empirical beam hardening correction
TRANSCRIPT
Empirical beam hardening correction „EBHC… for CTYiannis Kyriakou, Esther Meyer, Daniel Prell, and Marc Kachelrießa�
Institute of Medical Physics, University of Erlangen–Nürnberg, 91052 Erlangen, Germany
�Received 19 May 2010; revised 1 July 2010; accepted for publication 13 July 2010;published 8 September 2010�
Purpose: Due to x-ray beam polychromaticity and scattered radiation, attenuation measurementstend to be underestimated. Cupping and beam hardening artifacts become apparent in the recon-structed CT images. If only one material such as water, for example, is present, these artifacts canbe reduced by precorrecting the rawdata. Higher order beam hardening artifacts, as they result whena mixture of materials such as water and bone, or water and bone and iodine is present, require aniterative beam hardening correction where the image is segmented into different materials and thoseare forward projected to obtain new rawdata. Typically, the forward projection must correctly modelthe beam polychromaticity and account for all physical effects, including the energy dependence ofthe assumed materials in the patient, the detector response, and others. We propose a new algorithmthat does not require any knowledge about spectra or attenuation coefficients and that does not needto be calibrated. The proposed method corrects beam hardening in single energy CT data.Methods: The only a priori knowledge entering EBHC is the segmentation of the object intodifferent materials. Materials other than water are segmented from the original image, e.g., by usingsimple thresholding. Then, a �monochromatic� forward projection of these other materials is per-formed. The measured rawdata and the forward projected material-specific rawdata are monomiallycombined �e.g., multiplied or squared� and reconstructed to yield a set of correction volumes. Theseare then linearly combined and added to the original volume. The combination weights are deter-mined to maximize the flatness of the new and corrected volume. EBHC is evaluated using dataacquired with a modern cone-beam dual-source spiral CT scanner �Somatom Definition Flash,Siemens Healthcare, Forchheim, Germany�, with a modern dual-source micro-CT scanner �Tomo-Scope Synergy Twin, CT Imaging GmbH, Erlangen, Germany�, and with a modern C-arm CTscanner �Axiom Artis dTA, Siemens Healthcare, Forchheim, Germany�. A large variety of phantom,small animal, and patient data were used to demonstrate the data and system independence ofEBHC.Results: Although no physics apart from the initial segmentation procedure enter the correctionprocess, beam hardening artifacts were significantly reduced by EBHC. The image quality forclinical CT, micro-CT, and C-arm CT was highly improved. Only in the case of C-arm CT, wherehigh scatter levels and calibration errors occur, the relative improvement was smaller.Conclusions: The empirical beam hardening correction is an interesting alternative to conventionaliterative higher order beam hardening correction algorithms. It does not tend to over- or undercor-rect the data. Apart from the segmentation step, EBHC does not require assumptions on the spectraor on the type of material involved. Potentially, it can therefore be applied to any CTimage. © 2010 American Association of Physicists in Medicine. �DOI: 10.1118/1.3477088�
Key words: Computed tomography �CT�, beam hardening correction, x-ray, scatter
I. INTRODUCTION
X-rays typically form a polychromatic spectrum where thetotal attenuation does not follow the simple exponential lawbut is a superposition of various exponentials.
The log attenuation, as it is used for CT, is given as
q�L� = − ln � dEw�L,E�e−�0�d���E,s+���. �1�
Here, L is the line of integration corresponding to the raywhich is parameterized by s+��, where s is the vector of thesource and � is the direction vector of the ray L. ��E ,r� isthe energy-dependent spatial distribution of the linear attenu-ation coefficient. E is the photon energy, r the spatial vari-
able, and w�L ,E� is the detected spectrum �normalized toarea 1� comprising the distribution of the emitted x-rays,prefiltration, shaped filtration, and the sensitivity of the de-tector. Note that the detected spectrum is angular dependent,and thus a function of the ray position L. For convenience,we will drop L from the following equations.
In CT, one seeks to assess � by acquiring data over vari-ous lines L followed by image reconstruction. Due to thenonlinearity of Eq. �1�, one will observe significant artifactswhen approximating the measurement by the x-ray trans-form. Therefore, dedicated first, second, or higher order cor-rection algorithms have been designed and there is a vastamount of literature describing these, e.g., Refs. 1–23.
