int 119868011986411198891198641 represents a normalized x-raybeam energy spectrum [25 30] When metallic implants inpatients or some metallic interventional therapy apparatusexist in some CT scanned slices 119901(119903 120593) will include beam-hardening effects Thus projection data are not accurate forthe scanned slices and not equal to the ideal projections119877119891(119903 120593) above3 An Analytical MAR method

In this section we first analyze the forming cause of metalartifacts based on aforementioned knowledge then constructthe approximate function of an energy spectrumdistributionand finally give geometrical features of metal artifacts in 2Dfan-beam system

In CT scan system with a polychromatic x-ray sourcebeam-hardening leads to the significant difference betweenraw projection 119901(119903 120593) and Radon transform 119877119891(119903 120593) If weonly consider metal artifacts caused by beam-hardeningthe difference expression 119901(119903 120593) minus 119877119891(119903 120593) corresponds toprojections of metal artifacts

We remark the aforementioned function 119891(119883) as 1198911198641015840(119883)at fixed energy 1198641015840 for a polychromatic x-ray beam Then1198911198641015840(119883) is the corrected objective function Let 119877minus1 denotesome exact CT reconstruction algorithm such as filtering-back-projection (FBP) algorithm or backprojection-filtering(BPF) algorithm So 119877minus1119901(119883) is a reconstructed image withmetal artifacts and 119877minus1[119901 minus 1198771198911198641015840](119883) seen as metal artifactsis a reconstruction result from a difference part between pro-jection data and Radon transform data Obviously 119877minus1119901(119883)can be expressed as follows

119877minus1119901 (119883) = 1198911198641015840 (119883) + 119877minus1 [119901 minus 1198771198911198641015840] (119883) (5)

It is assumed that metallic objects have the same attenua-tion coefficient in a reconstruction image which is reasonablefor most cases in clinic applications In this case metal arti-facts mainly depend on some geometrical features of metalregions a normalized energy spectrum function and metalattenuations about x-ray Let 119891119864(119883) denote a reconstructionimage withmetal artifacts 119877minus1119901(119883) that is 119901(119903 120593) = 119877119891119864(119883)where 119864 is a variable According to (4) and (5) 119877minus1[119901 minus1198771198911198641015840](119883) can be written as follows

119877minus1 [119901 minus 1198771198911198641015840] (119883)= 119877minus1 minus ln(int120578 (119864) exp minus119877 (119891119864 minus 1198911198641015840) (119903 120593) 119889119864) (6)

The water and bone beam-hardening corrections canusually be used in current medical CT imaging systems

[30] In this paper we only consider beam-hardening effectsof metallic objects since their attenuation coefficients havemore dependency on x-ray energy than soft tissues andbones LetΩ denote a metal domain which can include somesmall metal regionsThen119891119864minus1198911198641015840 = 0 outside ofΩ Since theprobability of photoelectric interactions is roughly inverselyproportional to the cubic of excess photon energy [31]119891119864minus1198911198641015840in Ω can be approximated as

119891119864 minus 1198911198641015840 asymp 119908 (119864) (119864 minus 1198641015840) (7)

where 119908(119864) depends on 119864 and attenuation coefficients ofmetal regions

So according to (7) (6) can be rewritten as

119877minus1 [119901 minus 1198771198911198641015840] (119883)asymp 119877minus1 minus ln(int 120578 (119864) exp minus119908 (119864) (119864 minus 1198641015840)119877120582(Ω) (119903 120593) 119889119864) (8)

where 120582(Ω) is a characteristic function for the metal domainΩBecause the normalized energy spectrum function 120578(119864) is

unknown 119877minus1[119901 minus 1198771198911198641015840](119883) can not be accurately obtainedWe choose a following alternative function in order to give aspecific and relatively accurate expression of 120578(119864)

