Suppression of Cone-Beam Artefacts with Direct Iterative Reconstruction Computed Tomography Trajectories (DIRECTT)

Journal of Imaging

Article Suppression of Cone-Beam Artefacts with Direct Iterative Reconstruction Computed Tomography Trajectories (DIRECTT)

Sotirios Magkos 1,* , Andreas Kupsch 1 and Giovanni Bruno 1,2

1 Bundesanstalt für Materialforschung und -prüfung (BAM), Unter den Eichen 87, 12205 Berlin, Germany; [email protected] (A.K.); [email protected] (G.B.) 2 Institute of Physics and Astronomy, University of Potsdam, Karl-Liebknecht-Str. 24-25, 14476 Potsdam, Germany * Correspondence: [email protected]; Tel.: +49-30-81044463

Abstract: The reconstruction of cone-beam computed tomography data using ﬁltered back-projection algorithms unavoidably results in severe artefacts. We describe how the Direct Iterative Reconstruc- tion of Computed Tomography Trajectories (DIRECTT) algorithm can be combined with a model of the artefacts for the reconstruction of such data. The implementation of DIRECTT results in reconstructed volumes of superior quality compared to the conventional algorithms.

Keywords: iteration method; signal processing; X-ray imaging; computed tomography

1. Introduction

Citation: Magkos, S.; Kupsch, A.; Computed tomography (CT) systems for non-destructive testing and material analysis Bruno, G. Suppression of Cone-Beam generally use a cone beam on a sample that rotates in a circular orbit [1], with cylindrical Artefacts with Direct Iterative samples being among the most common [1–4]. The exact reconstruction of data acquired Reconstruction Computed during such a measurement is not possible because the geometry does not satisfy Tuy’s Tomography Trajectories (DIRECTT). sufficiency condition [5]. This is demonstrated in Figure1 with the reconstruction of a J. Imaging 2021, 7, 147. https:// concrete rod by the commonly used algorithm developed by Feldkamp, Davis and Kress doi.org/10.3390/jimaging7080147 (FDK) [6]. For higher cone angles, there is a decrease of the grey values, which represent the attenuation coefficient µ, and of the image quality in the direction of the rotation axis Academic Editors: Maria Pia Morigi (z-axis in Figure1). and Fauzia Albertin Several algorithms have been proposed to reduce such artefacts. Hsieh proposed a two-pass algorithm that estimates the cone-beam artefacts from the segmented high- Received: 5 July 2021 density material and then subtracts them from the FDK reconstruction [7]. Han and Accepted: 13 August 2021 Baek went further by devising a multi-pass approach that they tested for larger cone Published: 15 August 2021 angles and different material densities [8]. Maaß et al. proposed an iterative algorithm that also subtracts the estimated artefacts from the FDK reconstruction without requiring Publisher’s Note: MDPI stays neutral segmentation [9]. with regard to jurisdictional claims in Here, we will describe how we have adjusted the Direct Iterative Reconstruction published maps and institutional affil- of Computed Tomography Trajectories (DIRECTT) algorithm [10–12] to estimate such iations. artefacts and compensate for them.

Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

J. Imaging 2021, 7, 147. https://doi.org/10.3390/jimaging7080147 https://www.mdpi.com/journal/jimaging J. Imaging 2021, 7, 147 2 of 9 J.J. Imaging Imaging 2021 2021, 7, ,7 x, xFOR FOR PEER PEER REVIEW REVIEW 2 2of of 9 9

FigureFigureFigure 1. 1.1. OrthogonalOrthogonal Orthogonal slices slicesslices through throughthrough the thethe volume volumevolume of ofof a aa concrete concreteconcrete ro rodrodd as asas reconstructed reconstructedreconstructed by byby Feldkamp, Feldkamp,Feldkamp, Davis DavisDavis and andand Kress KressKress (FDK). (FDK).(FDK). TheTheThe orange orange lines lines indicate indicate the the relati relative relativeve position position of of the the cross cross sections. sections.

