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

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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 m, 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 J. Imaging 2021, 7, 147 3 of 9 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 pos- sible 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].
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