1205781015840 (119864) = 119860 sin (120596 (119864 minus 1198641015840 + 120579)) if 10038161003816100381610038161003816119864 minus 119864101584010038161003816100381610038161003816 le ℎ0 otherwise

(9)

where ℎ is a variable It is worth noting that the selectionsof an amplitude 119860 a frequency 120596 and a phase 120579 are generallyrelated toℎ An example 1205781015840(119864) is shown inFigure 2 by a dottedline where it is the smoother one In fact by choosing ℎ wecan make an approximate spectrum distribution 1205781015840(119864) satisfythe following relationship

ln(int [120578 (119864) minus 1205781015840 (119864)]sdot exp minus119908 (119864) (119864 minus 1198641015840) 119877120582(Ω) (119903 120593) 119889119864) asymp 1

(10)

This means that we can use ln(int 1205781015840(119864)expminus119908(119864)(119864 minus1198641015840)119877120582(Ω)(119903 120593)119889119864) to approximately replaceln(int 120578(119864)expminus119908(119864)(119864 minus 1198641015840)119877120582(Ω)(119903 120593)119889119864) So (8) canbe further approximated as

119877minus1 [119901 minus 1198771198911198641015840] (119883) asymp 119877minus1 minusln(int1205781015840 (119864)sdot exp minus119908 (119864) (119864 minus 1198641015840)119877120582(Ω) (119903 120593) 119889119864)

(11)

Through a series of mathematical derivations on theright side in (11) and supposing that 119908(119864) is approximately


Obtain the segmented metal image

Calculate the geometrical characterization image of metal artifacts

Through threshold segmentation

Solve the projection data of characteristic function

The detected object with metallic materials

Obtain the raw projection data

Reconstruct the CT image

Input

Give the characteristic function of metal domain

Output Give the corrected objective image

Figure 3 Implementation flow chart of the developed MARmethod

independent of 119864 119877minus1[119901 minus 1198771198911198641015840](119883) is finally deduced asfollows

119877minus1minus ln[[

[119860 exp minus119908ℎ1198771015840

radic(1199081198771015840)2 + 1205962 sin (120574 minus (120596ℎ + 120596120593)) minus 119860

sdot exp minus119908ℎ1198771015840radic(1199081198771015840)2 + 1205962 sin (120574 minus (minus120596ℎ + 120596120593))]]

]

(12)

where 1198771015840 = 119877120582(Ω)(119903 120593) cos 120574 = 1199081198771015840radic(1199081198771015840)2 + 1205962 Accord-ing to (5) the corrected objective function 1198911198641015840(119883) can beobtained by the following equation

1198911198641015840 (119883) = 119877minus1119901 (119883) minus 119877minus1 [119901 minus 1198771198911198641015840] (119883) (13)

Equation (13) is the analytical MAR algorithm expressiongiven by approximate geometrical features of metal artifactsin (12)

4 Implementation of MAR Method andNumerical Simulation Experiments

Based on the aforementioned derivation of MAR methodimplementations of the developed MAR technique mainly

includes three steps Firstly raw CT images are reconstructedusing FBP algorithm from the polychromatic projection dataThen metallic objects are segmented from the images toobtain a metal domain and its characteristic function and anapproximate artifact image can be reconstructed using FBPalgorithm Finally the corrected objective image is given bysubtracting the artifact image from the raw image Its specificimplementation steps are shown in Figure 3

To verify the effectiveness of the developedMARmethodnumerical simulations are performed on VC++ 60 platformwhere all steps in the method are coded in C languageFan-beam CT system parameters are as follows the distancebetween x-ray source and a center of rotation is 1100mmthe distance between x-ray source and a linear-array detectoris 1600mm and the length of a detector cell is 087mmProjection data are generated by using the circular trajectoryequidistant fan-beam CT scan mode where the number ofdetector cells is 512 and the number of 360 degree full-scanprojection angles is 660The size of image matrix is 512 times 512