2.2.2. Materials MaterialsMaterials and andand Methods MethodsMethods 2.1.2.1.2.1. Sample SampleSample Images ImagesImages AAA set setset of ofof 3000 30003000 cone-beam cone-beamcone-beam projections projectionsprojections of ofof the thethe concrete concreteconcrete rod rodrod of ofof Figure FigureFigure 1 1was was acquired acquired over over ◦ 360°360360° on onon an anan in-house in-housein-house GE GEGE v|tome|x v|tome|xv|tome|x L LL 300 300300 scanner. scanner.scanner. A AA 2024 20242024 × × 2024 2024 PerkinElmer PerkinElmer detector detectordetector with withwith aaa pixel pixelpixel sizesize ofof of 0.20.2 0.2 mmmm mm was was was used. used. used. Source-object Source-obj Source-objectect and and and source-detector source-detector source-detector distances distances distances of 81of of mm81 81 mm andmm and1018and 1018 1018 mm, mm, mm, respectively, respectively, respectively, resulted result result ineded a magnificationin in a a magnification magnification of 12.5 of of 12.5 for 12.5 a for voxelfor a a voxel voxel size size of size 0.016 of of 0.016 0.016 mm. mm. mm. The µ ThevoltageThe voltage voltage and and currentand current current settings settings settings of theof of th sourcethee source source were were were set set toset 140to to 140 140 kV kV kV and and and 80 80 80A, μ μA, respectively.A, respectively. respectively. A A0.5A 0.5 0.5 mm mm mm Cu Cu Cu prefilter prefilter prefilter was was was used. used. used. The The The acquisition acqu acquisitionisition time time time per per per projection projection projection was was was 6 s.6 6 s. s. TheTheThe geometry geometrygeometry of of the the CT CT CT scan scan scan of of of the the the cylindrical cylindrical cylindrical sample sample sample is is represented, isrepresented, represented, not not notto to scale, toscale, scale, by by by Figure2. The orange cone represents the field of view (FoV), while the blue dashed FigureFigure 2. 2. The The orange orange cone cone represents represents the the field field of of view view (FoV), (FoV), while while the the blue blue dashed dashed lines lines lines represent rays that traverse the front and rear edges of the sample. Near the lower representrepresent rays rays that that traverse traverse the the front front and and rear rear edges edges of of the the sample. sample. Near Near the the lower lower edge edge edge of the FoV, an inverse conical area of the sample is defined by the solid and dashed ofof the the FoV, FoV, an an inverse inverse conical conical area area of of the the sample sample is is defined defined by by the the solid solid and and dashed dashed orange orange orange lines. This is the part of the sample that lies within the FoV during only some of the lines.lines. This This is is the the part part of of the the sample sample that that lies lies within within the the FoV FoV during during only only some some of of the the projections and, therefore, is not fully reconstructible by FDK [13]. projectionsprojections and, and, therefore, therefore, is is not not fully fully reconstructible reconstructible by by FDK FDK [13]. [13].

FigureFigure 2. 2. Geometric Geometric representation representation of of the the computed computed tomography tomography (CT) (CT) scan scan (symmetric (symmetric with with Figure 2. Geometric representation of the computed tomography (CT) scan (symmetric with respect respectrespect to to the the central central plane plane SOD). SOD). to the central plane SOD). We consider the case that dimensions of the sample are not known precisely. It is WeWe considerconsider thethe casecase thatthat dimensionsdimensions ofof thethe samplesample areare notnot knownknown precisely.precisely. ItIt isis possible to determine them from the projections. The total height h of the sample within possiblepossible toto determinedetermine themthem fromfrom thethe projections.projections. TheThe totaltotal heightheight hh ofof thethe samplesample withinwithin 1 2 thethe FoV FoV is is the the sum sum of of its its parts parts h h 1and and h h 2that that extend extend respectively respectively above above and and below below the the plane plane the FoV is the sum of its parts h1 and h2 that extend respectively above and below the plane

