Skip to main content
SpringerLink
Log in

Introduction

  • Chapter
Book cover Advances in Discrete Tomography and Its Applications

Part of the book series: Applied and Numerical Harmonic Analysis ((ANHA))

Abstract

This chapter discuses the excellent progress made in discrete tomography (DT) during the last seven years and includes a comprehensive bibliography illustrating this progress. It also presents some of the fundamental definitions relevant to DT.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Alpers, A.: Instability and Stability in Discrete Tomography. Ph.D. Thesis, Technical Univ. of Munich, Shaker Verlag, Germany (2004).

    Google Scholar 

  2. Alpers, A., Brunetti, S.: On the stability of reconstructing lattice sets from X-rays along two directions. In: Andres, E., Damiand, G., Lienhardt, P. (eds.), Digital Geometry for Computer Imagery, Springer, Berlin, Germany, pp. 92–103(2005).

    Google Scholar 

  3. Alpers, A., Gritzmann, P.: On stability, error correction, and noise compensation in discrete tomography. SIAM J. Discr. Math., 20, 227–239 (2006).

    Article  MATH  MathSciNet  Google Scholar 

  4. Alpers, A., Gritzmann, P., Thorens, L.: Stability and instability in discrete tomography. In: Bertrand, G., Imiya, A., Klette, R. (eds.), Digital and Image Geometry, Springer, Berlin, Germany, pp. 175–186 (2001).

    Chapter  Google Scholar 

  5. Alpers, A., Knudsen, E., Poulsen, H., Herman, G.: Resolving ambiguities in reconstructed grain maps using discrete tomography. Electr. Notes Discr. Math., 20, 419–437 (2005).

    Article  MathSciNet  Google Scholar 

  6. Alpers, A., Poulsen, H., Knudsen, E., Herman, G.: A discrete tomography algorithm for improving the quality of 3DXRD grain maps. J. Appl. Crystallography, 39, 281–299 (2006).

    Google Scholar 

  7. Autrusseau, F., Guédon, J.: Chiffrement Mojette d’images m’medicales. Syst’emes d’Information de Santé (Ingénierie des Systémes d’Information-RSTI Série ISI), 8, 113–134 (2003).

    Google Scholar 

  8. Baake, M., Gritzmann, P., Huck, C., Langfeld, B., Lord, K.: Discrete tomography of planar model sets. Acta Crystallographica Section A., 62, 419–433 (2006).

    Article  MathSciNet  Google Scholar 

  9. Bakirov, V.F., Kline, R.A., Winfree, W.P.: Discrete variable thermal tomography.In: Thompson, D.O., Chimenti, D.E. (eds.): Review of Quantitative Nondestructive Evaluation, American Institute of Physics, 23, pp. 469–476 (2004).

    Google Scholar 

  10. Balaskó, M., Kuba, A., Nagy, A., Kiss, Z., Rodek, L., Ruskó, L.: Neutrongamma-and X-ray three-dimensional computer tomography at the Budapest research reactor, Nucl. Inst. & Meth., A, 542, 22–27 (2005).

    Article  Google Scholar 

  11. Balaskó, M., Sváb, E., Kuba, A., Kiss, Z., Rodek, L., Nagy, A.,: Pipe corrosion and deposit study using neutron-and gamma-radiation sources, Nucl. Inst. & Meth., A, 542, 302–308 (2005).

    Article  Google Scholar 

  12. Balázs, P.: Reconstruction of decomposable discrete sets from four projections: Strong decomposability. In: Andres, E., Damiand, G., Lienhardt, P. (eds.), Discrete Geometry in Computer Imagery, Springer, Berlin, Germany, pp. 104–114 (2005).

    Google Scholar 

  13. Balázs, P.: Reconstruction of discrete sets from four projections: Strong decomposability.Electr. Notes Discr. Math., 20, 329–345 (2005)

    Article  Google Scholar 

  14. Balazs, P., Balogh, E., Kuba, A.: A fast algorithm for reconstructing hvconvex 8-connected but not 4-connected discrete sets. In: Nyström, I., Sanniti di Baja, G., Svenson, S. (eds.), Discrete Geometry for Computer Imagery, Springer, Berlin, Germany, pp. 388–397 (2003).

    Google Scholar 

  15. Balázs, P., Balogh, E., Kuba, A.: Reconstruction of 8-connected but not 4-connected hv-convex discrete sets. Discr. Appl. Math., 147, 149–168 (2005).

    Article  MATH  Google Scholar 

  16. Balogh, E., Kuba, A.: Reconstruction algorithms for hv-convex 4-and 8-connected discrete sets. In: Loncaric, S., Babic, H. (eds.), Proc. 2nd Intl.Symp. on Image and Signal Processing and Analysis, ISPA 2001, Pula, Croatia, pp. 49–54 (2001).

    Google Scholar 

  17. Balogh, E., Kuba, A., Del Lungo, A., Nivat, M.: Reconstruction of binary matrices from absorbed projections. In Braquelaire, A., Lachaud, J.-O., Vialard, A. (eds.), Discrete Geometry for Computer Imagery, Springer, Berlin, Germany, pp. 392–403 (2002).