We select a modified digital mandibular phantom asshown in Figure 4 Its structure definitions are shown inTable 1 which are firstly given by Lemmens et al [32]Three circles with maximum gray values mean the implantedmercury objects Bone components and soft tissues in thephantom are set according to the parameters reported byICRU44 [33] Their attenuation coefficients are obtained


Figure 4 The digital mandibular phantom where the displaywindow is [00 45]

Figure 5 The projection data with beam-hardening effects

Table 1 Definitions of the digital mandibular phantom structures

Structure sequence Number Material Dense(gcm3)1 Mercury 1362 Bone 293 Bone 274 Soft tissue 085 Soft tissue 066 Soft tissue 02

by XCOM software [34] A polychromatic x-ray beam isgenerated by simulations where x-ray tube operates at 120kVp The polychromatic projection data are obtained shownin Figure 5

Figure 6 The reconstructed artifact image using (12)

According to the implementation flowchart in Figure 3we select the threshold value 3000HU [35 36] to segmentthe metallic parts and then calculate the projection data ofthe characteristic function According to (10) the parametersin the approximate energy spectrum function 1205781015840(119864) aredesigned as 119860 = 1205874ℎ 120596 = 1205872ℎ 1198641015840 = 65Kev ℎ =55Kev 120579 = ℎ and the optimized parameter 119908 = 0042According to (12) an artifact image is reconstructed as shownin Figure 6 where geometrical features of metal artifactsare clearly observed The objective image is obtained bysubtracting the artifact image as shown in Figure 7(a) whereCT image reconstruction takes about 85 seconds using theproposed MAR method on a computer with CPU 340GHzFor comparison we give a reconstruction result using thelinear interpolation MAR in Figure 7(b) We can find that thedeveloped methods can reduce most artifacts and preservemore original details In facts if the materials with lowerdensity than the mercury are implanted in the phantom thebetter corrected results can be obtained

5 Conclusion and Discussion

In this paper for CT scan system with a polychromatic x-ray source we propose the analytical method for reducingmetal artifacts This method can directly obtain the correctedimage by reconstructing features of metal artifacts withoutany prior information with ray energy spectrum largelyremove structural features of the artifacts and better retainthe original information So the developed method canobtain high-resolution CT images and provide more accurateinformation for clinical applications

A good alternative expression of the probability of thephotoelectric interaction is very important in the developedmethod However due to random noises from x-ray beamgeneration process and detection process of linear-array


(a) (b)

Figure 7The corrected images (a) is a reconstruction result using the developed analytical MARmethod in (13) and (b) is a reconstructionresult using linear interpolation MARmethod where the display window is [00 06] for displaying the details clearly

detectors in an actual CT system photoelectric interactionsare very difficult If more knowledge in random systemanalyzation could be applied in the photoelectric interactionthe better expression may be designed How to express thephotoelectric interactions using the random process will be achallenging and meaningful work

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported in part by National Science Foun-dation of China (no 61771328) Natural Science Foundationof Shandong Province (no ZR2017MF054) and SDUSTResearch Fund (no 2014TDJH102)

References

[1] B D Man J Nuyts P Dupont and G Marchal ldquoMetal streakartifacts in X-ray computed tomography A simulation studyrdquoIEEE Transactions on Nuclear Science vol 46 no 3 pp 691ndash696 1999

[2] S J Tang X Q Mou Q Xu Y B Zhan et al ldquoData consistencyconditionbased beam-hardening correctionrdquo Optical Engineer-ing vol 50 no 7 Article ID 076501 pp 1ndash13 2011

[3] M Chen and G Li ldquoForming Mechanism and Correction ofCT Image Artifacts Caused by the Errors ofThree System Para-metersrdquo Journal of Applied Mathematics vol 2013 Article ID545147 7 pages 2013

[4] J Y Huang J R Kerns J L Nute et al ldquoAn evaluation of threecommercially availablemetal artifact reductionmethods for CTimagingrdquo Physics in Medicine and Biology vol 60 no 3 pp1047ndash1067 2015

