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## RESEARCH ARTICLE 3D Algebraic Iterative Reconstruction for Cone-Beam X-Ray Differential Phase- Contrast Computed Tomography

### Citations

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Citation Context ...ns and developed a 3D algebraic iteration reconstruction (AIR) algorithm for it. This algorithm is based on the Kaczmarz method [46] or the later rediscovered algebraic reconstruction technique (ART) =-=[47]-=- and reconstructs the image ray by ray. Unlike the conventional iterative algorithms for absorption-based CT, it involves the derivative operation in the forward projections to take into account the d... |

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Citation Context ...ii) phase reconstruction. The first step can be accomplished by using a phase-stepping procedure [12, 14, 15], a reverse projection method [25], or a single-shot Fourier-based phase-extraction method =-=[26]-=-. The second step has so far been solved by using a FDK-type filtered back-projection (FBP) algorithm [22, 23], which is the direct extension of the FDK algorithm commonly used in absorptionbased CT [... |

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Citation Context ...arge. In the field of conventional x-ray absorption-based CT, iterative reconstruction algorithms have a wide spectrum of proven advantages [32–34], including dose reduction [35, 36], sparse sampling =-=[37, 38]-=-, and limited angle tomography [39, 40]. However, there were few studies on iterative reconstruction for DPC-CT in the past. Recently three kinds of iterative reconstruction techniques for fan-beam DP... |

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Citation Context ...]. The second step has so far been solved by using a FDK-type filtered back-projection (FBP) algorithm [22, 23], which is the direct extension of the FDK algorithm commonly used in absorptionbased CT =-=[27]-=- by replacing the ramp filter with a Hilbert imaginary filter to take into account the differential nature of DPC projection data. The FDK algorithm is computationally fast and relatively easy to impl... |

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Citation Context ...nstruction, it takes about 12 hours to converge. However, this is not an issue since there exist algorithm and hardware (graphics processing unit) to accelerate the algebraic reconstruction technique =-=[51, 52]-=-. The discretization can also be improved with other basis rather the voxel configuration in this work [42]. Future work Fig 4. 3D reconstruction results of the sample. (a) displays a stack of 2D diff... |

49 | Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,”
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Citation Context ...ificant interest [22–24]. It includes essentially two steps: (i) retrieval of the DPC projections and (ii) phase reconstruction. The first step can be accomplished by using a phase-stepping procedure =-=[12, 14, 15]-=-, a reverse projection method [25], or a single-shot Fourier-based phase-extraction method [26]. The second step has so far been solved by using a FDK-type filtered back-projection (FBP) algorithm [22... |

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Citation Context ... the absorption-based CT, DPC-CT involves the derivative operation in (5) in the forward projections. Although there are many iterative reconstruction methods for absorption-based CT (Please refer to =-=[33, 34, 48, 49]-=- for a review on this topic.), they are not applicable for DPC-CT since the simultaneous or block-iterative schemes involve weighted sums of all or multiple ray projections. They could blur the recons... |

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Citation Context ...ificant interest [22–24]. It includes essentially two steps: (i) retrieval of the DPC projections and (ii) phase reconstruction. The first step can be accomplished by using a phase-stepping procedure =-=[12, 14, 15]-=-, a reverse projection method [25], or a single-shot Fourier-based phase-extraction method [26]. The second step has so far been solved by using a FDK-type filtered back-projection (FBP) algorithm [22... |

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Citation Context ...the following we discuss the implementation procedure. Implementation Procedure Having specified the system matrixM and data a~ , we can describe the implementation of the algorithm in Eqs. (4), (5), =-=(6)-=- and (7). Depicted in Fig. 2, it includes an inner iteration and an outer iteration. This process is labeled in the following way: the outer iteration number is labeled by k, and the inner iterations ... |

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Citation Context ...e conebeam angle becomes large. In the field of conventional x-ray absorption-based CT, iterative reconstruction algorithms have a wide spectrum of proven advantages [32–34], including dose reduction =-=[35, 36]-=-, sparse sampling [37, 38], and limited angle tomography [39, 40]. However, there were few studies on iterative reconstruction for DPC-CT in the past. Recently three kinds of iterative reconstruction ... |

13 |
X-Ray Phase Tomography with LowBrilliance Sources
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Citation Context ...ay tube sources [15–21]. Several experimental studies reported in the literature have demonstrated that DPC-CT offers improved soft-tissue contrast compared to the conventional absorption-contrast CT =-=[16, 17, 19, 20]-=-. Most existing DPC-CT approaches are based on three kinds of scanning geometries, i.e., parallel-beam, fan-beam and cone-beam. Favored by the high imaging efficiency and the magnified spatial resolut... |

13 |
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Citation Context ... the absorption-based CT, DPC-CT involves the derivative operation in (5) in the forward projections. Although there are many iterative reconstruction methods for absorption-based CT (Please refer to =-=[33, 34, 48, 49]-=- for a review on this topic.), they are not applicable for DPC-CT since the simultaneous or block-iterative schemes involve weighted sums of all or multiple ray projections. They could blur the recons... |