    Chapter  Google Scholar 

  18. Balogh, E., Kuba, A., Dévényi, C., Del Lungo, A.: Comparison of algorithms for reconstructing hv-convex discrete sets. Lin. Algebra Appl., 339, 23–35 (2001).

    Article  MATH  Google Scholar 

  19. Barcucci, E., Brunetti, S., Del Lungo, A., Nivat, M.: Reconstruction of discrete sets from three or more X-rays. In: Bongiovanni, G., Gambosi, G., Petreschi, R. (eds.), Algorithms and Complexity, Springer, Berlin, Germany, pp. 199–210 (2000).

    Chapter  Google Scholar 

  20. Barcucci, E., Brunetti, S., Del Lungo, A., Nivat, M.: Reconstruction of lattice sets from their horizontal, vertical and diagonal X-rays. Discr. Math., 241, 65–78 (2001).

    Article  MATH  Google Scholar 

  21. Barucci, E., Del Lungo, A., Nivat, M., Pinzani, R.: X-rays characterizing some classes of discrete sets. J. Linear Algebra, 339, 3–21 (2001).

    Article  Google Scholar 

  22. Barcucci, E., Frosini, A., Rinaldi, S.: Reconstruction of discrete sets from two absorbed projections: An algorithm. Electr. Notes Discr. Math., 12 (2003)

    Google Scholar 

  23. Barcucci, E., Frosini, A., Rinaldi, S.: An algorithm for the reconstruction of discrete sets from two projections in presence of absorption. Discr. Appl. Math., 151, 21–35 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  24. Batenburg, K.J.: Analysis and optimization of an algorithm for discrete tomography.Electr. Notes Discr. Math., 12 (2003)

    Google Scholar 

  25. Batenburg, K.J.: A new algorithm for 3D binary tomography. Electr. Notes Discr. Math., 20, 247–261 (2005)

    Article  MathSciNet  Google Scholar 

  26. Batenburg, K.J.: An evolutionary algorithm for discrete tomography. Discr.Appl. Math., 151, 36–54 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  27. Batenburg, K.J.: Network Flow Algorithms for Discrete Tomography. Ph.D.Thesis, Leiden Univ., The Netherlands (2006)

    Google Scholar 

  28. Batenburg, K.J., Kosters, W.A.: A discrete tomography approach to Japanese Puzzles. In: Verbrugge, R., Taatgen, N., Schomaker, L. (eds.), Proc. 16th Belgian-Dutch Conf. Artificial Intelligence, Groningen, The Netherlands, pp. 243–250 (2004)

    Google Scholar 

  29. Batenburg, K.J., Kosters, W.A.: Neural networks for discrete tomography. In: Verbeeck, K., Tuyls, K., Nowé, A., Manderick, B., Kuijpers, B. (eds.), Proc.17th Belgian-Dutch Conf. Artificial Intelligence, Brussels, Belgium, pp. 21–27 (2005)

    Google Scholar 

  30. Batenburg, K.J., Palenstijn, W.J.: On the reconstruction of crystals through discrete tomography. In: Reulke, R., Eckardt, U., Flach, B., Knauer, U., Polthier, K. (eds.), Combinatorial Image Analysis, Springer, Berlin, Germany, pp. 23–37 (2006)

    Google Scholar 

  31. Batenburg, K.J., Sijbers, J.: Discrete tomography from micro-CT data: Application to the mouse trabecular bone structure. Proc. SPIE, 6142, pp. 1325–1335 (2006)

    Google Scholar 

  32. Battle, X.L., Bizais, Y.: 3D attenuation map reconstruction using geometrical models and free-form deformations. In: Beekman, F., Defrise, M., Viergever, M.(eds.), Proc. Fully Three-Dimensional Image Reconstruction in Radiology and Nuclear Medicine, Egmond aan Zee, The Netherlands, pp. 181–184 (1999)

    Google Scholar 

  33. Battle, X.L., Bizais, Y., Le Rest, C., Turzo, A.: Tomographic reconstruction using free-form deformation models. Proc. SPIE, 3661, 356–367 (1999)

    Article  Google Scholar 

  34. Battle, X.L., Le Rest, A., Turzo, C., Bizais, Y.: Three-dimensional attenuation map reconstruction using geometrical models and free-form deformations. IEEE Trans. Med. Imag., 19, 404–411 (2000)

    Article  Google Scholar 

  35. Battle, X.L., Le Rest, A., Turzo, C. Bizais, Y.: Free-form deformation in tmographic reconstruction. Application to attenuation map reconstruction. IEEE Trans. Nucl. Sci., 47, 1065–1071 (2000)

    Article  Google Scholar 

  36. Bebeacua, C., Mansour, T., Postnikov, A., Severini, S.: On the X-rays of permutations.Electr. Notes Discr. Math., 20, 193–203 (2005)

    Article  MathSciNet  Google Scholar 

  37. Beldiceanu, N., Katriel, I., Thiel, S.: Filtering algorithms for the Same and UsedBy constraints. In: Regin, J.C., Rueher, M. (eds.), Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimisation Problems Springer, Berlin, Germany, pp. 65–79 (2004)