All these approaches, including the one presented in this
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paper, require some a priori knowledge. The most commonway is to assume the decomposition ��E ,r�= f0�r��0�E�,where �0�E� is the energy dependence of the most prominentmaterial in the object. In our case this is the energy depen-dence of water and the correction that is applied in all clini-cal CT scanners is known as water precorrection.5,7,13,18,19,23
The aim of the water precorrection is to reconstruct f0�r�,where f0 is designed such that the measurement q is repro-duced by the polychromatic x-ray transform of f0,
q = − ln� dEw�E�e−p0�0�E� with p0 = �0
�
d�f0�s + ��� ,
�2�
being the �monochromatic� x-ray transform of f0�r�.In clinical CT, �0�E�=�water�E�. An important preprocess-
ing step before image reconstruction is the water precorrec-tion which inverts the function q�p0� to obtain p0 for a mea-sured value q. This inversion is usually performedempirically instead of doing an analytic water precorrection.The reasons are insufficient knowledge of w�E�, cupping dueto scattered radiation and general calibration issues. A verysimple but highly efficient empirical cupping correction al-gorithm is described in Ref. 24.
Since one can trade an arbitrary constant between thefunctions f0�r� and �0�E�, there are multiple ways of how tointerpret their physical meaning. Let us give three typicalexamples, while keeping in mind that the linear attenuationcoefficient ��E ,r�= f0�r��0�E� has units 1/length.
• Assume f0�r� shall be interpreted as a density image, inunits of mass/volume. Then the quantity �0�E� is themass attenuation coefficient �� /���E�, which has unitsof area/mass.
• If one desires f0�r� to be a density image relative to thedensity of a dominant material, water, for example, thenf0�r� is unitless and �0�E� needs to be interpreted asmass attenuation multiplied by the density of water, i.e.,�0�E� is the linear attenuation coefficient of water.
• To interpret f0�r� as a monochromatic linear attenuationimage at a certain energy E0, one has to interpret �0�E�as relative linear attenuation ��E� /��E0�. In this case,the units of f0 are 1/length, while �0 is unitless.
The second example is most similar to the Hounsfieldscale where all values are given relative to the density ofwater. The last example is what would be used in PET/CT toconvert the data to 511 keV for attenuation correction. Nomatter which interpretation is desired, there is no differencein how empirical beam hardening correction �EBHC� per-forms.
Whenever the assumption of having a water-equivalentobject is not valid, beam hardening artifacts will appear, evenwhen water precorrection was performed �Fig. 1�. Oneprominent example are CT images of the pelvis or of thehead. In those images dark streaks appear between bones ofhigh density. Especially in the femur region, images show adark band between both femurs. One reason for this behavior
is the different energy dependence of bone, i.e., the assump-tion �water is similar to �bone is not justified anymore.
EBHC is an empirical method to correct for such higherorder beam hardening artifacts that appear if two or moredifferent materials are present.25 EBHC does not make anyassumptions on attenuation coefficients, spectra, detector re-sponses, or other physical properties of the scanner. Sincemany of our examples use data from dual source CT scan-ners, it should be emphasized that EBHC is a single energycorrection method and that all our dual source scans wereperformed with the same tube voltage in both x-ray sources.
II. MATERIALS AND METHOD
This section provides the theory and motivates the EBHCidea. The different EBHC steps are then detailed in Secs.II A–II E. First, let us start with a short summary of whatEBHC does.
EBHC takes either water-precorrected rawdata p0 or animage f0 or a volume as input. If rawdata are the input, avolume is reconstructed from those rawdata. If images arethe input, rawdata can be obtained by a forward projection.The first EBHC step is to segment the image into low andhigh density materials. Our examples use water and bone.Subsequently, rawdata of the segmented high density mate-rial �bone� are obtained by forward projection. The obtainedrawdata are monomially combined with the original rawdata.Reconstruction of these new rawdata yields several basis im-ages �or basis volumes�, which are used to correct the beamhardening artifacts in the original image. The correction is alinear combination of the original image and the basis im-ages. The linear combination coefficients are chosen to maxi-mize the flatness and thereby minimize the artifact content ofthe resulting image.
