Abstract
We study various theoretical and algorithmic aspects of inverse problems in discrete tomography that are motivated by demands from material sciences for the reconstruction of crystalline structures from images produced by quantitative high resolution transmission electron microscopy.
In particular, we discuss questions related to the ill-posedness of the problem, determine the computational complexity of the basic underlying tasks and indicate algorithmic approaches in the presence of XXX-hardness.
Supported in part by a Max Planck Research Award and by the German Federal Ministry of Education, Science, Research and Technology Grant 03-GR7TM1.
Chapter PDF
Keywords
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.
References
R. Aharon, G.T. Herman, and A. Kuba, Binary vectors partially determined by linear equation systems, Discrete Math. 171 (1997), 1–16.
R.P. Anstee, The network flow approach for matrices with given row and column sums, Discrete Math. 44 (1983), 125–138.
E. Barcucci, A. Del Lungo, M. Nivat, and R. Pinzani, X-rays characterizing some classes of digital pictures, preprint.
-, Reconstructing convex polyominoes from their horizontal and vertical projections, Th. Comput. Sci. 155 (1996), 321–347.
G. Bianchi and M. Longinetti, Reconstructing plane sets from projections, Discrete Comput. Geom. 5 (1990), 223–242.
T. Burger and P. Gritzmann, Polytopes in combinatorial optimization, Geometry at Work (C. Gorim, ed.), Math. Assoc. Amer., 1997, in print.
S.K. Chang, The reconstruction of binary patterns from their projections, Comm. ACM 14 (1971), no. 1, 21–25.
J. Edmonds, Maximum matching and a polydedron with 0,1-vertices, J. Research National Bureau of Standards 69B (1965), no. 1 and 2, 125–130.
-, Matroid intersection, Ann. Discrete Math. 4 (1979), 39–49.
P.C. Fishburn, J.C. Lagarias, J. A. Reeds, and L.A. Shepp, Sets uniquely determined by projections on axes. II. discrete case, Discrete Math. 91 (1991), 149–159.
P.C. Fishburn, P. Schwander, L.A. Shepp, and J. Vanderbei, The discrete Radon transform and its approximate inversion via linear programming, Discrete Appl. Math. 75 (1997), 39–62.
R.J. Gardner and P. Gritzmann, Discrete tomography: Determination of finite sets by X-rays, Trans. Amer. Math. Soc 349 (1997), 2271–2295.
R.J. Gardner, P. Gritzmann, and D. Prangenberg, On the reconstruction of binary images from their discrete Radon transforms, Proc. Intern. Symp. Optical Science, Engineering, and Instrumentation (Bellingham, WA), SPIE, 1996, pp. 121–132.
—, Computational complexity issues in discrete tomography: On the reconstruction of finite lattice sets from their line X-rays, submitted; 1997.
—, On the computational complexity of determining polyatomic structures by X-rays, submitted, 1997.
J.J. Gerbrands and C.H. Slump, A network flow approach to reconstruction of the left ventricle from two projections, Computer Graphics and Image Processing 18 (1982), 18–36.
P. Gritzmann and S. de Vries, Polytopes for discrete tomography, in preparation.
P. Gritzmann, S. de Vries, and M. Wiegelmann, On the approximate reconstruction of binary images from their discrete X-rays, in preparation, 1997.
P. Gritzmann, D. Prangenberg, S. de Vries, and M. Wiegelmann, Success and failure of certain reconstruction and uniqueness algorithms in discrete tomography, submitted, 1997.
M. Grötschel and M. Padberg, Polyhedral theory, The Traveling Salesman Problem (E.L. Lawler, J.K. Lenstra, A.H.G. Rinnoy Kan, and D.B. Shmoys, eds.), Wiley-Interscience, Chichester, 1985, pp. 251–305.
R.W. Irving and M.R. Jerrum, Three-dimensional statistical data security problems, SIAM J. Comput. 23 (1994), 170–184.
C. Kisielowski, P. Schwander, F.H. Baumann, M. Seibt, Y. Kim, and A. Ourmazd, An approach to quantitative high-resolution transmission electron microscopy of crystalline materials, Ultramicroscopy 58 (1995), 131–155.
B. Korte and D. Hausmann, An analysis of the greedy heuristic for independence systems, Ann. Discrete Math. 2 (1978), 65–74.
D. Naddef, The Hirsch conjecture is true for (0,1)-polytopes, Math. Prog. 45 (1989), 109–110.
M. Padberg and M. Grötschel, Polyhedral computations, The Traveling Salesman Problem (E.L. Lawler, J.K. Lenstra, A.H.G. Rinnoy Kan, and D.B. Shmoys, eds.), Wiley-Interscience, Chichester, 1985, pp. 307–360.
A. Rényi, On projections of probability distributions, Acta Math. Acad. Sci. Hungar. 3 (1952), 131–142.
H.J. Ryser, Combinatorial properties of matrices of zeros and ones, Canad. J. Math. 9 (1957), 371–377.
-, Combinatorial mathematics, Mathem. Assoc. Amer. and Quinn & Boden, Rahway, NJ, 1963.
P. Schwander, C. Kisielowski, M. Seibt, F.H. Baumann, Y. Kim, and A. Ourmazd, Mapping projected potential, interfacial roughness, and composition in general crystalline solids by quantitative transmission electron microscopy, Physical Review Letters 71 (1993), 4150–4153.
Y. Vardi and D. Lee, The discrete Radon transform and its approximate inversion via the ELM algorithm, preprint, 1997.
C.J. Woeginger, The reconstruction of polyominoes from their orthogonal projections, preprint.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1997 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Gritzmann, P. (1997). On the reconstruction of finite lattice sets from their X-rays. In: Ahronovitz, E., Fiorio, C. (eds) Discrete Geometry for Computer Imagery. DGCI 1997. Lecture Notes in Computer Science, vol 1347. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0024827
Download citation
DOI: https://doi.org/10.1007/BFb0024827
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-63884-1
Online ISBN: 978-3-540-69660-5
eBook Packages: Springer Book Archive