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SOD. The plane is deﬁned by the source (S), the centre of rotation (O) and the centre of the detector (D). The plane SOD is assumed to be perpendicular to the detector. We can see from Figure2 that,

SO − r h = 1 (1) SD DF and, SO + r h = 1 (2) SD DR where r is the radius of the sample, and DF and DR the distance between the central detector row and the detector rows where the front (F) and rear (R) edge of the concrete rod are respectively projected. Dividing Equation (1) by Equation (2) and rearranging, we obtain:

DF/DR − 1 r = · SO. (3) DF/DR + 1

The length h1 can be calculated now from either Equation (1) or Equation (2), while the length h2 is: SO + r H h = · (4) 2 SD 2 where H is the height of the detector. The part of the sample that, as mentioned above, does not always lie within the FoV has been accounted for through the inclusion of the radius r in Equation (4).

2.2. The Direct Iterative Reconstruction of Computed Tomography Trajectories (DIRECTT) Algorithm The DIRECTT algorithm was first proposed for the reconstruction of two-dimensional (2D) images by Lange et al. and, in a previous article [12], we introduced a new, more efficient, and fully 3D version. The algorithm operates on finding the best solution possible by mimicking the actual physical projection process, instead of directly solving the inverse problem. It only reconstructs certain voxels during each iteration, simulating the projection of the partial reconstruction, and repeating the workflow for the difference between measured and simulated projections until this difference is sufficiently close to zero [10–12]. Although the concrete rod of Figure1 was one of the two datasets that were used to showcase the performance of DIRECTT, the algorithm was implemented only on the slice that corresponds to the cross section of the sample with the plane SOD [12]. That slice will be hereafter referred to as the central slice. Attempting to implement DIRECTT on the whole dataset does not lead to an improvement over the FDK reconstruction of Figure1 . On the contrary, it results in severe artefacts and missing data because the algorithm fails to predict the decreasing grey values along the z-axis. However, these artefacts can be reduced by modelling them based on the shape of the sample.

2.3. Software For this work, the forward- and back-projection operations involved in DIRECTT were performed using the Python programming language and the open-source ASTRA (All Scale Tomographic Reconstruction Antwerp) toolbox [14]. Via ASTRA, computation- ally demanding operations are ofﬂoaded to a graphics processing unit using the CUDA (Compute Uniﬁed Device Architecture) language. The toolbox also includes several reconstruction algorithms, such as the FDK, the simultaneous iterative reconstruction technique (SIRT) [15], and a conjugate gradient (CG) method based on the Krylov subspace [16], that can run with little input from the user [14]. J. Imaging 2021, 7, x FOR PEER REVIEW 4 of 9

3. Results A single measured projection of the concrete and the corresponding simulated projection of a virtual homogeneous cylinder of height h and radius r are shown in Figure 3a,b, respectively. The two horizontal lines in the former indicate the detector rows that correspond to points F and R of Figure 2. The two rows are identified automatically from the projections. Specifically, F is the lowermost detector row containing exclusively values that correspond to the background. Similarly, R is the lowermost row containing exclusively values lower than the mode of all absorption projections, which roughly corre- J. Imaging 2021, 7, 147 sponds to the absorption of rays that penetrate the sample perpendicularly.4 of 9 By simulating the geometry of the scan, projecting and back-projecting the virtual cylinder, and normalizing for the central slice, a 3D model M of the artefacts that arise during the reconstruction is computed. Normalizing for the central slice ensures that 3. Results hardly any artefacts are considered for the parts of the volume that always lie within the A single measuredFoV and projectioncan be accurately of the concretereconstructed and theby FDK. corresponding Note that the simulated model M projec- needs to be com- tion of a virtualputed homogeneous just once. cylinderThe slices of of height M thath correspondand radius tor arethose shown in Figure in Figure 1 are 3showna,b, in the top row of Figure 4. The volume resulting from the back-projection of the unfiltered projec- respectively. The two horizontal lines in the former indicate the detector rows that corre- tions of the concrete rod, the first step during each iteration of DIRECTT [12], is shown in spond to points F and R of Figure2. The two rows are identiﬁed automatically from the the bottom row of Figure 4. Both volumes of Figure 4 comprise more slices along the z- projections. Speciﬁcally, F is the lowermost detector row containing exclusively values that axis than that of Figure 1. The extra slices have been included because it is essential for correspond to the background. Similarly, R is the lowermost row containing exclusively the implementation of any iterative reconstruction algorithm, such as DIRECTT, that the values lower than the mode of all absorption projections, which roughly corresponds to parts of the sample that do not lie within the FoV for every projection are reconstructed the absorption of rays that penetrate the sample perpendicularly. too.