For convenience, we restrict the derivation of EBHC tothe situation of two materials. The method can be general-ized to handling more than two materials. This may be im-portant when water, bone, and iodine are present in theobject.26,27
Let the object be composed of two materials f1�r��1�E�+ f2�r��2�E�, which we may call water and bone in the fol-
FIG. 1. Due to beam polychromaticity dark streaks �marked with arrows�appear between objects that are of higher atomic number than water. Thesebeam hardening artifacts are of higher order than what the typical waterprecorrection can account for. The image on the left hand side shows a watercylinder with two inserts: on the right hand side, an image of the base of theskull is shown.
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lowing. We assume that �1�E�=�0�E�, which means that onematerial’s energy dependence is identical to the energy de-pendence, the precorrection was performed for �preferably, itshould be the most prominent material�. We further assumethat �i�E0�=1 at the desired display energy E0. From Sec. I,we know that this can be always achieved by scaling f i and�i appropriately while keeping the product f i�i constant.
Since the images or rawdata are water precorrected �i.e.,precorrected for the energy dependence �1=�0�, the follow-ing decomposition into an effective water-equivalent density
f̂1�r� of the object and into an effective energy dependence
�̂2�E� of bone is helpful,
��r,E� = f1�r��1�E� + f2�r��2�E�
= �f1�r� + f2�r���1�E� + f2�r���2�E� − �1�E��
= f̂1�r��1�E� + f2�r��̂2�E� .
Now, the polychromatic x-ray transform of our object isgiven as
q = − ln� dEw�E�e−p̂1�1�E�−p2�̂2�E�, �3�
where p̂1 and p2 are the line integrals through f̂1�r� and f2�r�,respectively.
Combining Eqs. �2� and �3� yields
� dEw�E�e−p0�0�E� =� dEw�E�e−p̂1�1�E�−p2�̂2�E�.
What we would like to display is the beam hardening
artifact-free image f̂1�r�= f1�r�+ f2�r�, where we used�1�E0�=�2�E0�=1. Hence, we need to solve for p̂1. A seriesexpansion yields
p̂1�p0,p2� = �ij
cijp0i p2
j .
Regard a ray which penetrates no material other than water-equivalent tissue. Then p2=0 for this ray. Since we defined�1=�0, and since we assumed the initial CT image to bewater precorrected, we must have p̂1=p0. This implies thatc10=1 and that ci0=0 for i�1. What remains is
p̂1�p0,p2� = p0 + c01p2 + c11p0p2 + c02p22 + ¯ , �4�
where the ellipsis denote terms of order three or higher.We assume that one can segment the initial image f0�r�
into contributions of water and bone. Forward projecting thesegmented bone image will then yield a good approximationto the rawdata p2. From Eq. �4� we know that we must com-pute the rawdata sets p0p2 and p2
2. We reconstruct them andobtain two additional volumes f11�r� and f02�r�, as illustratedin Fig. 2. We then empirically determine the unknown coef-ficients c01, c11, and c02 that yield the best image,
f̂1�r� = f0�r� + c01f01�r� + c11f11�r� + c02f02�r� . �5�
The image f̂1 is the corrected and true density image�given that we interpret the images as density images�, whilein f0 the water density is quite accurate, but the bones andthe dark streaks between them are regions with wrong den-sity information.
The next subsections detail the EBHC steps: segmenta-tion, forward projection, rawdata combination, reconstruc-tion of the basis images, and linear combination to maximizeflatness.
II.A. Segmentation
To demonstrate the performance we have to agree uponsome segmentation algorithm. For simplicity, we use thesame soft threshold-based weighting approach as we did pre-viously use in Ref. 28. Voxels with CT values below a cer-tain threshold T1 are assumed to be water equivalent, voxelsabove a higher threshold T2 are assumed to be bone equiva-lent, and voxels falling between T1 and T2 are assumed to bea mixture of water and bone. The weighting functions areunitless and take values between 0 and 1. Figure 3 shows thecorresponding weighting functions wwater�f� and wbone�f�,which we use to segment the initial image f0�r� into a waterimage f0�r�wwater�f0�r�� and a bone image f0�r�wbone�f0�r��.
