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Histogram tomography

Department of Mathematics, University of Manchester, Manchester, UK

This contribution is part of the Special Issue: Inverse problems in imaging and engineering science
Guest Editors: Lauri Oksanen; Mikko Salo
Link: https://www.aimspress.com/newsinfo/1270.html

Special Issues: Inverse problems in imaging and engineering science

In many tomographic imaging problems the data consist of integrals along lines or curves. Increasingly we encounter “rich tomography” problems where the quantity imaged is higher dimensional than a scalar per voxel, including vectors tensors and functions. The data can also be higher dimensional and in many cases consists of a one or two dimensional spectrum for each ray. In many such cases the data contain not just integrals along rays but the distribution of values along the ray. If this is discretized into bins we can think of this as a histogram. In this paper we introduce the concept of “histogram tomography”. For scalar problems with histogram data this holds the possibility of reconstruction with fewer rays. In vector and tensor problems it holds the promise of reconstruction of images that are in the null space of related integral transforms. For scalar histogram tomography problems we show how bins in the histogram correspond to reconstructing level sets of function, while moments of the distribution are the x-ray transform of powers of the unknown function. In the vector case we suggest a reconstruction procedure for potential components of the field. We demonstrate how the histogram longitudinal ray transform data can be extracted from Bragg edge neutron spectral data and hence, using moments, a non-linear system of partial differential equations derived for the strain tensor. In x-ray diffraction tomography of strain the transverse ray transform can be deduced from the diffraction pattern the full histogram transverse ray transform cannot. We give an explicit example of distributions of strain along a line that produce the same diffraction pattern, and characterize the null space of the relevant transform.
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Keywords tomography; distribution; histogram; spectrum; diffraction; strain; Radon transform; tensor tomography; vector tomography; x-ray diffraction; Bragg edge; neutron diffraction; Doppler transform

Citation: William R. B. Lionheart. Histogram tomography. Mathematics in Engineering, 2020, 2(1): 55-74. doi: 10.3934/mine.2020004