    Google Scholar 

  38. Boufkhad, Y., Dubois, O., Nivat, M.: Reconstructing (h,v)-convex 2-dimensional patterns of objects from approximate horizontal and vertical projections.Theor. Comput. Sci., 290, 1647–1664 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  39. Brewbaker, C.R.: Lonesum (0,1)-Matrices and Poly-Bernoulli Numbers of Negative Index. MSc. Thesis, Iowa State Univ. (2005)

    Google Scholar 

  40. Brimkov, V.E., Barneva, R.P.: Exact image reconstruction from a single projection through real computation. Electr. Notes Discr. Math., 20, 233–246 (2005)

    Article  MathSciNet  Google Scholar 

  41. Brualdi, R.A.: Minimal nonnegative integral matrices and uniquely determined (0,1)-matrices. Lin. Algebra Appl., 341, 351–356 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  42. Brualdi, R.A., Dahl, G.: Matrices of zeros and ones with given line sums and a zero block. Lin. Algebra Appl., 371, 191–207 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  43. Brualdi, R.A., Hwang, S.-G.: A Bruhat order for the class of (0,1)-matrices with row sum vector R and column sum vector.S. Electr. J. Linear Algebra, 12, 6–16 (2004)

    MathSciNet  Google Scholar 

  44. Bruandet, J.P., Peyrin, F., Dinten, J.M., Amadieu, O., Barlaud, M.: Binary objects tomographic reconstruction from few noisy X-ray radiographs using a region based curve evolution method. IEEE Nuclear Science Symposium Conference Record, San Diego, USA, pp. 1717–1719 (2001)

    Google Scholar 

  45. Brunetti, S.: Convexity and Complexity in Discrete Tomography. Ph.D. Thesis, Univ. of Florence, Italy (2001)

    Google Scholar 

  46. Brunetti, S., Daurat, A.: Reconstruction of discrete sets from two or more X-rays in any direction. Proc. 7th Intl. Workshop on Combinatorial Image Analysis, Caen, France, pp. 241–258 (2000)

    Google Scholar 

  47. Brunetti, S., Daurat, A.: An algorithm reconstructing lattice convex sets. Theoret.Comput. Sci., 304, 35–57 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  48. Brunetti, S., Daurat, A.: Stability in discrete tomography: Linear programming, additivity and convexity. In: Nystrom, I., Sanniti di Baja, G., Svensson, S. (eds.), Discrete Geometry for Computer Imagery, Springer, Berlin, Germany, pp. 398–407 (2003)

    Google Scholar 

  49. Brunetti, S., Daurat, A.: Determination of Q-convex bodies by X-rays. Electr.Notes Discr. Math., 20, pp. 67–81 (2005)

    Article  MathSciNet  Google Scholar 

  50. Brunetti, S., Daurat, A.: Stability in discrete tomography: Some positive results.Discr. Appl. Math., 147, pp. 207–226 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  51. Brunetti, S., Daurat, A., Del Lungo, A.: Approximate X-rays reconstruction of special lattice sets. Pure Math. Appl., 11, 409–425 (2000)

    MATH  MathSciNet  Google Scholar 

  52. Brunetti, S., Daurat, A., Del Lungo, A.: An algorithm for reconstructing special lattice sets from their approximate X-rays, In: Borgefors, G., Nyström, I., Sanniti di Baja, G. (eds.), Discrete Geometry for Computer Imagery, Springer, Berlin, Germany, pp. 113–125 (2000)

    Chapter  Google Scholar 

  53. Brunetti, S., Del Lungo, A., Gerard, Y.: On the computational complexity of reconstructing three-dimensional lattice sets from their two dimensional X-rays.Lin. Algebra Appl. 339, 59–73 (2001)

    Article  MATH  Google Scholar 

  54. Brunetti, S., Del Lungo, A., Del Ristoro, F., Kuba, A., Nivat, M.: Reconstruction of 4-and 8-connected convex discrete sets from row and column projections. Lin. Algebra Appl. 339, 37–57 (2001)

    Article  MATH  Google Scholar 

  55. Capricelli, T.D., Combettes, P.L.: Parallel block-iterative reconstruction algorithms for binary tomography. Electr. Notes Discr. Math., 20, 263–280 (2005)

    Article  MathSciNet  Google Scholar 

  56. Carvalho, B.M., Herman, G.T., Matej, S., Salzberg, C., Vardi, E.: Binary tomography for triplane cardiography. In: Kuba, A., Samal, M., ToddPokropek, A. (eds.), Information Processing in Medical Imaging, Springer, Berlin, Germany, pp. 29–41 (1999)

    Chapter  Google Scholar 

  57. Castiglione, G., Frosini, A., Restivo, A., Rinaldi, S.: A tomographical characterization of L-convex polyominoes. In: Rangarajan, A., Vemuri, B.C., Yuille, A.L. (eds.), Discrete Geometry in Computer Imagery, Springer, Berlin, Germany, pp. 115–125 (2005)

    Google Scholar 

  58. Castiglione, G., Restivo, A.: Reconstruction of L-convex polyominoes. Electr. Notes Discr. Math., 12 (2003).

    Google Scholar 

  59. Censor, Y.: Binary steering in discrete tomography reconstruction with sequential and simultaneous iterative algorithms. Lin. Algebra Appl., 339 111–124 (2001).

