Uniqueness and Complexity in Discrete Tomography
We study the discrete inverse problem of reconstructing finite subsets of the n-dimensional integer lattice ℤn that are only accessible via their line sums (discrete X-rays) in a finite set of lattice directions. Special emphasis is placed on the question of when such sets are uniquely determined by the data and on the difficulty of the related algorithmic problems. Such questions are motivated by demands from the material sciences for the reconstruction of crystalline structures from images produced by quantitative high-resolution transmission electron microscopy.
KeywordsLattice Line Finite Subset Truth Assignment Cross Ratio Discrete Tomography
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- 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, 4150–4153 (1993).CrossRefPubMedGoogle Scholar
- R. J. Gardner, Geometric Tomography (Cambridge University Press, New York) 1995.Google Scholar
- E. Barcucci, A. Del Lungo, M. Nivat, and R. Pinzani, “X-rays characterizing some classes of digital pictures,” (Technical Report RT 4/96, Dipartimento di Sistemi e Informatica, Universita di Firenze, Firenze), 1996.Google Scholar
- R. J. Gardner, “Geometric tomography,” Notices Amer. Math. Soc. 42, 422–429 (1995).Google Scholar
- A. Schrijver, Theory of Linear and Integer Programming (Wiley, New York), 1987.Google Scholar
- 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, SPIE, pp. 121–132 (1996).Google Scholar
- P. Gritzmann and M. Wiegelmann, “On combinatorial patterns given by cross-characteristics: Uniqueness versus additivity,” in preparation.Google Scholar
- R. J. Gardner, P. Gritzmann, and D. Prangenberg, “On the computational complexity of inverting higher-dimensional discrete X-ray transforms,” in preparation.Google Scholar
- R. J. Gardner, P. Gritzmann, and D. Prangenberg, “On the computational complexity of determining polyatomic structures by X-rays,” Theor. Comput. Sci., to appear.Google Scholar
- M. Chrobak and C. Dürr, “Reconstructing polyatomic structures from discrete X-rays: NP-completeness proof for three atoms,” preprint.Google Scholar