Abstract
In this chapter we present an algebraic theory of patterns that can be applied in discrete tomography for any dimension. We use that the difference of two such patterns yields a configuration with vanishing line sums. We show by introducing generating polynomials and applying elementary properties of polynomials that such so-called switching configurations form a linear space. We give a basis of this linear space in terms of the so-called switching atom, the smallest nontrivial switching configuration. We do so both in case that the material does not absorb light and absorbs light homogeneously. In the former case we also show that a configuration can be constructed with the same line sums as the original and with entries of about the same size, and we provide a formula for the number of linear dependencies between the line sums. In the final section we deal with the case that the transmitted light does not follow straight lines.
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References
Barcucci, E., Del Lungo, A., Nivat, M., Pinzani, R.: X-rays characterizing some classes of discrete sets. Lin. Algebra Appl., 339, 3–21 (2001).
Barcucci, E., Frosini, A., Rinaldi, S.: Reconstruction of discrete sets from two absorbed projections: An algorithm. Electr. Notes Discr. Math., 12 (2003).
Batenburg, K.J.: Reconstruction of binary images from discrete X-rays. CWI, Technical Report PNA-E0418, ftp.cwi.nl/CWIreports/PNA/PNA-E0418.pdf (2004).
Batenburg, K.J.: A new algorithm for 3D binary tomography. Electr. Notes Discr. Math., 20, 247–261 (2005).
Cohen, H.: A Course in Computational Algebraic Number Theory. Springer, Berlin, Germany (1993).
Del Lungo, A., Gronchi, P., Herman, G.T. (eds.): Proceedings of the Workshop on Discrete Tomography: Algorithms and Applications. Lin. Algebra Appl., 339, 1–219 (2001).
Gardner, R.J.: Geometric Tomography. Cambridge University Press, Cambridge, UK (1995).
Gardner, R.J., Gritzmann, P.: Discrete tomography: Determination of finite sets by X-rays. Trans. Amer. Math. Soc., 349, 2271–2295 (1997).
Gardner, R.J., Gritzmann, P.: Uniqueness and complexity in discrete tomography. In: Herman, G.T., Kuba, A. (eds.), Discrete Tomography: Foundations, Algorithms, and Applications. Birkhäuser, Boston, MA, pp. 85–113 (1999).
Gardner, R.J., Gritzmann, P., Prangenberg, D.: On the computational complexity of reconstructing lattice sets from their X-rays. Discr. Math., 202, 45–71 (1999).
Hajdu, L.: Unique reconstruction of bounded sets in discrete tomography. Electr. Notes Discr. Math., 20, 15–25 (2005).
Hajdu, L., Tijdeman, R.: Algebraic aspects of discrete tomography. J. Reine Angew. Math., 534, 119–128 (2001).
Hajdu, L., Tijdeman, R.: An algorithm for discrete tomography. Lin. Algebra Appl., 339, 147–169 (2001).
Hajdu, L., Tijdeman, R.: Algebraic aspects of emission tomography with absorption. Theoret. Comput. Sci., 290, 2169–2181 (2003).
Herman, G.T., Kuba, A. (eds.): Discrete Tomography: Foundations, Algorithms, and Applications. Birkhäuser, Boston, MA (1999).
Herman, G.T., Kuba, A. (eds.): Proceedings of the Workshop on Discrete Tomography and Its Applications. Electr. Notes Discr. Math., 20, 1–622 (2005).
Kong, T.Y., Herman, G.T.: 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).
Kong, T.Y., Herman, G.T.: Tomographic equivalence and switching operations. In: Herman, G.T., Kuba, A. (eds.), Discrete Tomography: Foundations, Algorithms, and Applications. Birkhäuser, Boston, MA, pp. 59–84 (1999).
Kuba, A., Herman, G.T.: Discrete tomography: A historical overview. In: Herman, G.T., Kuba, A. (eds.), Discrete Tomography: Foundations, Algorithms, and Applications. Birkhauser, Boston, MA, pp. 3–34 (1999).
Kuba, A., Nivat, M.: Reconstruction of discrete sets with absorption. In: Borge-fors, G., Nystrm, I., Sanniti di Baja, G. (eds.), Discrete Geometry in Computer Imagery, Springer, Berlin, Germany, pp. 137–148 (2000).
Kuba, A., Nivat, M.: Reconstruction of discrete sets with absorption. Lin. Algebra Appl., 339, 171–194 (2001).
Lang, S.: Algebra. Addison-Wesley, Reading, MA (1984).
Ryser, H.J.: Combinatorial properties of matrices of zeros and ones. Canad. J. Math., 9, 371–377 (1957).
Schinzel, A.: Polynomials with Special Regard to Reducibility. Cambridge University Press, Cambridge, UK (2000).
Shliferstein, H.J., Chien, Y.T.: Switching components and the ambiguity problem in the reconstruction of pictures from their projections. Pattern Recognition, 10, 327–340 (1978).
Zopf, S., Kuba, A.: Reconstruction of measurable sets from two generalized projections. Electr. Notes Discr. Math., 20, 47–66 (2005).
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Hajdu, L., Tijdeman, R. (2007). Algebraic Discrete Tomography. In: Herman, G.T., Kuba, A. (eds) Advances in Discrete Tomography and Its Applications. Applied and Numerical Harmonic Analysis. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4543-4_4
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DOI: https://doi.org/10.1007/978-0-8176-4543-4_4
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