    Article  MATH  MathSciNet  Google Scholar 

  60. Chrobak, M., Couperus, P., Dürr, C., Woeginger, G.: On tiling under tomographic constraints. Theor. Comp. Sci. 290, 2125–2136 (2003).

    Article  MATH  Google Scholar 

  61. Chrobak, M., Dürr, C.: Reconstructing hv-convex polyominoes from orthogonal projections. Inform. Process. Lett., 69, 283–289 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  62. Chrobak, M., Dürr, C.: Reconstructing polyatomic structures from X-rays: NPcompleteness proof for three atoms. Theor. Comp. Sci., 259, 81–98 (2001).

    Article  MATH  Google Scholar 

  63. Costa, M. C., Jarray, F., Picouleau, C.: Reconstruction of binary matrices under adjacency constraints. Electr. Notes Discr. Math. 20, 281–297 (2005).

    Article  MathSciNet  Google Scholar 

  64. Costa, M. C., de Werra, D., Picouleau, C.: Using graphs for some discrete tomography problems. Discr. Appl. Math., 154, 35–46 (2006).

    Article  MATH  Google Scholar 

  65. Costa, M. C., de Werra, D., Picouleau, C., Schindl, D.: A solvable case of image reconstruction in discrete tomography. Discr. Appl. Math., 148, 240–245 (2005).

    Article  MATH  Google Scholar 

  66. Dahl, G., Brualdi, R.A.: Matrices of zeros and ones with given line sums and a zero block. Electr. Notes Discr. Math., 20, 83–97 (2003).

    Article  MathSciNet  Google Scholar 

  67. Dahl, G., Flatberg, T.: Optimization and reconstruction of hv-convex (0, 1)matrices. Electr. Notes Discr. Math., 12 (2003).

    Google Scholar 

  68. Dahl, G., Flatberg, T.: Optimization and reconstruction of hv-convex (0, 1)matrices. Discr. Appl. Math., 151, 93–105 (2005).

    Article  MATH  MathSciNet  Google Scholar 

  69. Daurat, A.: Convexité dans le Plan Discret. Application à la Tomographie, Ph.D. Thesis, LLAIC1, and LIAFA, Université Paris 7, France (2000).

    Google Scholar 

  70. Daurat, A.: Determination of Q-convex sets by X-rays. Theoret. Comput. Sci.,332 19–45 (2005).

    Article  MATH  MathSciNet  Google Scholar 

  71. Daurat, A., Del Lungo, A., Nivat, M.: Medians of discrete sets according to a linear distance. Discrete Comput. Geom., 23, 465–483 (2000).

    MATH  MathSciNet  Google Scholar 

  72. Debled-Rennesson, I., Remy, J.-L., Rouyer-Degli, J.: Detection of the discrete convexity of polyominoes. In: Borgefors, G., Nyström, I., Sanniti di Baja, G. (eds.), Disc. Geometry in Computer Imagery Springer, Berlin, Germany, pp. 491–504 (2000).

    Chapter  Google Scholar 

  73. Debled-Rennesson, I., Remy, J.-L., Rouyer-Degli, J.: Detection of the discrete convexity of polyominoes. Discr. Appl. Math., 125, 115–133 (2003).

    Article  MATH  MathSciNet  Google Scholar 

  74. Del Lungo, A.: Reconstructing permutation matrices from diagonal sums. Theor. Comput. Sci., 281, 235–249 (2002).

    Article  MATH  Google Scholar 

  75. Del Lungo, A., Frosini, A., Nivat, M., Vuillon, L.: Discrete tomography: Reconstruction under periodicity constraints. In: Widmayer, P., Triguero, F., Morales, R., Hennessy, M., Eidenbenz, S., Conejo, R. (eds.), Automata, Languages and Programming, Springer, Berlin, Germany, pp. 38–56 (2002).

    Chapter  Google Scholar 

  76. 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).

    Article  Google Scholar 

  77. Di Gesu, V.D., Valenti, C.: The stability problem and noisy projections in discrete tomography. J. Visual Languages and Computing, 15 361–371 (2004).

    Article  Google Scholar 

  78. Dulio, P., Gardner, R.J., Peri, C.: Discrete point X-rays of convex lattice sets. Electronic Notes in Discrete Math., 20, 1–13 (2005).

    Article  MathSciNet  Google Scholar 

  79. Dulio, P., Gardner, R.J., Peri, C.: Discrete point X-rays. SIAM J. Discrete Math., 20, 171–188 (2006).

    Article  MATH  MathSciNet  Google Scholar 

  80. Dürr, C., Goles, E., Rapaport, I., Remila, E.: Tiling with bars under tomographic constraints. Theor. Comput. Sci., 290, 1317–1329 (2003).

