Discrete Tomography pp 163-188 | Cite as

# Reconstruction of Connected Sets from Two Projections

## Abstract

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.

## Keywords

Reconstruction Problem Vertical Projection Disjoint Cycle Median Column Filling Operation## Preview

Unable to display preview. Download preview PDF.

## References

- [1]A. R. Shliferstein and Y. T. Chien, “Switching components and the ambiguity problem in the reconstruction of pictures from their projections,”
*Pattern Recognition*,**10**, 327–340 (1978).CrossRefGoogle Scholar - [2]R. W. Irving and M. R. Jerrum, “Three-dimensional statistical data security problems,”
*SIAM Journal of Computing*,**23**, 170–184 (1994).CrossRefGoogle Scholar - [3]G. P. M. Prause and D. G. W. Onnasch, “Binary reconstruction of the heart chambers from biplane angiographic image sequence,”
*IEEE Trans. Medical Imaging*,**15**, 532–559 (1996).CrossRefGoogle Scholar - [4]R. P. Anstee, “Invariant sets of arcs in network flow problems,”
*Discrete Applied Mathematics*,**13**, 1–7 (1986).CrossRefGoogle Scholar - [5]P. Fishburn, P. Schwander, L. Shepp, and R. J. Vanderbei, “The discrete Radon transform and its approximate inversion via linear programming,”
*Discrete Applied Mathematics*,**75**, 39–61 (1997).CrossRefGoogle Scholar - [6]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 - [7]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).CrossRefGoogle Scholar - [8]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 - [9]H. J. Ryser, “Combinatorial properties of matrices of zeros and ones,”
*Canad. J. Math.*,**9**, 371–377 (1957).CrossRefGoogle Scholar - [10]S. K. Chang, “The reconstruction of binary patterns from their projections,”
*Communications ACM*,**14**, 21–24 (1971).CrossRefGoogle Scholar - [11]R. A. Brualdi, “Matrices of zeros and ones with fixed row and column sum vectors,”
*Linear Algebra and Applications*,**33**, 159–231 (1980).CrossRefGoogle Scholar - [12]A. Del Lungo, “Polyominoes defined by two vectors,”
*Theoretical Computer Science*,**127**, 187–198 (1994).CrossRefGoogle Scholar - [13]A. Kuba, “The reconstruction of two-directionally connected binary patterns from their two orthogonal projections,”
*Computer Vision*,*Graphics*,*and Image Processing*,**27**, 249–265 (1984).CrossRefGoogle Scholar - [14]S. K. Chang and C. K. Chow, “The reconstruction of three-dimensional objects from two orthogonal projections and its application to cardiac cineangiography,”
*IEEE Trans. on Computers*,**22**, 18–28 (1973).CrossRefGoogle Scholar - [15]S. W. Golomb,
*Polyominoes*, Revised and Expanded Edition, (Princeton University Press, Princeton, NJ), 1994.Google Scholar - [16]E. Barcucci, A. Del Lungo, M. Nivat, and R. Pinzani, “Reconstructing convex polyominoes from horizontal and vertical projections,”
*Theoretical Computer Science*,**155**, 321–347 (1996).CrossRefGoogle Scholar - [17]M. R. Garey and D.S. Johnson,
*Computers and intractability: A guide to the theory of NP-completeness*, (Freeman, New York), 1979.Google Scholar - [18]G. J. Woeginger, “The reconstruction of polyominoes from their orthogonal projections,”
*Technical Report SFB-65*,*TU Graz*,*Graz*, (1996).Google Scholar - [19]E. Barcucci, A. Del Lungo, M. Nivat, and R. Pinzani, “Medians of polyominoes: a property for the reconstruction,”
*International Journal of Imaging Systems and Technology*,**8**, 69–77 (1998).CrossRefGoogle Scholar - [20]B. Aspvall, M. F. Plass, and R. E. Tarjan, “A linear-time algorithm for testing the truth of certain quantified Boolean formulas,”
*Information Processing Letters*,**8**, 121–123 (1979).CrossRefGoogle Scholar