[5] S Tang K Huang Y Cheng X Mou and X Tang ldquoOpti-mization based beam-hardening correction in CT under dataintegral invariant constraintrdquoPhysics inMedicineampBiology vol63 no 13 Article ID 135015 15 pages 2018

[6] B De Man Iterative reconstruction for reduction of metalartifacts in computed tomography Publication Katholieke Uni-versity Leuven 2001

[7] R M Lewitt and R H T Bates ldquoImage-reconstruction fromprojections III Projection completion methods (theory)rdquoOptik (Jena) vol 50 no 3 pp 189ndash204 1978

[8] Y Zhang Y F Pu J R Hu Y Liu Q L Chen and J L ZhouldquoEfficient CT Metal Artifact Reduction Based on Fractional-Order Curvature Diffusionrdquo Computational and MathematicalMethods in Medicine vol 2011 Article ID 173748 9 pages 2011

[9] Y Zhang Y F Pu J R Hu and Y Liu ldquoA new CT metal arti-facts reduction algorithm based on fractional-order sonograminpaintingrdquo Journal of X-Ray Science and Technology vol 19 no3 pp 373ndash384 2011

[10] W A Kalender R Hebel and J Ebersberger ldquoReduction of CTartifacts caused by metallic implantsrdquo Radiology vol 164 no 2pp 576-577 1987

[11] E Meyer R Raupach M Lell B Schmidt et al ldquoNormalizedmetal artifact reduction (NMAR) in computed tomographyrdquoMedical Physics vol 37 no 10 pp 5482ndash5493 2010

[12] J Yu M Li Y Wang and G He ldquoA decomposition method forlarge-scale box constrained optimizationrdquoAppliedMathematicsand Computation vol 231 no 12 pp 9ndash15 2014

[13] J Wang S Wang Y Chen J Wu J Coatrieux and L LuoldquoMetal artifact reduction in CT using fusion based prior imagerdquoMedical Physics vol 40 no 8 Article ID 081903 2013

[14] D Jiang X Wang G Xu and J Lin ldquoA Denoising-Decom-position Model Combining TV Minimisation and Fractional


Derivativesrdquo East Asian Journal on Applied Mathematics vol 8no 3 pp 447ndash462 2018

[15] Y Zhang X Mou and H Yan ldquoWeighted Total Variationconstrained reconstruction for reduction of metal artifact inCTrdquo in Proceedings of the 2010 IEEENuclear Science SymposiumMedical Imaging Conference NSSMIC 2010 and 17th Interna-tional Workshop on Room-Temperature Semiconductor X-rayand Gamma-ray Detectors RTSD 2010 pp 2630ndash2634 USANovember 2010

[16] J Zhu and B Hao ldquoA New Noninterior Continuation Methodfor Solving a System of Equalities and Inequalitiesrdquo Journal ofAppliedMathematics vol 2014 Article ID 592540 6 pages 2014

[17] L Gjesteby Q Yang Y Xi B Claus et al ldquoReducing metalstreak artifacts in CT images via deep learningPilot resultsrdquoin Proceedings of the 14th International Meeting on Fully Three-Dimensional Image Reconstruction in Radiology andNuclear pp611ndash614 2017

[18] Y Zhang and H Yu ldquoConvolutional Neural Network BasedMetal Artifact Reduction in X-Ray Computed TomographyrdquoIEEE Transactions on Medical Imaging vol 37 no 6 pp 1370ndash1381 2018

[19] J Hsieh R C Molthen C A Dawson and R H Johnson ldquoAniterative approach to the beam hardening correction in conebeam CTrdquoMedical Physics vol 27 no 1 pp 23ndash29 2000

[20] L-G Huang L Yin Y-L Wang and X-L Lin ldquoSome Wilkerand Cusa type inequalities for generalized trigonometric andhyperbolic functionsrdquo Journal of Inequalities and ApplicationsPaper No 52 8 pages 2018