    Article  MATH  Google Scholar 

  81. Frosini, A.: Complexity Results and Reconstruction Algorithms for Discrete Tomography, Ph.D. Thesis, Univ. of Siena, Italy (2003).

    Google Scholar 

  82. Frosini, A., Barcucci, E., Rinaldi, S.: An algorithm for the reconstruction of discrete sets from two projections in present of absorption. Disc. Appl. Math., 151, 21–35 (2005).

    Article  MATH  MathSciNet  Google Scholar 

  83. Frosini, A., Nivat, M.: Binary matrices under the microscope. A tomographical problem. In: Klette, R, Zunic, J.D. (eds.), Combinatorial Image Analysis, Springer, Berlin, Germany, pp. 1–22 (2004).

    Google Scholar 

  84. Frosini, A., Nivat, M., Vuillon, L.: An introductive analysis of periodical discrete sets from a tomographical point of view. Theor. Comput. Sci., 347, 370–392 (2005).

    Article  MATH  MathSciNet  Google Scholar 

  85. Frosini, A., Rinaldi, S.: The complexity of the reconstruction of (r,h,v) from two projections and an approximation algorithm. Pure Math. Appl., 11, 485–496 (2000).

    MATH  MathSciNet  Google Scholar 

  86. Frosini, A., Rinaldi, S., Barcucci, E., Kuba, A.: An efficient algorithm for reconstructing binary matrices from horizontal and vertical absorbed projections.Electr. Notes Discr. Math., 20, 347–363 (2005).

    Article  MathSciNet  Google Scholar 

  87. Frosini, A., Simi, G.: The reconstruction of a bicolored domino tiling from two projections. In: Widmayer, P., Triguero, F., Morales, R., Hennessy, M., Eidenbenz, S., Conejo, R. (eds.), Automata, Languages and Programming Springer, Berlin, Germany, pp. 136–144 (2002).

    Google Scholar 

  88. Frosini, A., Simi, G.: Reconstruction of low degree domino tilings. Electr. Notes Discr. Math., 12 (2003).

    Google Scholar 

  89. Frosini, A., Simi, G.: The NP-completeness of a tomographical problem on bicolored domino tilings. Theor. Comput. Sci., 319, 447–454 (2004).

    Article  MATH  MathSciNet  Google Scholar 

  90. Gardner, R.J.: Geometric Tomography 2nd edition, Cambridge University Press, New York, NY (2006).

    MATH  Google Scholar 

  91. Gardner, R.J., Gritzmann, P.: Discrete tomography: Determination of finite sets by X-rays. J. Linear Algebra, 339, 3–21 (2001).

    Article  Google Scholar 

  92. Gardner, R.J., Gritzmann, P., Prangenberg, D.: On the computational complexity of reconstructing lattice sets from their X-rays. Discrete Math. 202, 45–71 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  93. Gardner, R.J., Gritzmann, P., Prangenberg, D.: On the computational complexity of determining polyatomic structures from by X-rays. Theor. Comput. Sci., 233, 91–106 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  94. Gerard, Y.: Reduction from three-dimensional discrete tomography to multicommodity flow problem. Theor. Comput. Sci., 346, 300–306 (2005).

    Article  MATH  MathSciNet  Google Scholar 

  95. Gerard, Y., Feschet, F.: Application of a discrete tomography algorithm to computerized tomography. Electr. Notes Discr. Math., 20, 501–517 (2005).

    Article  MathSciNet  Google Scholar 

  96. Gcebala, M.: The reconstruction of polyominoes from approximately orthogonal projections. In: Pacholski, L., Ruika, P. (eds.), ’Current Trends in Theory and Practice of Informatics, Springer, Berlin, Germany, pp. 253–260 (2001).

    Google Scholar 

  97. Gcebala, M.: The reconstruction of some 3D convex polyominoes from orthogonal projections. In: Grosky, W.I., Plásil, F. (eds.), Current Trends in Theory and Practice of Informatics, Springer, Berlin, Germany, pp. 262–272 (2002).

    Google Scholar 

  98. Gritzmann, P., de Vries, S.: On the algorithmic inversion of the discrete Radon transform. Theor. Comp. Science, 281, 455–469 (2001).

    Article  Google Scholar 

  99. Gritzmann, P., de Vries, S.: Reconstructing crystalline structures from few images under high resolution transmission electron microscopy. In: Jäger, W. (ed.), Mathematics: Key Technology for the Future, Springer, Berlin, Germany, pp. 441–459 (2003).

    Google Scholar 

  100. Gritzmann, P, de Vries, S., Wiegelmann, M.: Approximating binary images from discrete X-rays. SIAM J. Optim., 11, 522–546 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  101. Guédon, J., Normand, N.: The mojette transform: The first ten years. In: Andres, Ê., Damiand, G., Lienhardt, P. (eds.), Discrete Geometry for Computer Imagery, Springer, Berlin, Germany, pp. 79–91 (2005).

    Google Scholar 

  102. Hajdu, L.: Unique reconstruction of bounded sets in discrete tomography. Electr. Notes Discr. Math., 20, 15–25 (2005).