The threshold for water can be fixed because of the watercorrection, for example, to 100 HU. The threshold for bonewas chosen, depending on the application, roughly at 500HU. However, the choice is not as critical as if a hard thresh-old was chosen.
FIG. 2. Initial image and the corresponding correction images for the example of a rat head. X−1 is the reconstruction operator. Correction images: X−1�p2�,X−1�p0p2�, X−1�p2
2�. �W=0 HU /C=1000 HU�.
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It should be noted that more sophisticated segmentationmethods can be used together with EBHC because EBHCdoes not depend on a specific type of segmentation algo-rithm. For example, distinguishing between bone and iodinecannot be reliably done using simple thresholds and requiresother segmentation approaches, such as the dynamic ap-proach proposed in Refs. 26 and 27, for example.
II.B. Forward projection
The bone image f0�r�wbone�f0�r�� is forward projected toobtain the new rawdata p2. The water image does not need tobe forward projected. For forward projection we use a three-dimensional generalization of the two-dimensional Josephforward projector.29
II.C. Rawdata combination
Having the original, water-precorrected rawdata p0 andthe forward-projected rawdata of the second material p2
available, we can go into Eq. �4� and compute the productsp0p2 and p2
2. This multiplication has to be understood el-ementwise, i.e., each entry of the sinogram p0 is multipliedby the corresponding entry of the sinogram p2. To considerhigher order terms or more materials, which we do not dohere, additional rawdata combinations have to be computed.
II.D. Reconstruction of the basis images
The rawdata combinations are reconstructed to obtain thebasis images, an example of how such images look like isgiven in Fig. 2. In our notation—see Eqs. �4� and �5�—thebasis image f11 is computed by reconstructing p0p2 and f02
by reconstructing p22. The image f01 is already available, as it
is the segmented bone image. For the reconstruction of thebasis images, and as well for the initial reconstruction of theuncorrected images, we are using the extended parallel back-projection algorithm of Ref. 30. It can be used to reconstructspiral, circle, and sequence data and guarantees full dose
usage. For circle and sequence data, our reconstruction cor-responds to a Feldkamp-type reconstruction. For spiral data,it corresponds to the reconstruction algorithm that is imple-mented in the clinical scanner. EBHC does not depend on thespecific reconstruction algorithm. Possible cone-beam arti-facts do not influence the EBHC method. EBHC is not de-signed to remove cone-beam artifacts, but it does not makethem more severe either.
II.E. Linear combination and maximization of flatness
We want to choose the unknown coefficients c01, c11, and
c02 of Eq. �5� in a way that gives a corrected volume f̂1�r�with the least possible total variation. The rationale for thisdesire is that the beam hardening artifacts appear as bright ordark streaks or bands that increase the variation of the im-ages. We have used and compared different figures of merit,with respect to their performance in estimating artifact con-tent in the images and found the total variation �as found inRef. 31� of the volume to be an adequate cost function.Hence, we aim at minimizing the L1 norm of the absolutevalue of the gradient of a linear combination with unknowncoefficients,
T�c01,c11,c02� =� d3r�� f̂1�r��
=� d3r���f0�r� + c01f01�r� + c11f11�r�
+ c02f02�r��� .
This objective function is convex, but it is not strictly convex�see Appendix�. As a consequence, a local solution of theminimization problem is a global minimizer, but its unique-ness is not guaranteed.34 Our implementation realizes theintegral over x, y, and z as a sum over the voxel indices, andthe partial derivatives are computed using the finite differ-ence of a voxel and its lower neighbors in the x-, y-, and
FIG. 3. We use a simple image-based segmentation technique that assigns material-dependent weights to each voxel by soft thresholding. The weightingfunctions, which are shown in the graph on the left hand side, are unitless and take values between 0 and 1. Right hand side: the original image is multipliedwith the weighting functions to obtain a water image and a bone image.
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z-directions, respectively. We use the simplex algorithm32 todetermine the parameters c01, c11, and c02 that yield a mini-mum value of T.
The EBHC-corrected image is computed by adding thebasis images multiplied with their coefficients to the initialimage, according to Eq. �5�.