Figure 3. Figure(a) Projection 3. (a) Projectionof the concrete of the rod. concrete The detector rod. Therows detector on which rows the front on which (F) and the rear front (R) (F)edge and of the rear concrete (R) rod are respectivelyedge of projected the concrete are marked rod are by respectively horizontal lines. projected (b) Simulated are marked projection by horizontal of a homogeneous lines. (b) Simulatedcylinder of dimensions equal to the rod. The greyscale values in both figures correspond to the attenuation integral values μd. projection of a homogeneous cylinder of dimensions equal to the rod. The greyscale values in both ﬁgures correspond to the attenuation integral values µd.

By simulating the geometry of the scan, projecting and back-projecting the virtual cylinder, and normalizing for the central slice, a 3D model M of the artefacts that arise during the reconstruction is computed. Normalizing for the central slice ensures that hardly any artefacts are considered for the parts of the volume that always lie within the FoV and can be accurately reconstructed by FDK. Note that the model M needs to be computed just once. The slices of M that correspond to those in Figure1 are shown in the top row of Figure4. The volume resulting from the back-projection of the unfiltered projections of the concrete rod, the first step during each iteration of DIRECTT [12], is shown in the bottom row of Figure4. Both volumes of Figure4 comprise more slices along the z-axis than that of Figure1. The extra slices have been included because it is essential for the implementation of any iterative reconstruction algorithm, such as DIRECTT, that the parts of the sample that do not lie within the FoV for every projection are reconstructed too. There is an obvious qualitative relation between the two rows of Figure4. The pixel- by-pixel division of the reconstructed volume by the model M results in a “corrected” volume, on which DIRECTT can be successfully implemented. The threshold values, based on which the voxels to be reconstructed during each iteration are selected, are calculated from the central slice as described in [12] but are applied simultaneously on the whole volume. The algorithm terminates when the projections of the reconstruction array match the measured ones. The final reconstructed volume is shown in Figure5. J. Imaging 2021, 7, x FOR PEER REVIEW 5 of 9

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Figure 4. Top row: Orthogonal slices through the model M of the artefacts that arise during the reconstruction; Bottom row: Orthogonal slices through the volume resulting from the back-projection of the unfiltered projections of the concrete rod. The extent of the volume of Figure 1 along the z-axis is indicated by the red dashed lines. Apart from that, the position of the slices shown in either row corresponds precisely to that of the slices in Figure 1.

There is an obvious qualitative relation between the two rows of Figure 4. The pixel-

