Success and failure of certain reconstruction and uniqueness algorithms in discrete tomography

International Journal of Imaging Systems and Technology

Volumn 9, Issue 2-3, 1998, Pages 101-109

  Gritzmann, P ^a   Prangenberg, D ^b   De Vries, S ^a   Wiegelmann, M ^a  

^a TECHNICAL UNIVERSITY OF MUNICH   (Germany)

^b NONE

Author keywords

[No Author keywords available]

Indexed keywords

ALGORITHMS; APPROXIMATION THEORY; ERROR ANALYSIS; IMAGE RECONSTRUCTION; MATHEMATICAL MODELS;

DISCRETE TOMOGRAPHY;

COMPUTERIZED TOMOGRAPHY;

EID: 0031673351     PISSN: 08999457     EISSN: None     Source Type: Journal    
DOI: 10.1002/(sici)1098-1098(1998)9:2/3<101::aid-ima6>3.0.co;2-f     Document Type: Article

References (19)

1. 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, 131-155 (1995).
2. 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," Phys. Rev. Lett. 71, 4150-4153 (1993).
3. R. J. Gardner, P. Gritzmann, and D. Prangenberg, "Computational complexity issues in discrete tomography: On the reconstruction of finite lattice sets from their line X-rays" (preprint).
4. A. Schrijver. Theory of Linear and Integer Programming (Wiley-Interscience, Chichester), 1986.
5. H. J. Ryser. "Combinatorial properties of matrices of zeros and ones," Can. J. Math. 9, 371-377 (1957).
6. S. K. Chang. "The reconstruction of binary patterns from their projections," Commun. ACM 14, 21-25 (1971).
7. J. J. Gerbrands and C. H. Slump. "A network flow approach to reconstruction of the left ventricle from two projections," Comput. Graphics Image Process. 18, 18-36 (1982).
8. M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness (Freeman, San Francisco), 1979.
9. D. B. Shmoys, "Computing near-optimal solutions to combinatorial optimization problems," in Combinatorial Optimization, DIMACS Series in Discrete Math. Theor. Comput. Sci., Vol. 20. W. Cook, Ed., AMS, Providence, 1995, pp. 355-397.
10. P. M. Salzberg, P. I. Rivera-Vega, and A. Rodriguez. "A network flow model for binary tomography on lattices," Int. J. Imaging Syst. Technol. 9, 147-154 (1998).
11. K. G. Murty, C. Spera, and T. Yi. "Matchings in colored networks," tech. rep., University of Michigan, 1993.
12. C. H. Papadimitriou and M. Yannakakis, "Optimization, approxima tion, and complexity classes," J. Comput. System Sci. 43, 425-440 (1991).
13. 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, 149-159 (1991).
14. P. C. Fishbum, P. Schwander, L. A. Shepp, and J. Vanderbei. "The discrete Radon transform and its approximate inversion via linear programming," Discrete Appl. Math. 75, 39-62 (1997).
15. T. Y. Kong and G. T. Herman. "On which grids can tomographic equivalence of binary pictures be characterized in terms of elementary switching operations," Int. J. Imaging Syst. Technol. 9, 118-125 (1998).
16. P. Gritzmann, S. de Vries, and M. Wiegelmann. "On the approximate reconstruction of binary images from their discrete X-rays" (in preparation).
17. R. Aharoni, G. T. Herman, and A. Kuba. "Binary vectors partially determined by linear equation systems," Discrete Math. 171, 1-16 (1997).
18. Y. Vardi and D. Lee. "The discrete Radon transform and its approximate inversion via the ELM algorithm," Int. J. Imaging Syst. Techno. 9, 155-173 (1998).
19. P. Gritzmann and S. de Vries. "On the facial structure of tomography polytopes" (in preparation).