[21] M Kachelrieszlig O Watzke and W A Kalender ldquoGeneralizedmulti-dimensional adaptive filtering for conventional and spiralsingle-slice multi-slice and cone-beam CTrdquo Medical Physicsvol 28 no 4 pp 475ndash490 2001

[22] L Yuan Q Hu and H An ldquoParallel preconditioners for planewave Helmholtz and Maxwell systems with large wave num-bersrdquo International Journal of Numerical Analysis amp Modelingvol 13 no 5 pp 802ndash819 2016

[23] J A OrsquoSullivan and J Benac ldquoAlternating minimization algo-rithms for transmission tomographyrdquo IEEE Transactions onMedical Imaging vol 26 no 3 pp 283ndash297 2007

[24] Q Hu and L Yuan ldquoA plane wave method combined with localspectral elements for nonhomogeneous Helmholtz equationand time-harmonic Maxwell equationsrdquo Advances in Compu-tational Mathematics vol 44 no 1 pp 245ndash275 2018

[25] H S Park D Hwang and J K Seo ldquoMetal artifact reduction forpolychromatic X-rayCTbased on a beam-hardening correctorrdquoIEEE Transactions on Medical Imaging vol 35 no 2 pp 480ndash487 2016

[26] G L Zeng Medical image reconstruction Higher EducationPress and Springer New York NY USA 2009

[27] A Beer ldquoBestimmung der Absorption des rothen Lichts infarbigen Flussigkeitenrdquo Annalen der Physik vol 162 no 5 pp78ndash88 1852

[28] J H Lambert Photometria sive de mensura de gradibus luminiscolorum et umbrae Eberhard Klett Augsberg Germany 1993

[29] J Hsieh Computed Tomography Principles Design Artifactsand Recent Advances SPIE press Bellingham WashingtonUSA 2nd edition 2009

[30] J K Seo and E J Woo Nonlinear Inverse Problems in ImagingJohn Wiley amp Sons Ltd NY USA 2012

[31] H E Johns and J R Cunningham The physics of radiologyCharles CThomas Publisher Ltd Springfield IL 1983

[32] C Lemmens D Faul and J Nuyts ldquoSuppression of metalartifacts in CT using a reconstruction procedure that combinesMAP andprojection completionrdquo IEEETransactions onMedicalImaging vol 28 no 2 pp 250ndash260 2009

[33] ICRUReport44 Tissue substitutes in radiation dosimetry andmeasurements International Commission on Radiation Unitsand Measurements 1989

[34] M J Berger J H Hubbell S Seltzer et al XCOM photon crosssections database NIST Standard Reference Database 1998

[35] F E Boas andD Fleischmann ldquoEvaluation of two iterative tech-niques for reducing metal artifacts in computed tomographyrdquoRadiology vol 259 no 3 pp 894ndash902 2011

[36] OWatzke andW A Kalender ldquoA pragmatic approach to metalartifact reduction in CT Merging of metal artifact reducedimagesrdquo European Radiology vol 14 no 5 pp 849ndash856 2004

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reduce beam-hardening effects The former required longerpostprocessing and higher doses of radiation while the latterneeded more prior information about the energy spectrumof the incident x-ray and the energy-dependent attenuationcoefficient of the materials Park et al [25] put forwarda MAR method by giving the approximate expression ofmetal artifacts where choosing the approximate alternativeof energy spectrum is very critical to obtain more accurategeometrical characterizations of artifacts

For a polychromatic x-ray spectrum CT imaging thispaper analyzes the geometrical features of metal artifacts ina two-dimensional fan-beam system constructs an approx-imate energy spectrum function and gives an approximateexpression ofmetal artifacts by using an excess photon energyrelation The artifacts can be analytically expressed as theapproximate mathematical equation in this method