by-pixel division of the reconstructed volume by the model M results in a “corrected” volume,FigureFigure 4.4. on TopTop which row:row: OrthogonalOrthogonalDIRECTT slicesslicescan be throughthrough successfu thethe modelmodellly implemented. MM ofof thethe artefactsartefacts The thatthatthreshold arisearise duringduring values, thethe reconstruction; Bottom row: Orthogonal slices through the volume resulting from the back-projec- basedreconstruction; on which Bottom the voxels row: Orthogonal to be reconstructed slices through during the volume each resultingiteration from are theselected, back-projection are cal- tion of the unfiltered projections of the concrete rod. The extent of the volume of Figure 1 along the culatedof the unﬁltered from the projections central slice of the as concrete described rod. Thein [12] extent but of are the applied volume of simultaneously Figure1 along the on z-axis the z-axis is indicated by the red dashed lines. Apart from that, the position of the slices shown in either wholeisrow indicated corresponds volume. by the The precisely red algorithm dashed to that lines. terminates of Apartthe slices from when in that, Figure the the projections1. position of theof the slices reconstruction shown in either array row matchcorresponds the measured precisely toones. that The of the final slices reconstructed in Figure1. volume is shown in Figure 5. There is an obvious qualitative relation between the two rows of Figure 4. The pixel- by-pixel division of the reconstructed volume by the model M results in a “corrected” volume, on which DIRECTT can be successfully implemented. The threshold values, based on which the voxels to be reconstructed during each iteration are selected, are calculated from the central slice as described in [12] but are applied simultaneously on the whole volume. The algorithm terminates when the projections of the reconstruction array match the measured ones. The final reconstructed volume is shown in Figure 5.

FigureFigure 5. 5. FinalFinal reconstruction reconstruction of of the the concrete concrete rod rod as as computed computed by by DIRECTT. DIRECTT. The volume has has the the same same size size as as the the volume volume shown in Figure 1 through omission of its upper and lower slices. The position of the slices shown corresponds precisely shown in Figure1 through omission of its upper and lower slices. The position of the slices shown corresponds precisely to to that of the slices in Figure 1. that of the slices in Figure1. 4. Discussion and Conclusions 4. Discussion and Conclusions The reconstruction by DIRECTT is an improvement over that by FDK in Figure 1 as The reconstruction by DIRECTT is an improvement over that by FDK in Figure1 as therethere are are no no artefacts artefacts linked linked to to the the increasing increasing cone cone angle. angle. La Lackingcking the the ground ground truth truth for for the the Figure 5. Final reconstructionreconstructedreconstructed of the concrete volume, rod as thecomputed evaluationevaluation by DIRECTT. ofof thethe twotw Theo algorithms algorithmsvolume has using theusing same a a full-reference full-referencesize as the volume metric metric is shown in Figure 1 throughis meaningless.omission meaningless. of its Nevertheless,upper Nevertheless, and lower a quantitativeslices. a quantitati The position evaluationve ofevaluation the ofslices the shown algorithmsof the corresponds algorithms can be precisely undertaken can be to that of the slices in Figureby 1.calculating the histogram entropy of the respective volumes. The histogram entropy (HE) is a global metric deﬁned according to the relation: 4. Discussion and Conclusions Z The reconstruction by DIRECTTHE = is pan(µ improvement) log[p(µ)]dµ, over that by FDK in Figure 1 (5) as there are no artefacts linked to the increasing cone angle. Lacking the ground truth for the reconstructed volume, the evaluation of the two algorithms using a full-reference metric is meaningless. Nevertheless, a quantitative evaluation of the algorithms can be

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J. Imaging 2021, 7, 147 undertaken by calculating the histogram entropy of the respective volumes.6 of 9 The histogram entropy (HE) is a global metric defined according to the relation:

HE =𝑝𝜇 log𝑝𝜇 d𝜇, (5) where p(µ) is the distribution function of the grey values. The value of the HE increases if homogeneouslywhere distributed p(μ) is the noise distribution is present function in the image, of the andgrey decreases values. The if sharpvalue edgesof the areHE increases if present [17]. Therefore,homogeneously a low valuedistributed is an noise indication is present of a goodin the balanceimage, and between decreases noise if andsharp edges are blur. The FDK-present and DIRECTT-reconstructed [17]. Therefore, a low value volumes is an (Figuresindication1 and of a5 ,good respectively) balance between have noise and histogram entropiesblur. The of 2.86FDK- and and 1.89, DIRECTT-reconstructed respectively. volumes (Figures 1 and 5, respectively) have An alternativehistogram way toentropies evaluate of the 2.86 results and 1.89, of the respectively. algorithms is to simulate the projection of the reconstructedAn volumesalternative and way calculate to evaluate the the Pearson results correlationof the algorithms coefﬁcient is to simulate [12,18] the projec- between thesetion projections of the reconstructed and the measured volumes ones.and calculat The Pearsone the Pearson correlation correlation coefﬁcient coefficient [12,18] (PCC), the valuebetween of which these can projections range between and the 1 for measured total linear ones. correlation The Pearson and − correlation1 for total coefficient linear anti-correlation,(PCC), the is value calculated of which according can range to the between relation: 1 for total linear correlation and −1 for total linear anti-correlation, is calculated according to the relation: σ M,P σ PCCM,P = ,M,P (6) σMσPCCP M,P = , (6) σMσP