III. EXPERIMENTS
EBHC is evaluated using phantom and patient data ac-quired with a modern cone-beam dual-source spiral CT scan-ner �Somatom Definition Flash, Siemens Healthcare, Forch-heim, Germany�. Additionally a modern micro-CT scannerwith a flat-detector �CT Imaging GmbH, Erlangen, Ger-many� and a C-arm CT scanner �Axiom Artis dTA, SiemensHealthcare, Forchheim, Germany� were used. The C-armFDCT scanner, equipped with a large flat detector of 40�30 cm2, is afflicted with a higher scatter fraction as com-pared to the other two systems.33 Measurements were per-formed at a tube voltage of 120 kV for the clinical CT, 40 kVfor the micro-CT, and 70 kV in the case of the C-arm FDCT.No assumptions regarding spectrum, detector, or geometrywere necessary for the application of EBHC on the recon-structed images.
A variety of phantoms, animals, and patient data wereused to demonstrate the data and system independence of themethod. For the clinical CT, a water phantom with a diam-eter of 20 cm was used in combination with different inserts.The inserts are made of aluminum or hydroxyapatite �HA-200� and 25 mm in diameter. Additionally, a 20 cm PE diskphantom with HA-800 inserts �25 mm� and a 20 mm tubefilled with an iodine solution were used. Finally, a cranialscan of a patient was used for beam hardening correction, asthe skull base is a prominent example where beam hardeningartifacts occur. For comparison reasons, we scanned the samewater phantom with the C-arm CT and used previously ac-quired patient head scan. In the case of the micro-CT scan-ner, we present the correction of CT measurements of two ratheads.
IV. RESULTS
IV.A. General image quality evaluation
EBHC was evaluated using phantom and patient data,which were acquired with a cone-beam spiral CT scanner, amicro-CT scanner, and a C-arm flat-detector CT �FD-CT�scanner.
Figure 4 shows the correction results for the cylindricalwater phantom with aluminum and HA inserts and for the PEdisk phantom containing HA-800 inserts and iodine solution.In both phantoms, the prominent streaks are removed and thehomogeneity of water and PE is restored. Differences be-tween the original and the beam hardening corrected imagesare better emphasized in the subtraction image. The beamhardening correction also restores the true CT values. For thewater phantom the mean CT value in ROI1 was correctedfrom �48 HU to 2 HU. The CT values of the PE phantom,whose density is 0.93 g /cm3, were corrected in both ROIs
from �85 HU to �70 HU and �87 HU to �71 HU forROI2 and ROI3, respectively. As demonstrated with the dif-ference images, spatial resolution is preserved since no ob-ject edges or other spatial information apart from the beamhardening artifacts are visible.
Figure 5 shows two examples of micro-CT data generatedafter cranial CT scans of two rats. Because of this low energylevel, beam hardening artifacts are very prominent especially
FIG. 4. Water phantom and PE disk phantom with inserts scanned at theclinical spiral CT scanner without and with correction. Difference imagesbetween the uncorrected and the corrected images show that there is no lossof resolution. The EBHC algorithm reduced higher order beam hardeningartifacts such as dark and bright streaks in both cases. The depicted regionsof interest �#1,#2,#3� were used for the evaluation of CT numbers �C=0 HU /W=200 HU for the CT images and C=0 HU /W=100 HU for thedifference images�.
FIG. 5. Original and corrected images of two rat heads. The dark streaks,which are marked with arrows in the uncorrected images, are almost com-pletely removed by EBHC, with noise being the only remaining artifact.�C=50 HU /W=800 HU for the images on the left hand side and C=0 HU /W=1000 HU for the images on the right hand side�.
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between bony structures. The EBHC-corrected images showa significant reduction of the artifacts in both cases, confirm-ing the generality of the method.
IV.B. Influence of acquisition parameters
In this section we want to address the differences in cor-rection quality if the original data are highly disrupted byscatter and other issues such as calibration errors. This ismostly the case in C-arm CT, where water-precorrected im-ages do not achieve the image quality of clinical CT data.
