##
**Analyzing reconstruction artifacts from arbitrary incomplete X-ray CT data.**
*(English)*
Zbl 1439.44004

This article studies nonsmooth artifacts in the reconstruction of a
two-dimensional function by a filtered backprojection algorithm for
X-ray computed tomography with arbitrary incomplete data.

It is shown that all singular artifacts arise from points on the boundary of the data set. They can only be one of two types.

Object-dependent artifacts are generated by singularities of the object being scanned; these artifacts extend along lines, and generalize the streak artifacts observed in limited-angle tomography. Object-independent artifacts take one of two forms: streaks or lines if the boundary of the data set is not smooth at a point, and curved artifacts if the boundary is locally smooth.

The results are interpreted and illustrated numerically in various settings, including the motivating example: a synchrotron data set in which artifacts appear on lines that have no relation to the object.

It is shown that all singular artifacts arise from points on the boundary of the data set. They can only be one of two types.

Object-dependent artifacts are generated by singularities of the object being scanned; these artifacts extend along lines, and generalize the streak artifacts observed in limited-angle tomography. Object-independent artifacts take one of two forms: streaks or lines if the boundary of the data set is not smooth at a point, and curved artifacts if the boundary is locally smooth.

The results are interpreted and illustrated numerically in various settings, including the motivating example: a synchrotron data set in which artifacts appear on lines that have no relation to the object.

Reviewer: Fritz Keinert (Ames)

### Keywords:

X-ray tomography; incomplete data tomography; reconstruction artifact; wavefront set; microlocal analysis
PDF
BibTeX
XML
Cite

\textit{L. Borg} et al., SIAM J. Imaging Sci. 11, No. 4, 2786--2814 (2018; Zbl 1439.44004)

### References:

[1] | L. L. Barannyk, J. Frikel, and L. V. Nguyen, On artifacts in limited data spherical radon transform: Curved observation surface, Inverse Problems, 32 (2016), 015012. · Zbl 1332.35391 |

[2] | R. H. T. Bates and R. M. Lewitt, Image reconstruction from projections. III: Projection completion methods (theory), Optik, 50 (1978), pp. 189–204. |

[3] | R. H. T. Bates and R. M. Lewitt, Image reconstruction from projections. IV: Projection completion methods (computational examples), Optik, 50 (1978), pp. 269–278. |

[4] | F. E. Boas and D. Fleischmann, CT artifacts: Causes and reduction techniques, Imaging Med., 4 (2012), pp. 229–240. |

[5] | J. Boman and E. T. Quinto, Support theorems for real analytic Radon transforms, Duke Math. J., 55 (1987), pp. 943–948. · Zbl 0645.44001 |

[6] | L. Borg, J. S. Jørgensen, J. Frikel, and J. Sporring, Reduction of variable-truncation artifacts from beam occlusion during in situ x-ray tomography, Meas. Sci. Technol., 28 (2017), 124004, . |

[7] | L. Borg, J. S. Jørgensen, and J. Sporring, Towards Characterizing and Reducing Artifacts Caused by Varying Projection Truncation, Technical Report, Department of Computer Science, University of Copenhagen, Copenhagen, Denmark, 2017, . |

[8] | R. Chityalad, K. R. Hoffmann, S. Rudina, and D. R. Bednareka, Region of interest (ROI) computed tomography (CT): Comparison with full field of view (FFOV) and truncated CT for a human head phantom, in Proceedings of the International Society for Optical Engineering, Proc. SPIE Int. Soc. Opt. Eng. 5745(1), 2005, pp. 583–590. |

[9] | J. K. Choi, H. S. Park, S. Wang, Y. Wang, and J. K. Seo, Inverse problem in quantitative susceptibility mapping, SIAM J. Imaging Sci., 7 (2014), pp. 1669–1689, . · Zbl 1361.94010 |

[10] | J. J. Duistermaat, Fourier Integral Operators, Progr. Math. 130, Birkhäuser Boston, Boston, MA, 1996. · Zbl 0841.35137 |

[11] | M. E. Eldib, M. Hegazy, Y. J. Mun, M. H. Cho, M. H. Cho, and S. Y. Lee, A ring artifact correction method: Validation by micro-CT imaging with flat-panel detectors and a \(2\) D photon-counting detector, Sensors, 17 (2017), 269. |

[12] | A. Faridani, D. V. Finch, E. L. Ritman, and K. T. Smith, Local tomography II, SIAM J. Appl. Math., 57 (1997), pp. 1095–1127, . · Zbl 0897.65083 |

[13] | A. Faridani, E. L. Ritman, and K. T. Smith, Local tomography, SIAM J. Appl. Math., 52 (1992), pp. 459–484, . · Zbl 0758.65081 |

