Discrete Tomography for Generating Grain Maps of Polycrystals

  • A. Alpers
  • L. Rodek
  • H.F. Poulsen
  • E. Knudsen
  • G.T. Herman
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)


The determination of crystalline structures is a demanding and fundamental task of crystallography. Most crystalline materials, natural or artificial, are in fact polycrystals, composed of tiny crystals called grains. Every grain has an associated average orientation that determines the spatial configuration of the crystalline lattice. Typically, the structure of a polycrystal is rendered via an orientation map or a grain map, in which individual pixels/voxels are assigned a grain orientation or a grain label. We present two related approaches to reconstructing a 2D grain map of a polycrystal from X-ray diffraction patterns. The first technique makes the assumption that each grain is actually a perfect crystal, i.e., that the specimen is not deformed. The other method can be applied also when the sample has been exposed to moderate levels of deformation. In both cases, the grain map is produced by a Bayesian discrete tomographic algorithm that uses Gibbs priors. The optimization of the objective function is accomplished via the Metropolis algorithm. The efficacy of the techniques is demonstrated by simulation experiments.


Crystal Symmetry Average Orientation White Pixel Detector Pixel Metropolis Algorithm 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Alpers, A., Knudsen, E., Poulsen, H.F., Herman, G.T.: Resolving ambiguities in reconstructed grain maps using discrete tomography. Electr. Notes Discr. Math., 20, 419–437 (2005).CrossRefMathSciNetGoogle Scholar
  2. 2.
    Alpers, A., Poulsen, H.F., Knudsen, E., Herman, G.T.: A discrete tomography algorithm for improving the quality of 3DXRD grain maps. J. Appl. Cryst., 39, 281–299 (2006).CrossRefGoogle Scholar
  3. 3.
    Altmann, S.L.: Rotations, Quaternions, and Double Groups. Clarendon Press, Oxford, UK (1986).zbMATHGoogle Scholar
  4. 4.
    Bass0mann, H., Besslich, P.W.: Bildverarbeitung Ad Oculos. Springer, Berlin,Germany (1991).Google Scholar
  5. 5.
    Bremaud, P.: Markov Chains: Gibbs Fields, Monte Carlo Simulations, and Queues. Springer, New York, NY (1999).Google Scholar
  6. 6.
    Carvalho, B.M., Herman, G.T., Matej, S., Salzberg, C., Vardi, E.: Binary tomography for triplane cardiography. In: Kuba, A., Samal, M., Todd-Pokropek, A. (eds.), Information Processing in Medical Imaging. Springer, Berlin, Germany, pp. 29-41 (1999).Google Scholar
  7. 7.
    Chan, M.T., Herman, G.T., Levitan, E.: Probabilistic modeling of discrete images. In: Herman, G.T., Kuba, A. (eds.), Discrete Tomography: Foundations,Algorithms, and Applications. Birkhauser, Boston, MA, pp. 213–235 (1999).Google Scholar
  8. 8.
    Conway, J.H., Smith, D.A.: On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry. A. K. Peters, Natick, MA (2003).zbMATHGoogle Scholar
  9. 9.
    Frank, F.C.: Orientation mapping. Met. Trans., A19 403–408 (1988).Google Scholar
  10. 10.
    Geman, S., Geman, D.: Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Trans. PAMI, 6, 721–741 (1984).zbMATHGoogle Scholar
  11. 11.
    Gordon, R., Bender, R., Herman, G.T.: Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and X-ray photography. J. Theor. Biol., 29 471–482 (1970).CrossRefGoogle Scholar
  12. 12.
    Grimmer, H.: Disorientations and coincidence rotations for cubic lattices. Acta Cryst., A30, 685–688 (1974).Google Scholar
  13. 13.
    Hansen, L., Pospiech, J., Liicke, K.: Tables of Texture Analysis of Cubic Crystals. Springer, Berlin, Germany (1978).Google Scholar
  14. 14.
    Heinz, A., Neumann, P.: Representation of orientation and disorientation data for cubic, hexagonal, tetragonal and orthorombic crystals. Acta Cryst., A47,780–789 (1991).MathSciNetGoogle Scholar
  15. 15.
    Herman, G.T., Kuba, A. (eds.): Discrete Tomography: Foundations, Algorithms, and Applications. Birkhäuser, Boston, MA (1999).zbMATHGoogle Scholar
  16. 16.
    Kirkpatrick, S., Gelatt, C.D., Vecchi, M.P.: Optimization by simulated annealing. Science, 220, 671–680 (1983).CrossRefMathSciNetGoogle Scholar
  17. 17.
    Kocks, U.F., Tome, C.N., Wenk, H.R.: Texture and Anisotropy. Cambridge University Press, Cambridge, UK (1998).zbMATHGoogle Scholar
  18. 18.
    Kuipers, J.B.: Quaternions and Rotation Sequences: A Primer with Applications to Orbits, Aerospace and Virtual Reality. Princeton University Press, Princeton, NJ (1999).zbMATHGoogle Scholar
  19. 19.
    Liao, H.Y., Herman, G.T.: Automated estimation of the parameters of Gibbs priors to be used in binary tomography. Discrete Appl. Math., 139 149–170 (2004).zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Markussen, T., Fu, X., Margulies, L., Lauridsen, E.M., Nielsen, S.F., Schmidt, S., Poulsen, H.F.: An algebraic algorithm for generation of three-dimensional grain maps based on diffraction with a wide beam of hard X-rays. J. Appl. Cryst., 37, 96–102 (2004).CrossRefGoogle Scholar
  21. 21.
    Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H., Teller, E.:Equation of state calculations by fast computing machines. J. Chem. Phys., 21,1087–1092 (1953).CrossRefGoogle Scholar
  22. 22.
    Morawiec, A.: Orientations and Rotations. Computations in Crystallographic Textures. Springer, Berlin, Germany (2004).zbMATHGoogle Scholar
  23. 23.
    Poulsen, H.F.: A six-dimensional approach to microstructure analysis. Phil.Mag. 83, 2761–2778 (2003).CrossRefGoogle Scholar
  24. 24.
    Poulsen, H.F.: Three-Dimensional X-Ray Diffraction Microscopy: Mapping Polycrystals and Their Dynamics. Springer, Berlin, Germany (2004).Google Scholar
  25. 25.
    Poulsen, H.F., Fu, X.: Generation of grain maps by an algebraic reconstruction technique. J. Appl. Cryst., 36, 1062–1068 (2003).CrossRefGoogle Scholar
  26. 26.
    Rodek, L., Knudsen, E., Poulsen, H.F., Herman, G.T.: Discrete tomographic reconstruction of 2D polycrystal orientation maps from X-ray diffraction projections using Gibbs priors. Electr. Notes Discr. Math., 20, 439–453 (2005).CrossRefMathSciNetGoogle Scholar

Copyright information

© Birkhäuser Boston 2007

Authors and Affiliations

  • A. Alpers
    • 1
  • L. Rodek
    • 2
  • H.F. Poulsen
    • 3
  • E. Knudsen
    • 3
  • G.T. Herman
    • 4
  1. 1.School of Operations Research and Industrial Engineering, Cornell UniversityNYUSA
  2. 2.Dept. of Image Processing and Computer GraphicsUniversity of SzegedSzegedHungary
  3. 3.Materials Research DepartmentRisø National LaboratoryRoskildeDenmark
  4. 4.Dept. of Computer ScienceThe Graduate Center, City Univ. of New YorkNew YorkUSA

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