Figure 6 shows a cylindrical water phantom �20 cm diam-eter� with two aluminum inserts acquired on both systemswithout and with correction. The images are rotated in orderto show the inserts at the same position in the phantom. Inthe case of spiral CT �Fig. 6�, a complete removal of beamhardening artifacts is achieved. Although a very large streakreduction is achieved in the case of the C-arm FDCT scanner�Fig. 6�, there are some differences in performance due to theincreased presence of other artifact sources in the C-arm datasuch as scatter, calibration errors, etc. For example, the pe-riphery of the water phantom is not shown at this windowingdue to overexposure effects at the detector and two darkstreaks in the upper part of the phantom are due to the trun-cation of the patient table in some projections. Overexposurealso affects the patient head scan, where the periphery isdepicted with lower CT values. As shown in Fig. 6 for theexample of head scans, the beam hardening artifacts areclearly reduced.
For the same phantoms and patient data sets in the clinicalCT case, we counted the number of iterations necessary toachieve a sufficient convergence. One iteration consists ofthe linear combination of the volumes and the calculation ofthe total variation. After approximately 25–50 iterations, a
reliable convergence was achieved in the examined cases. Ofcourse the time needed for the generation of the correctionimages, which requires one forward projection of the bonevoxels and two filtered backprojections, has to be taken intoaccount as well. Thanks to the availability of modern hard-ware architectures, such as the CBE, the CPU, the GPU, orthe Larrabee, this additional computational overhead shouldnot be a restriction in practice.
IV.C. Comparison to iterative BHC „IBHC…
Figure 7 compares the performance of our empirical ap-proach EBHC with the gold standard iterative beam harden-ing correction �IBHC�. For IBHC we use our implementationthat was published in Ref. 28. A water phantom containingthree HA-800 inserts was simulated. The figure displays theuncorrected and corrected CT images along with the corre-sponding difference images. In the simulation, beam harden-ing and noise were considered; the rawdata were prepro-cessed with an analytical water precorrection. In the EBHC-corrected image in Fig. 7, dark and bright streaks due tobeam hardening are accurately removed, while they areclearly visible in the uncorrected image. The EBHC resultsare nearly identical to the IBHC method shown in Fig. 7.
In this simulated case, where we exactly know the spec-trum, the detector efficiency and the attenuation coefficientsinvolved, and where we provide this knowledge to IBHC, theiterative beam hardening correction cannot be outperformedby any other method. Nevertheless, any slight deviation fromthis exact knowledge will cause an insufficient IBHC correc-tion. To demonstrate this we deliberately modified the spec-trum that we made available to IBHC by adding 6 mm alu-minum to the prefiltration. In our opinion this is a very docilechange compared to the information typically available from
FIG. 6. Phantom and patient head images without correction and with correction for spiral CT and C-arm FD-CT. The dark streaks between the aluminuminserts and the bones at the skull base are almost completely removed by the EBHC. Even with correction, the image quality is still lower in the case of C-armCT due to scatter and other calibration issues �C=0 HU /W=400 HU for the phantom acquired with spiral CT, C=50 HU /W=180 HU for the headacquired with spiral CT, and C=0 HU /W=800 HU for the images acquired with C-arm CT�.
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real physical scanners. Even this slightly wrong assumptionon the spectrum can cause an insufficient IBHC correction asshown in Fig. 7, denoted as IBHC�. Other error sourceswhich are present in measured images, such as scatter, wouldlead to even worse results for the IBHC.
V. SUMMARY AND DISCUSSION
We have developed a beam hardening correction algo-rithm that empirically corrects for higher order beam hard-ening artifacts. The advantage of EBHC is the correction ofbeam hardening artifacts without the need of a priori infor-mation of tube spectra, prefiltration, detector efficiency, oreven attenuation coefficients. EBHC provides results compa-rable to or better than the gold standard iterative beam hard-ening correction �which fails in the case of inadequate as-sumptions on the spectra and attenuation coefficients�. Thesegmentation scheme used in this paper is simple and onlytwo materials were considered, here. For realistic applica-tions it may be replaced by more sophisticated segmentationtechniques such as segmentation of contrast agent, bone, wa-ter, and even metals.