where σM,P iswhere the covariance σM,P is the and covarianceσM and σandP are σM the and standard σP are the deviation standard of deviation the measured of the measured and simulatedand projections, simulated respectively. projections, respectively. Such coefﬁcients Such are coefficients calculated are for calculated each detector for each detector pixel and are plottedpixel and in are Figure plotted6. While in Figure the PCC 6. While values the decrease PCC values for largedecrease cone for angles large incone angles in the case of FDK,the they case remainof FDK, near they 1 remain regardless near of1 regard the angleless inof thethe caseangle of in DIRECTT. the case of DIRECTT.

Figure 6. Pearson correlation coefficient between measured projections and simulated projections Figure 6. Pearson correlation coefficient between measured projections and simulated projections of the volumes recon- of the volumes reconstructed by: (a) FDK, and (b) Direct Iterative Reconstruction of Computed structed by: (a) FDK, and (b) Direct Iterative Reconstruction of Computed Tomography Trajectories (DIRECTT). The Pear- son correlationTomography coefficient Trajectories (PCC) has been (DIRECTT). calculated The for Pearson each detector correlation pixel. coefficient (PCC) has been calculated for each detector pixel. The comparison between the two algorithms can also be done locally on parts of the The comparison between the two algorithms can also be done locally on parts of the volume that are affected in a greater degree by the cone-beam artefacts. For instance, the volume that areprofiles affected along in athe greater x-axis degree through by the the centres cone-beam of the artefacts. xy-slices For in both instance, Figures the 1 and 5 are x xy profiles alongplotted the -axis in Figure through 7. It the is evident centres that, of the in the-slices case of in FDK, both the Figures attenuation1 and 5values are away from plotted in Figurethe7 centre. It is evident of the sample that, in are the underestimated. case of FDK, the attenuation values away from the centre of the sample are underestimated. Finally, the reconstruction of the volume has also been performed using SIRT and CG, which were programmed to perform a fixed number of 600 iterations. The results are shown in Figure8. The values of the metrics for each reconstruction algorithm are listed in Table1. While the simulated projections in the case of both SIRT and CG have a high correlation to the measured ones, the respective histogram entropy values are significantly higher than that of DIRECTT. Moreover, although CG has, for the better part, suppressed the artefacts that are linked to the increasing cone angle, it has, at the same time, resulted in ring-like artefacts exactly where the regions that lie fully within the FoV border the regions that are only partially within it.

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J. Imaging 2021, 7, x FOR PEER REVIEW 8 of 9 FigureFigure 7.7. ProfileProfile alongalong thethe xx-axis-axis ofof thethe attenuationattenuation coefficientcoefficient valuesvalues ofof thethe firstfirst sliceslice shownshown inin bothboth FiguresFigures1 1and and5. 5.

Finally, the reconstruction of the volume has also been performed using SIRT and CG, which were programmed to perform a fixed number of 600 iterations. The results are shown in Figure 8. The values of the metrics for each reconstruction algorithm are listed in Table 1. While the simulated projections in the case of both SIRT and CG have a high correlation to the measured ones, the respective histogram entropy values are significantly higher than that of DIRECTT. Moreover, although CG has, for the better part, suppressed the artefacts that are linked to the increasing cone angle, it has, at the same time, resulted in ring-like artefacts exactly where the regions that lie fully within the FoV border the regions that are only partially within it. To sum up, we have described how the DIRECTT algorithm is adjusted for the reconstruction of cone-beam CT data. The artefacts that normally arise during such an op- eration have been suppressed resulting in a reconstruction that is a clear improvement over that computed by FDK. The performance of DIRECTT has been quantified by both global and local metrics and compared to the performance of other iterative reconstruction algorithms.