    Article  MathSciNet  Google Scholar 

  103. Hajdu, L., Tijdeman, R.: Algebraic aspects of discrete tomography. J. Reine Angew. Math., 534, 119–128 (2001).

    MATH  MathSciNet  Google Scholar 

  104. Hajdu, L., Tijdeman, R.: An algorithm for discrete tomography. J. Linear Algebra, 339, 147–169 (2001).

    Article  MATH  MathSciNet  Google Scholar 

  105. Hajdu, L., Tijdeman, R.: Algebraic aspects of emission tomography with absorption. Theoret. Comput. Sci., 290, 2169–2181 (2003).

    Article  MATH  MathSciNet  Google Scholar 

  106. Herman, G.T., Kuba, A. (eds.): Discrete Tomography: Foundations, Algorithms, and Applications. Birkhäuser, Boston, MA (1999).

    MATH  Google Scholar 

  107. Herman, G.T., Kuba, A.: Discrete tomography in medical imaging. Proc. IEEE, 91, 380–385 (2003).

    Article  Google Scholar 

  108. Herman, G.T., Kuba, A. (eds.): Proceedings of the Workshop on Discrete Tomography and Its Applications. Electronic Notes in Discrete Math., 20, 1–622 (2005).

    Google Scholar 

  109. Huck, C., Baake, M., Langfeld, B., Gritzmann, P., Lord, K.: Discrete tomography of mathematical quasicrystals: A primer. Electr. Notes Discr. Math., 20, 179–191 (2005).

    Article  MathSciNet  Google Scholar 

  110. Imiya, A., Tirii, A., Sato, K.: Tomography on finite graphs. Electr. Notes Discr. Math., 20, 217–232 (2005).

    Article  Google Scholar 

  111. Jarray F.: R’esolution de Problemes de Domographie Discrète. Applications à la Planification de Ppersonnel. Ph.D. Thesis, CNAM, Paris, France (2004).

    Google Scholar 

  112. Jinschek, J.R., Batenburg, K.J., Calderon, H.A., Van Dyck, D., Chen, F.R., Kisielowski, C.: Prospects for bright field and dark field electron tomography on a discrete grid. Microscopy and Microanalysis; Cambridge J. Online, 10, Suppl. 3 (2004).

    Google Scholar 

  113. Jinschek, J.R., Calderon, H.A., Batenburg, K.J., Radmilovic, V., Kisielowski, C.: Discrete tomography of Ga and InGa particles from HREM image simulation and exit wave reconstruction. In: Martin, D.C., Muller, D.A., Midgley, P.A., Stach, E.A. (eds.), Electron Microscopy of Molecular and Atom-Scale Mechanical Behavior, Chemistry and Structure, Materials Research Society, Warrendale, PA, pp. 4.5.1–4.5.6 (2004).

    Google Scholar 

  114. Kaneko, A., Nagahama, R: Structure of total reconstructed sets from given two projection data. Electr. Notes Discr. Math., 20, 27–46 (2005).

    Article  MathSciNet  Google Scholar 

  115. Kaneko, A., Nagahama, R: Switching graphs and digraphs associated with total reconstructed sets from two projection data. Nat. Sci. Report Ochanomizu Univ., 56, 33–45 (2005).

    MathSciNet  Google Scholar 

  116. Kingston, A.M.: Extension and Application of Finite Discrete Radon Projection Theory. Ph.D. Thesis, Monash Univ., Australia (2005).

    Google Scholar 

  117. Kingston, A., Svalbe, I.: A discrete modulo N projective Radon transform for N×N images. In: Andres, É., Damiand, G., Lienhardt, P. (eds.), Discrete Geometry for Computer Imagery, Springer, Berlin, Germany, pp. 136–147 (2005).

    Google Scholar 

  118. Kiss, Z., Rodek, L., Kuba, A.: Image reconstruction and correction methods in neutron and X-ray tomography. Simulation and physical experiments, Acta Cybernetica, 17, 557–587 (2006).

    MATH  MathSciNet  Google Scholar 

  119. Kiss, Z., Rodek, L., Nagy, A., Kuba, A., Balaskó, M.: Reconstruction of pixelbased and geometric objects by discrete tomography. Simulation and physical experiments, Electr. Notes Discr. Math., 20, 475–491 (2005).

    Article  Google Scholar 

  120. Krimmel, S., Baumann, J., Kiss, Z., Kuba, A., Nagy, A., Stephan, J.: Discrete tomography for reconstruction from limited view angles in non-destructive testing. Electr. Notes Discr. Math., 20, 455–474 (2005).

    Article  MathSciNet  Google Scholar 

  121. Kuba, A.: Reconstruction in different classes of 2D discrete sets. In: Bertrands, G., Couprie, M., Perroton, L. (eds.): Discrete Geometry for Computer Imagery, Springer, Berlin, Germany, pp. 1153–1163 (1999).

    Google Scholar 

  122. Kuba, A., Balogh, E.: Reconstruction of convex 2D discrete sets in polynomial time. Theor. Comput. Sci., 283, 223–242 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  123. Kuba, A., Herman, G.T.: Discrete tomography: A historical overview. In: Herman, G.T., Kuba, A. (eds.), Discrete Tomography: Foundations, Algorithms, and Applications, Birkhäuser, Boston, MA, pp. 3–34 (1999).