11 |
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Citation Context ... the absorption-based CT, DPC-CT involves the derivative operation in (5) in the forward projections. Although there are many iterative reconstruction methods for absorption-based CT (Please refer to =-=[33, 34, 48, 49]-=- for a review on this topic.), they are not applicable for DPC-CT since the simultaneous or block-iterative schemes involve weighted sums of all or multiple ray projections. They could blur the recons... |

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Citation Context ...y absorption-based CT, iterative reconstruction algorithms have a wide spectrum of proven advantages [32–34], including dose reduction [35, 36], sparse sampling [37, 38], and limited angle tomography =-=[39, 40]-=-. However, there were few studies on iterative reconstruction for DPC-CT in the past. Recently three kinds of iterative reconstruction techniques for fan-beam DPC-CT were proposed, namely, statistical... |

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Citation Context ...ay tube sources [15–21]. Several experimental studies reported in the literature have demonstrated that DPC-CT offers improved soft-tissue contrast compared to the conventional absorption-contrast CT =-=[16, 17, 19, 20]-=-. Most existing DPC-CT approaches are based on three kinds of scanning geometries, i.e., parallel-beam, fan-beam and cone-beam. Favored by the high imaging efficiency and the magnified spatial resolut... |

5 |
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Citation Context ...ay tube sources [15–21]. Several experimental studies reported in the literature have demonstrated that DPC-CT offers improved soft-tissue contrast compared to the conventional absorption-contrast CT =-=[16, 17, 19, 20]-=-. Most existing DPC-CT approaches are based on three kinds of scanning geometries, i.e., parallel-beam, fan-beam and cone-beam. Favored by the high imaging efficiency and the magnified spatial resolut... |

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Citation Context ...r iteration within the kth outer iteration. The ray-by-ray reconstruction is called the inner iteration. Once all rays are used, it will start another iteration, which is called the outer iteration. δ=-=[1, 0]-=- is the initial guess of the reconstructed image, which is chosen to be the zero at each voxel. The parameter d is the relax coefficient and can be from 0 to 2 in theory from the convergence theory fo... |

4 |
Pogany A, Stevenson AW
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Citation Context ...Þ where the function floor(r) gives the largest integer less than r and the function mod(r,s) the remainder for the division rs. Now the imaging system, namely, Eq. (1) is discretized by Eqs. (4) and =-=(5)-=-. The problem of DPC-CT is then to reconstruct δ in Eq. (4) from the measured cone-beam DPC projections α in Eq. (5). Obviously, unlike the absorption-based CT, DPC-CT involves the derivative operatio... |

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Citation Context ...ificant interest [22–24]. It includes essentially two steps: (i) retrieval of the DPC projections and (ii) phase reconstruction. The first step can be accomplished by using a phase-stepping procedure =-=[12, 14, 15]-=-, a reverse projection method [25], or a single-shot Fourier-based phase-extraction method [26]. The second step has so far been solved by using a FDK-type filtered back-projection (FBP) algorithm [22... |

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Citation Context ...ntially two steps: (i) retrieval of the DPC projections and (ii) phase reconstruction. The first step can be accomplished by using a phase-stepping procedure [12, 14, 15], a reverse projection method =-=[25]-=-, or a single-shot Fourier-based phase-extraction method [26]. The second step has so far been solved by using a FDK-type filtered back-projection (FBP) algorithm [22, 23], which is the direct extensi... |

3 |
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Citation Context ...ized the cone-beam DPC-CT imaging modality as a system of linear equations and developed a 3D algebraic iteration reconstruction (AIR) algorithm for it. This algorithm is based on the Kaczmarz method =-=[46]-=- or the later rediscovered algebraic reconstruction technique (ART) [47] and reconstructs the image ray by ray. Unlike the conventional iterative algorithms for absorption-based CT, it involves the de... |

2 |
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Citation Context ... OPEN ACCESS Citation: Fu J, Hu X, Velroyen A, Bech M, Jiang M, Pfeiffer F (2015) 3D Algebraic Iterative Reconstruction for Cone-Beam X-Ray Differential Phase-Contrast Computed Tomography. PLoS ONE 10=-=(3)-=-: e0117502. doi:10.1371/journal.pone.0117502 Academic Editor: Qinghui Zhang, University of Nebraska Medical Center, UNITED STATES Received: July 17, 2014 Accepted: December 23, 2014 Published: March 1... |

2 |
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Citation Context ...owing we discuss the implementation procedure. Implementation Procedure Having specified the system matrixM and data a~ , we can describe the implementation of the algorithm in Eqs. (4), (5), (6) and =-=(7)-=-. Depicted in Fig. 2, it includes an inner iteration and an outer iteration. This process is labeled in the following way: the outer iteration number is labeled by k, and the inner iterations by i in ... |

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Citation Context ...hardware (graphics processing unit) to accelerate the algebraic reconstruction technique [51, 52]. The discretization can also be improved with other basis rather the voxel configuration in this work =-=[42]-=-. Future work Fig 4. 3D reconstruction results of the sample. (a) displays a stack of 2D differential phase contrast projections retrieved from the recorded moiré fringe images by detector. (b), (c) a... |