  • 1. Natterer F (2001) The Mathematics of Computerized Tomography, Philadelphia: Society for Industrial and Applied Mathematics.
  • 2. Sales M, Strobl M, Shinohara T, et al. (2018) Three dimensional polarimetric neutron tomography of magnetic fields. Sci Rep 8: 2214.
  • 3. Desai NM, Lionheart WRB, Sales M, et al. (2019) Polarimetric neutron tomography of magnetic fields: Uniqueness of solution and reconstruction. Inverse Probl DOI: https://doi.org/10.1088/1361-6420/ab44e0.
  • 4. An X, Kraetschmer T, Takami K, et al. (2011) Validation of temperature imaging by H2O absorption spectroscopy using hyperspectral tomography in controlled experiments. Appl Opt 50: A29-A37.
  • 5. Ma L, Li X, Sanders S, et al. (2013) 50-kHz-rate 2D imaging of temperature and H2O concentration at the exhaust plane of a J85 engine using hyperspectral tomography. Opt Express 21: 1152-1162.    
  • 6. Andersson F (2005) The Doppler moment transform in Doppler tomography. Inverse Probl 21: 1249.
  • 7. Sharafutdinov VA (1994) Integral Geometry of Tensor Fields, Walter de Gruyter.
  • 8. Lionheart WRB, Withers PJ (2015) Diffraction tomography of strain. Inverse Probl 31: 045005.
  • 9. Boman J, Sharafutdinov V (2018) Stability estimates in tensor tomography. Inverse Probl Imaging 12: 1245-1262.    
  • 10. Bogachev VI (2007) Measure Theory, Berlin: Springer Science & Business Media.
  • 11. Akhiezer NI (1965) The Classical Moment Problem: And Some Related Questions in Analysis, Edinburgh: Oliver & Boyd.
  • 12. Gardner RJ, McMullen P (1980) On Hammer's X-Ray Problem. J Lond Math Soc 2: 171-175.
  • 13. Gardner RJ, Gritzmann P (1997) Discrete tomography: Determination of finite sets by X-rays. T Am Math Soc 349: 2271-2295.
  • 14. Gardner RJ, Kiderlen M (2007) A solution to Hammer's X-ray reconstruction problem. Adv Math 214: 323-343.    
  • 15. Gardner RJ (2006) Geometric Tomography, 2 Eds., Cambridge University Press.
  • 16. Faridani A, Ritman EL, Smith KT (1992) Local tomography. SIAM J Appl Math 52: 459-484.
  • 17. Herman GT, Kuba A (2012) Discrete Tomography: Foundations, Algorithms, and Applications, Springer Science & Business Media.
  • 18. Batenburg KJ, Sijbers J (2011) DART: A practical reconstruction algorithm for discrete tomography. IEEE T Image Process 20: 2542-2553.
  • 19. Bentz C, Costa MC, De Werra D, et al. (2008) On a graph coloring problem arising from discrete tomography. Networks 51: 256-267.
  • 20. Schuster T (2008) 20 years of imaging in vector field tomography: A review, In: Math. Methods in Biomedical Imaging and Intensity-Modulated Radiation Therapy (IMRT). Ser. Publications of the Scuola Normale Superiore, 7: 389-424.
  • 21. Sparr G, Strahlen K, Lindstrom K, et al. (1995) Doppler tomography for vector fields. Inverse Probl 11: 1051.
  • 22. Kravtsov YA (1968) "Quasi-isotropic" approximation of geometrical optics. Dokl Akad Nauk SSSR 183: 74-76.
  • 23. Aben H, Errapart A, Ainola L, et al. (2005) Photoelastic tomography for residual stress measurement in glass. Opt Eng 44: 093601.
  • 24. Tomlinson RA, Yang H, Szotten D, et al. (2006) The design and commissioning of a novel tomographic polariscope, In: SEM Annual Conf. and Exposition on Experimental and Applied Mechanics, 1141-1147.
  • 25. Lionheart W, Sharafutdinov V (2009) Reconstruction algorithm for the linearized polarization tomography problem with incomplete data. Contemp Math 14: 137.
  • 26. Johnstone D, van Helvoort A, Midgley P (2017) Nanoscale strain tomography by scanning precession electron diffraction. Microsc Microanal 23: 1710-1711.
  • 27. Woracek R, Santisteban J, Fedrigo A, et al. (2018) Diffraction in neutron imaging-A review, In: Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 878: 141-158.
  • 28. Georgievskii D (2016) Generalized compatibility equations for tensors of high ranks in multidimensional continuum mechanics. Russ J Math Phys 23: 475-483.
  • 29. Sklar M (1959) Fonctions de répartition à n dimensions et leurs marges. Publ Inst Statist Univ Paris 8: 229-231.
  • 30. Nelsen RB (2007) An Introduction to Copulas, Springer Science & Business Media.
  • 31. Santisteban JR, Edwards L, Fitzpatrick ME, et al. (2002) Strain imaging by Bragg edge neutron transmission, In: Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 481: 765-768.
  • 32. Abbey B, Zhang SY, Vorster WJJ, et al. (2009) Feasibility study of neutron strain tomography. Procedia Eng 1: 185-188.
  • 33. Knops RJ, Payne LE (1971) Modern Uniqueness Theorems in Three-Dimensional Elastostatics, In: Uniqueness Theorems in Linear Elasticity, Berlin: Springer, 32-60.
  • 34. Gregg AWT, Hendriks JN, Wensrich CM, et al. (2017) Tomographic reconstruction of residual strain in axisymmetric systems from Bragg-edge neutron imaging. Mech Res Commun 85: 96-103.
  • 35. Wensrich CM, Hendriks JN, Gregg A, et al. (2016) Bragg-edge neutron transmission strain tomography for in situ loadings, In: Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms, 383: 52-58.
  • 36. Hendriks JN, Gregg AWT, Wensrich CM, et al. (2017) Bragg-edge elastic strain tomography for in situ systems from energy-resolved neutron transmission imaging. Phys Rev Mater 1: 053802.
  • 37. Desai NM, Lionheart WRB (2016) An explicit reconstruction algorithm for the transverse ray transform of a second rank tensor field from three axis data. Inverse Probl 32: 115009.


This article has been cited by

  • 1. Lauri Oksanen, Mikko Salo, Inverse problems in imaging and engineering science, Mathematics in Engineering, 2020, 2, 2, 287, 10.3934/mine.2020014

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