    Google Scholar 

  124. Kuba, A., Herman, G.T., Matej, S., Todd-Pokropek, A.: Medical applications of discrete tomography. In: Du, D.Z., Pardalos, P.M., Wang, J. (eds.), Discrete Mathematical Problems in Medical Applications; DIM ACS Series in Discrete Mathematics and Theoretical Computer Science. AMS, Providence, RI, 55, pp. 195–208 (2000).

    Google Scholar 

  125. Kuba, A., Nagy, A.: Reconstruction of hv-convex binary matrices from their absorbed projections. Electr. Notes Theor. Comput. Sci., 46 (2001).

    Google Scholar 

  126. Kuba, A., Nagy, A., Balogh, E.: Reconstruction of hv-convex binary matrices from their absorbed projections. Discr. Appl. Math., 139, 137–148 (2004).

    Article  MATH  MathSciNet  Google Scholar 

  127. Kuba, A., Nivat, M.: Reconstruction of discrete sets with absorption. In: Borgefors, G., Nyström, I., Sanniti di Baja, G. (eds.), Discrete Geometry in Computed Imagery, Springer, Berlin, Germany, pp. 137–148 (2000).

    Google Scholar 

  128. Kuba, A., Nivat, M.: Reconstruction of discrete sets with absorption. Lin. Algebra Appl., 339, 171–194 (2001).

    Article  MATH  MathSciNet  Google Scholar 

  129. Kuba, A., Nivat, M.: A sufficient condition for non-uniqueness in binary tomography with absorption. Discr. Appl. Math., 346, 335–357 (2005).

    MATH  MathSciNet  Google Scholar 

  130. Kuba, A., Rodek, L., Kiss, Z., Ruskó, L., Nagy, A., Balaskó, M.: Discrete tomography in neutron radiography. Nucl. Inst. & Meth., A, 542, 376–382 (2005).

    Article  Google Scholar 

  131. Kuba, A., Ruskó, L., Kiss, Z., Nagy, A.: Discrete reconstruction techniques, Electr. Notes Discr. Math., 20, 385–398 (2005).

    Article  Google Scholar 

  132. Kuba, A., Ruskó, L., Rodek, L., Kiss, Z.: Preliminary studies of discrete tomography in neutron imaging, IEEE Trans. Nuclear Sci., 52, 380–385 (2005).

    Article  Google Scholar 

  133. Kuba, A., Ruskó, L., Rodek, L., Kiss, Z.: Application of discrete tomography in neutron imaging, In: Chirco, P. et al. (eds.): Proc. 7th World Conf. Neutron Radiography, Rome, Italy, pp. 361–371 (2005).

    Google Scholar 

  134. Kuba, A., Woeginger, G.: Two remarks on reconstructing binary matrices from their absorbed projections. In: Andres, Ê., Damiand, G., Lienhardt, P. (eds.), Discrete Geometry for Computer Imagery, Springer, Berlin, Germany, pp. 79–91 (2005).

    Google Scholar 

  135. Kudo, H., Nakamura, H.: A new approach to SPECT attenuation correction without transmission measurements. In: Proc. IEEE Nuclear Sci. Symp. and Medical Imaging Conf., pp. 13.58–13.62 (2000).

    Google Scholar 

  136. Liao, H.Y.: Reconstruction of Label Images Using Gibbs Priors. Ph.D. Thesis, City University of New York, New York, USA (2005).

    Google Scholar 

  137. Liao, H.Y., Herman, G.T.: Automated estimation of the parameters of Gibbs priors to be used in binary tomography. Electr. Notes Theor. Comput. Sci., 46 (2001).

    Google Scholar 

  138. Liao, H.Y., Herman, G.T.: Reconstruction of label images from a few projections as motivated by electron microscopy. In: IEEE 28th Annual Northeast Bioengineering Conf., Philadephia, PA, pp. 205–206 (2002).

    Google Scholar 

  139. Liao, H.Y., Herman, G.T.: Tomographic reconstruction of label images from a new projections. Electr. Notes Discr. Math., 12 (2003).

    Google Scholar 

  140. Liao, H.Y., Herman, G.T.: Automated estimation of the parameters of Gibbs priors to be used in binary tomography. Discr. Appl. Math., 139, 149–170 (2004).

    Article  MATH  MathSciNet  Google Scholar 

  141. Liao, H.Y., Herman, G.T.: A method for reconstructing label images from a few projections, as motivated by electron microscopy. Proc. IEEE Intl. Symp. on Biomedical Imaging, Arlington, VA, pp. 551–554 (2004).

    Google Scholar 

  142. Liao, H.Y., Herman, G.T.: Discrete tomography with a very few views, using Gibbs priors and a marginal posterior mode. Electr. Notes Discr. Math., 20, 399–418 (2005).

    Article  MathSciNet  Google Scholar 

  143. Liao, H.Y., Herman, G.T.: Reconstruction by direct labeling in discrete tomography, using Gibbs priors and a marginal posterior mode approach. Proc. IEEE 31st Northeast Bioengineering Conf., Philadelphia, PA, pp. 134–135 (2005).