Just as other higher order beam hardening correction al-gorithms, EBHC requires a forward projection and a filteredbackprojection step which, sometimes, are believed toslightly reduce spatial resolution. While this potential loss inspatial resolution can be avoided by fine tuning the forwardand backprojectors and the convolution kernel involved inthe filtered backprojection step, we did not do so because thecorrection image that is added to the original image is onlyof small magnitude compared to the CT values in the originalimage. Therefore, spatial resolution and object edges are wellpreserved. EBHC can therefore be applied to any high reso-lution reconstruction without compromising image resolutionor details.
EBHC can be generalized to more than two materials eas-ily. In the case of three materials, water, bone and iodine, forexample, Eq. �4� becomes
p̂1�p0,p2,p3� = p0 + c010p2 + c110p0p2 + c020p22 + c001p3
+ c101p0p3 + c002p32 + c011p2p3 + ¯ ,
and Eq. �5� would generalize accordingly. The extension tofour or more materials is straightforward now.
EBHC is designed to remove higher order beam harden-ing artifacts but not the global cupping artifacts, which arealready removed by the water precorrection. Thus, it is im-portant to use water-precorrected rawdata or images as inputfor EBHC, which is the case for data from clinical CT scan-ners. One may extend EBHC to also remove cupping, i.e., toincorporate the water precorrection into EBHC. This couldbe done by allowing more coefficients in the polynomialequation as degrees of freedom instead of setting them to 0or 1.
VI. CONCLUSIONS
EBHC provides an efficient reduction of beam hardeningartifacts, without any assumptions on the spectra or the typeof materials involved. Therefore, it can also be appliedpurely image based to any CT image without knowledgeabout the scanner, e.g., as a standalone application. The onlya priori information required for EBHC is the segmentationof the object into water and bone. This, however, is not criti-cal in medical CT where a simple thresholding techniqueappears to be sufficient for most cases. Note that the samesegmentation is required for all other higher order beamhardening correction techniques published so far.
Potential applications for EBHC are manifold. In clinicalCT all quantitative studies and many qualitative studies will
FIG. 7. Simulated water phantom with HA-800 inserts without correction and with different correction methods. The EBHC algorithm reduced higher orderbeam hardening artifacts such as dark and bright streaks and is comparable to the optimally configured IBHC. If the assumptions on the spectrum are slightlywrong, IBHC� does not provide satisfactory results. The difference images show the corrected images minus the uncorrected image �C=0 HU /W=200 HU for both the CT images and the difference images�.
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benefit from a robust removal of beam hardening artifacts. Inradiation therapy treatment planning, the planning CTs aswell as the cone-beam CT scans acquired during planningwill definitely profit from the improved CT value accuracy.Also interventional CT studies can benefit from reducedbeam hardening artifacts. Additionally, last but not least,many dual energy CT exams that are based on a linear com-bination of reconstructed images, because the manufacturerdoes not provide any rawdatabased dual energy material de-composition, can definitely profit from a routine method toremove beam hardening from these basis images.
ACKNOWLEDGMENTS
This work was supported in parts by CT Imaging GmbH,Erlangen, Germany, and in parts by the DeutscheForschungsgemeinschaft �DFG� under Grant No. FOR 661.
APPENDIX: CONVEXITY OF THE OBJECTIVEFUNCTION
A local minimum of a convex function is also a globalminimum.34 We prove the convexity of the objective func-tion of Sec. II E in a general form, independent of the num-ber of unknown linear combination coefficients. For c�RN
being the vector of N unknowns, the objective function T isdefined as
T�c� =� d3r�n=1
N
cn � fn�r� .
To prove convexity, let w� �0,1� and a ,b�RN. Making useof the triangle inequality, we find
T�wa + �1 − w�b� =� d3rw�n
an � fn�r�
+ �1 − w��n
bn � fn�r�� d3rw�
n
an � fn�r�+� d3r�1 − w��
n
bn � fn�r�= w� d3r�
n
an � fn�r�+ �1 − w�� d3r�
n
bn � fn�r�= wT�a� + �1 − w�T�b� .
Hence, T�c� is convex.
a�Author to whom correspondence should be addressed. Electronic mail:[email protected]; Telephone: 49 �9131� 8522957; Fax: 49 �9131� 85 22824.
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