[14] | D. V. Finch, I.-R. Lan, and G. Uhlmann, Microlocal analysis of the restricted X-ray transform with sources on a curve, in Inside Out, Inverse Problems and Applications, G. Uhlmann, ed., Math. Sci. Res. Inst. Publ. 47, Cambridge University Press, Cambridge, UK, 2003, pp. 193–218. · Zbl 1086.35134 |

[15] | F. G. Friedlander, Introduction to the Theory of Distributions, 2nd ed., Cambridge University Press, Cambridge, UK, 1998. |

[16] | J. Frikel and E. T. Quinto, Characterization and reduction of artifacts in limited angle tomography, Inverse Problems, 29 (2013), 125007. · Zbl 1284.92044 |

[17] | J. Frikel and E. T. Quinto, Artifacts in incomplete data tomography with applications to photoacoustic tomography and sonar, SIAM J. Appl. Math., 75 (2015), pp. 703–725, . · Zbl 1381.44006 |

[18] | J. Frikel and E. T. Quinto, Limited data problems for the generalized Radon transform in \(\mathbb{R}^n\), SIAM J. Math. Anal., 48 (2016), pp. 2301–2318, . · Zbl 1344.44002 |

[19] | A. Greenleaf and G. Uhlmann, Non-local inversion formulas for the X-ray transform, Duke Math. J., 58 (1989), pp. 205–240. · Zbl 0668.44004 |

[20] | V. Guillemin, On Some Results of Gelfand in Integral Geometry, Proc. Sympos. Pure Math. 43, AMS, Providence, RI, 1985, pp. 149–155. · Zbl 0576.58028 |

[21] | V. Guillemin and S. Sternberg, Geometric Asymptotics, AMS, Providence, RI, 1977. |

[22] | M. G. Hahn and E. T. Quinto, Distances between measures from \(1\)-dimensional projections as implied by continuity of the inverse Radon transform, Z. Wahrsch. Verw. Gebiete, 70 (1985), pp. 361–380. · Zbl 0555.28005 |

[23] | L. Hörmander, Fourier integral operators, I, Acta Math., 127 (1971), pp. 79–183. |

[24] | L. Hörmander, The Analysis of Linear Partial Differential Operators: Distribution Theory and Fourier Analysis I, Classics Math., Springer-Verlag, Berlin, 2003. |

[25] | A. Katsevich, Local tomography for the limited-angle problem, J. Math. Anal. Appl., 213 (1997), pp. 160–182. · Zbl 0894.65065 |

[26] | A. I. Katsevich and A. G. Ramm, Pseudolocal tomography, SIAM J. Appl. Math., 56 (1996), pp. 167–191, . |

[27] | V. P. Krishnan and E. T. Quinto, Microlocal analysis in tomography, in Handbook of Mathematical Methods in Imaging, O. Scherzer, ed., Springer-Verlag, New York, Berlin, 2015, pp. 847–902. · Zbl 1395.94042 |

[28] | P. Kuchment, The Radon Transform and Medical Imaging, CBMS-NSF Regional Conf. Ser. in Appl. Math. 85, SIAM, Philadelphia, 2014, . |

[29] | P. Kuchment, K. Lancaster, and L. Mogilevskaya, On local tomography, Inverse Problems, 11 (1995), pp. 571–589. · Zbl 0832.44001 |

[30] | F. Lauze, Y. Quéau, and E. Plenge, Simultaneous reconstruction and segmentation of CT scans with shadowed data, in Scale Space and Variational Methods in Computer Vision, F. Lauze, Y. Dong, and A. B. Dahl, eds., Springer, Cham, 2017, pp. 308–319. |

[31] | A. K. Louis, Picture reconstruction from projections in restricted range, Math. Methods Appl. Sci., 2 (1980), pp. 209–220. · Zbl 0457.65081 |

[32] | A. K. Louis, Incomplete data problems in X-ray computerized tomography I. Singular value decomposition of the limited angle transform, Numer. Math., 48 (1986), pp. 251–262. · Zbl 0578.65131 |

[33] | F. Natterer, Efficient implementation of “optimal” algorithms in computerized tomography, Math. Methods Appl. Sci., 2 (1980), pp. 545–555. · Zbl 0454.65083 |

[34] | F. Natterer, The Mathematics of Computerized Tomography, Classics Appl. Math. 32, SIAM, Philadelphia, 2001, . · Zbl 0973.92020 |

[35] | F. Natterer and F. Wübbeling, Mathematical Methods in Image Reconstruction, SIAM, Philadelphia, 2001, . · Zbl 0974.92016 |

[36] | L. V. Nguyen, How strong are streak artifacts in limited angle computed tomography?, Inverse Problems, 31 (2015), 055003. · Zbl 1364.65287 |