The rest of this paper is organized as follows In Section 2we summarize some background knowledge about Radontransform and the projection expression for a polychromaticx-ray spectrum In Section 3 an analytical MAR method isdeveloped by looking for geometrical characterizations ofmetal artifacts In Section 4 we give numerical simulationsto verify the effectiveness of the developed method Finallyconclusions will be made and some related issues will bediscussed

2 Background Knowledge

In this section we give Radon transform of a two-dimensional (2D) function and projection expressions for apolychromatic x-ray spectrum This section will give theo-retical basis for finding the relations between single-energyprojection data and polychromatic projection data in nextsection

Mathematically 2D Radon transforms can be seen as aline integral process from one 2D function to another Let119877 denote Radon transform For 119891(119883) and a line 119871 119877 can beexpressed as follows

119877119891 (119871) = int119871119891 (119883) 119889119897 (1)

or

119877119891 (119903 120593) = int119871(119903120593)

119891 (119909 119910) 119889119897 (2)

where 119883 = (119909 119910) is a reconstructed point 119871 = 119871(119903 120593) 119883 sdot 0 = 119903 0 = (cos120593 sin 120593) 119889119897 is a arc length on 119871 120593 is aprojection angle and 119903 is a sampling variable along a detectordirection In fact (2) is also called ray sum line integral orthe projection under single energy [26]

When 120593 is fixed and 119903 is (minusinfin+infin) in (2) 119877119891(119903 120593) isa set of parallel projections shown in Figure 1 For all 120593 isin[0 120587) 119877119891(119903 120593) is usually called parallel-beam projections inCT imaging The expression of a fan-beam projection forsingle energy is similar to (2) by converting the parametersof parallel-beam scan mode to those of fan-beam scan mode

For CT system with a polychromatic x-ray spectrum theprojection expression above will become difficult Let 1198680119864 and

x

y

r

O

rdetector

f(x y)

Rf(r )

Figure 1 Radon transform of 119891(119909 119910) under 120593

0 20 40 60 80 100 120minus0005

0

0005

001

0015

002

0025

003

0035

Figure 2 An example of x-ray energy spectrum when x-ray tubeoperates at 120 KV where a horizontal axis is x-ray energy and alongitudinal axis is a normalized output

119868119864 denote the incident x-ray intensity and the transmittedintensity at an energy 119864 respectively Let 119891(119883 119864) denotethe linear attenuation coefficient of the detected object at119864 To emphasize an energy-dependent nature of materialattenuation Lambert-Beer law [27 28] can be written asfollows

119868119864 = 1198680119864 exp minusint119871119891 (119883 119864) 119889119897 (3)

X-ray beam produced by x-ray tube operating at 120 KVgenerally covers a broad spectrum which is with two sharppeaks in an example shown in Figure 2 by a solid line Thespectrum distribution shows that the range of the output x-ray photon energy is [20 keV 120 keV] where energy photonsless than 20 keV are absorbed by somematerials in x-ray tube[29]





(4)

















(9)



(10)



(11)










Input






[119860 exp minus119908ℎ1198771015840



]

(12)





















(a) (b)



Data Availability




Acknowledgments


References














































Journal of












Journal of







Volume 2018






Volume 2018












(4)

















(9)



(10)



(11)










Input






[119860 exp minus119908ℎ1198771015840



]

(12)





















(a) (b)



Data Availability




Acknowledgments


References














































Journal of












Journal of







Volume 2018






Volume 2018
















Input






[119860 exp minus119908ℎ1198771015840



]

(12)





















(a) (b)



Data Availability




Acknowledgments


References














































Journal of












Journal of







Volume 2018






Volume 2018




















(a) (b)



Data Availability




Acknowledgments


References














































Journal of












Journal of







Volume 2018






Volume 2018







































Journal of












Journal of







Volume 2018






Volume 2018








an analytical method for reducing metal artifacts in x-ray...

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