1 |
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Citation Context ...B;AÞ; ð6Þ where the function floor(r) gives the largest integer less than r and the function mod(r,s) the remainder for the division rs. Now the imaging system, namely, Eq. (1) is discretized by Eqs. =-=(4)-=- and (5). The problem of DPC-CT is then to reconstruct δ in Eq. (4) from the measured cone-beam DPC projections α in Eq. (5). Obviously, unlike the absorption-based CT, DPC-CT involves the derivative ... |

1 | ZQ, Ding F, et al. (2009) Alternative method for differential phasecontrast imaging with weakly coherent hard x-rays. Phys Rev A 79: 013815. doi: 10.1103/PhysRevA - ZF, KJ, et al. |

1 | Nagatomi T, Oshima KC, et al. (2013) Hard x-ray phase contrast imaging using a tabletop talbot-lau interferometer with multiline embedded x-ray targets. Opt Lett 38: 157–9. doi: 10.1364/OL.38.000157 PMID: 23454947 - Shimura, Morimoto, et al. |

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Citation Context ...15], a reverse projection method [25], or a single-shot Fourier-based phase-extraction method [26]. The second step has so far been solved by using a FDK-type filtered back-projection (FBP) algorithm =-=[22, 23]-=-, which is the direct extension of the FDK algorithm commonly used in absorptionbased CT [27] by replacing the ramp filter with a Hilbert imaginary filter to take into account the differential nature ... |

1 | A reconstruction method for cone-beam differential x-ray phase-contrast computed tomography. Opt Express 20: 21512–19. doi: 10.1364/OE.20 - Fu, Velroyen, et al. - 2012 |

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Citation Context ...e conebeam angle becomes large. In the field of conventional x-ray absorption-based CT, iterative reconstruction algorithms have a wide spectrum of proven advantages [32–34], including dose reduction =-=[35, 36]-=-, sparse sampling [37, 38], and limited angle tomography [39, 40]. However, there were few studies on iterative reconstruction for DPC-CT in the past. Recently three kinds of iterative reconstruction ... |

1 |
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(Show Context)
Citation Context ...arge. In the field of conventional x-ray absorption-based CT, iterative reconstruction algorithms have a wide spectrum of proven advantages [32–34], including dose reduction [35, 36], sparse sampling =-=[37, 38]-=-, and limited angle tomography [39, 40]. However, there were few studies on iterative reconstruction for DPC-CT in the past. Recently three kinds of iterative reconstruction techniques for fan-beam DP... |

1 |
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(Show Context)
Citation Context ...y absorption-based CT, iterative reconstruction algorithms have a wide spectrum of proven advantages [32–34], including dose reduction [35, 36], sparse sampling [37, 38], and limited angle tomography =-=[39, 40]-=-. However, there were few studies on iterative reconstruction for DPC-CT in the past. Recently three kinds of iterative reconstruction techniques for fan-beam DPC-CT were proposed, namely, statistical... |

1 | Study of OSEMwith different subsets in grating-based x-ray differential phase-contrast imaging. Anal Bioanal Chem 401: 837–44. doi: 10 - Zhang, YL, et al. - 2011 |

1 |
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(Show Context)
Citation Context ...t. Recently three kinds of iterative reconstruction techniques for fan-beam DPC-CT were proposed, namely, statistical iterative algorithms [32, 41–43], simultaneous algebraic reconstruction technique =-=[44]-=- and algebraic iteration reconstruction [45]. Those studies demonstrated the feasibility of iterative reconstruction for DPC-CT. Although iterative reconstruction exhibits great advantages over FBP-ty... |

1 |
An algebraic iterative reconstruction technique for differential x-ray phase-contrast computed tomography
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(Show Context)
Citation Context ...ruction techniques for fan-beam DPC-CT were proposed, namely, statistical iterative algorithms [32, 41–43], simultaneous algebraic reconstruction technique [44] and algebraic iteration reconstruction =-=[45]-=-. Those studies demonstrated the feasibility of iterative reconstruction for DPC-CT. Although iterative reconstruction exhibits great advantages over FBP-type methods, there is currently no in-depth d... |

1 |
Noo F, Defrise M
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Citation Context ... the algorithm stops when there is no appreciable change. Numerical study To test the proposed method, numerical simulations were performed. The phantom is adopted from the well-known Defrise phantom =-=[50]-=-, which is known as cone-beam killer. It consists of nine discs of diameter 55 mm, vertically separated at different heights of 0, 26.1, 52.4, 78.7 and 105.1 mm (height was measured from the axial mid... |

1 |
Algebraic reconstruction techniques can be made computationally efficient
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Citation Context ...nstruction, it takes about 12 hours to converge. However, this is not an issue since there exist algorithm and hardware (graphics processing unit) to accelerate the algebraic reconstruction technique =-=[51, 52]-=-. The discretization can also be improved with other basis rather the voxel configuration in this work [42]. Future work Fig 4. 3D reconstruction results of the sample. (a) displays a stack of 2D diff... |