[37] | L. V. Nguyen, On artifacts in limited data spherical Radon transform: Flat observation surfaces, SIAM J. Math. Anal., 47 (2015), pp. 2984–3004, . · Zbl 1349.42027 |

[38] | L. V. Nguyen, On the strength of streak artifacts in filtered back-projection reconstructions for limited angle weighted x-ray transform, J. Fourier Anal. Appl., 23 (2017), pp. 712–728. · Zbl 1383.92041 |

[39] | B. Palacios, G. Uhlmann, and Y. Wang, Reducing streaking artifacts in quantitative susceptibility mapping, SIAM J. Imaging Sci., 10 (2017), pp. 1921–1934, . · Zbl 1397.35346 |

[40] | B. Palacios, G. Uhlmann, and Y. Wang, Quantitative analysis of metal artifacts in X-ray tomography, SIAM J. Math. Anal., 50 (2018), 4914–4936, . · Zbl 1409.35236 |

[41] | V. Palamodov, Nonlinear artifacts in tomography, Soviet Phys. Dokl., 31 (1986), pp. 888–890. · Zbl 0638.65092 |

[42] | V. Palamodov, A method of reduction of artifacts of quantitative susceptibility mapping, SIAM J. Imaging Sci., 9 (2016), pp. 481–489, . · Zbl 1347.94009 |

[43] | X. Pan, E. Y. Sidky, and M. Vannier, Why do commercial CT scanners still employ traditional, filtered back-projection for image reconstruction?, Inverse Problems, 25 (2009), 123009. · Zbl 1185.68811 |

[44] | H. S. Park, J. K. Choi, and J. K. Seo, Characterization of metal artifacts in x-ray computed tomography, Comm. Pure Appl. Math., 70 (2017), pp. 2191–2217. · Zbl 1379.92027 |

[45] | B. Petersen, Introduction to the Fourier Transform and Pseudo-Differential Operators, Pitman, Boston, MA, 1983. · Zbl 0523.35001 |

[46] | E. T. Quinto, The dependence of the generalized Radon transform on defining measures, Trans. Amer. Math. Soc., 257 (1980), pp. 331–346. · Zbl 0471.58022 |

[47] | E. T. Quinto, Singularities of the X-ray transform and limited data tomography in \(\mathbb{R}^2\) and \(\mathbb{R}^3\), SIAM J. Math. Anal., 24 (1993), pp. 1215–1225, . · Zbl 0784.44002 |

[48] | E. T. Quinto, Exterior and limited angle tomography in non-destructive evaluation, Inverse Problems, 14 (1998), pp. 339–353. · Zbl 0908.65125 |

[49] | E. T. Quinto, An introduction to x-ray tomography and radon transforms, in The Radon Transform and Applications to Inverse Problems, G. Ólafsson and E. T. Quinto, eds., Proc. Sympos. Appl. Math. 63, AMS, Providence, RI, 2006, pp. 1–23. · Zbl 1118.44002 |

[50] | E. T. Quinto, Support Theorems for the Spherical Radon Transform on Manifolds, Int. Math. Res. Not., 2006 (2006), 67205. · Zbl 1125.44002 |

[51] | A. G. Ramm and A. I. Zaslavsky, Singularities of the Radon transform, Bull. Amer. Math. Soc. (N.S.), 28 (1993), pp. 109–115. · Zbl 0767.44002 |

[52] | G. Rigaud, On analytical solutions to beam-hardening, Sens. Imaging, 18 (2017), 5. |

[53] | W. Rudin, Functional Analysis, McGraw–Hill, New York, 1973. · Zbl 0253.46001 |

[54] | L. Shen, E. T. Quinto, S. Wang, and M. Jiang, Simultaneous reconstruction and segmentation with the Mumford-Shah functional for electron tomography, in Proceedings of the 38th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC), 2016, pp. 5909–5912. · Zbl 1408.94590 |

[55] | P. Stefanov and G. Uhlmann, Is a curved flight path in SAR better than a straight one?, SIAM J. Appl. Math., 73 (2013), pp. 1596–1612, . · Zbl 1291.35448 |

[56] | F. Trèves, Introduction to Pseudodifferential and Fourier Integral Operators, Volume 2: Fourier Integral Operators, Plenum Press, New York, London, 1980. |

[57] | J. Vogelgesang and C. Schorr, Iterative region-of-interest reconstruction from limited data using prior information, Sens. Imaging, 18 (2017), 16. |

[58] | F. W. Warner, Foundations of Differentiable Manifolds and Lie Groups, Grad. Texts in Math. 94, Springer-Verlag, New York, Berlin, 1983. |

[59] | Y. Yang, S. S. Hakim, S. Bruns, K. N. Dalby, K. Uesugi, S. L. S. Stipp, and H. O. Sørensen, Effect of cumulative surface on pore development in chalk, submitted. |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.