FigureFigure 8. Orthogonal 8. Orthogonal slices slices through through the the volume volume of a of concrete a concrete rod rod as reconstructed as reconstructed by bysimultaneous simultaneous iterative iterative reconstruction reconstruction techniquetechnique (SIRT, (SIRT, top top row) row) and and conjugate conjugate gradient gradient (CG, (CG, bottom bottom row). row). The The volume volume has has the the same same size size as the volume shownshown in in Figure 1 through omission of its upper and lower slices. The position of the slices shown corresponds precisely to that Figure1 through omission of its upper and lower slices. The position of the slices shown corresponds precisely to that of the of the slices in Figure 1. slices in Figure1.

Table 1.To Comparison sum up, of we the have performance described of the how reconstruction the DIRECTT algorithms. algorithm is adjusted for the Reconstructionreconstruction ofNumber cone-beam of Itera- CT data.Average The artefactsTime Histogram that normally En- ariseMean during Value such of an operationAlgorithm have been suppressedtions resultingper Iteration in a reconstruction(s) 1 tropy that is a clear improvementPCC FDK 1 54.85 ± 0.26 2.86 0.75 DIRECTT 562 14.51 ± 0.18 1.89 0.92 SIRT 600 6.15 ± 0.05 2.61 0.97 CG 600 6.41 ± 0.01 2.33 0.97 1 On a computer equipped with an NVIDIA GeForce GTX 1080 Ti GPU.

Author Contributions: Conceptualization, S.M. and A.K.; methodology, S.M. and A.K.; software, S.M.; validation, S.M., A.K. and G.B.; formal analysis, S.M.; investigation, S.M.; resources, G.B.; data curation, S.M.; writing—original draft preparation, S.M.; writing—review and editing, S.M., A.K. and G.B.; visualization, S.M. All authors have read and agreed to the published version of the manuscript. Funding: This work was part of the MUMMERING Innovative Training Network. The project has received funding from the European Union’s Horizon 2020 research and innovation programme under Marie Skłodowska-Curie Grant Agreement No. 765604. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Restrictions apply to the availability of these data. Data were obtained from BAM and are available from the authors with the permission of BAM.

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over that computed by FDK. The performance of DIRECTT has been quantiﬁed by both global and local metrics and compared to the performance of other iterative reconstruction algorithms.

Table 1. Comparison of the performance of the reconstruction algorithms.

Reconstruction Number of Average Time per Histogram Mean Value of Algorithm Iterations Iteration (s) 1 Entropy PCC FDK 1 54.85 ± 0.26 2.86 0.75 DIRECTT 562 14.51 ± 0.18 1.89 0.92 SIRT 600 6.15 ± 0.05 2.61 0.97 CG 600 6.41 ± 0.01 2.33 0.97 1 On a computer equipped with an NVIDIA GeForce GTX 1080 Ti GPU.

Author Contributions: Conceptualization, S.M. and A.K.; methodology, S.M. and A.K.; software, S.M.; validation, S.M., A.K. and G.B.; formal analysis, S.M.; investigation, S.M.; resources, G.B.; data curation, S.M.; writing—original draft preparation, S.M.; writing—review and editing, S.M., A.K. and G.B.; visualization, S.M. All authors have read and agreed to the published version of the manuscript. Funding: This work was part of the MUMMERING Innovative Training Network. The project has received funding from the European Union’s Horizon 2020 research and innovation programme under Marie Skłodowska-Curie Grant Agreement No. 765604. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Restrictions apply to the availability of these data. Data were obtained from BAM and are available from the authors with the permission of BAM. Acknowledgments: The authors kindly acknowledge Dietmar Meinel (BAM) for providing the experimental data. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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