Reconstruction of Connected Sets from Two Projections
The problem of reconstructing a two-dimensional discrete set from its projections has been studied in discrete mathematics and applied in several areas. It has some interesting applications in image processing, electron microscopy, statistical data security, biplane angiography and graph theory. This chapter presents the computational complexity results of the problem of reconstructing a set from its horizontal and vertical projections with respect to some classes of sets on which some connectivity constrzints are imposed. We show that this reconstruction problem can be solved in polynomial time in a class of discrete sets, and is NP-complete otherwise.
KeywordsReconstruction Problem Vertical Projection Disjoint Cycle Median Column Filling Operation
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- R. J. Gardner, P. Gritzmann, and D. Prangenberg, “On the computational complexity of reconstructing lattice sets from their X-rays,” Technical Report 970.05012, Techn. Univ. Miinchen, Fak. f. Math.,Miinchen, (1997).Google Scholar
- 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
- S. W. Golomb, Polyominoes, Revised and Expanded Edition, (Princeton University Press, Princeton, NJ), 1994.Google Scholar
- M. R. Garey and D.S. Johnson, Computers and intractability: A guide to the theory of NP-completeness, (Freeman, New York), 1979.Google Scholar
- G. J. Woeginger, “The reconstruction of polyominoes from their orthogonal projections,” Technical Report SFB-65, TU Graz,Graz, (1996).Google Scholar