    Chapter  Google Scholar 

  144. Liao, H.Y., Herman, G.T.: A coordinate ascent approach to tomographic reconstruction of label images from a few projections. Discr. Appl. Math., 139, 184–197 (2005).

    Article  MathSciNet  Google Scholar 

  145. Masilamani, V., Dersanambika, K.S., Krithivasan, K.: Binary 3D matrices under the microscope: A tomographical problem. Electr. Notes Discr. Math., 20, 573–586 (2005).

    Article  MathSciNet  Google Scholar 

  146. Masilamani, V., Krithivasan, K.: An efficient reconstruction of 2D-tiling with t 1,2, t 2,1, t 1,1 tiles. In: Eckardt, U., Flach, B., Knauer, U., Polthier, K. (eds.), Combinatorial Image Analysis, Springer, Berlin, Germany, pp. 474–480 (2006).

    Google Scholar 

  147. Movassaghi, B., Rasche, V., Grass, M., Viergever, M.A., Niessen, W.J.: A quantitative analysis of 3-D coronary modeling from two or more projection images. IEEE Trans. Medical Imaging, 23, 1517–1531 (2004).

    Article  Google Scholar 

  148. Nagy, A., Kuba, A.: Reconstruction of binary matrices from fan-beam projections. Acta Cybernetica, 17, 359–385 (2005).

    MATH  MathSciNet  Google Scholar 

  149. Nagy, A., Kuba, A.: Parameter settings for reconstructing binary matrices from fan-beam projections. J. Comput. Info. Tech., 14, 101–110 (2006).

    Article  Google Scholar 

  150. Nagy, A., Kuba, A., Samal, M.: Reconstruction of factor structures using discrete tomography method. Electr. Notes Comput. Sci., 20, 519–534 (2005).

    MathSciNet  Google Scholar 

  151. Nam, Y.: Integral matrices with given row and column sums. Ars Combinatorica, 52, 141–151 (1999).

    MATH  MathSciNet  Google Scholar 

  152. Nivat, M.: On a tomographic equivalence between (0,1)-matrices. In: Karhumäki, J., Maurer, H., Paun, G., Rozenberg, G. (eds.), Theory Is Forever, Springer, Berlin, Germany, pp. 216–234 (2004).

    Google Scholar 

  153. Normand, N., Guédon, J.: Spline mojette transform. Application in tomography and communication. Proc. EUSIPCO, 2, 407–410 (2002).

    Google Scholar 

  154. Picouleau, C.: Reconstruction of domino tiling from its two orthogonal projections. Theor. Comp. Sci., 255, 437–447 (2001).

    Article  MATH  MathSciNet  Google Scholar 

  155. Picouleau, C., Brunetti, S., Frosini, A.: Reconstructing a binary matrix under timetabling constraints. Electr. Notes Discr. Math., 20, 99–112 (2005).

    Article  MathSciNet  Google Scholar 

  156. Popa, C., Zdunek, R.: Penalized least-squares image reconstruction for borehole tomography. Proc. ALGORITMY 2005, Podbanske, Slovakia, pp. 407–410 (2005).

    Google Scholar 

  157. Rodek, L., Knudsen, E., Poulsen, H., Herman, G.: Discrete tomographic reconstruction of 2D polycrystal orientation maps from X-ray diffraction projections using Gibbs priors. Electr. Notes Discr. Math., 20, 439–453 (2005).

    Article  MathSciNet  Google Scholar 

  158. Ruskó, L., Kuba, A.: Multi-resolution method for binary tomography. Electr. Notes Discr. Math., 20, 299–311 (2005).

    Article  Google Scholar 

  159. Schillinger, B.: Proposed combination of CAD data and discrete tomography for the detection of coking and lubricants in turbine blades or engines. Electr. Notes Discr. Math., 20, 493–499 (2005).

    Article  MathSciNet  Google Scholar 

  160. Schiile, T., Schnörr, C., Weber, S., Hornegger, J.: Discrete tomography by convex-concave regularization and D.C. programming. Discr. Appl. Math., 151, 229–243 (2005).

    Article  Google Scholar 

  161. Schüle, T., Weber, S., Schnörr, C.: Adaptive reconstruction of discrete-valued objects from few projections. Electr. Notes in Discr. Math., 20, 365–384 (2005).

    Article  Google Scholar 

  162. Sensali, M., Gamero, L., Herment, A., Mousseaux, E.: 3D reconstruction of vessel lumen from very few angiograms by dynamic contours using a stochastic approach, Graph. Models, 62, 105–127 (2000).

    Article  Google Scholar 

  163. Servières, M.: Reconstruction Tomographique Mojette. Thèse de doctorat, Universitéde Nantes, France (2005).

    Google Scholar 

  164. Servières, M., Guédon, J., Normand, N.: A discrete tomography approach to PET reconstruction. In: Bizais, Y.J. (ed.), Proc. Fully 3D Reconstruction in Radiology and Nuclear Medicine, University of Brest, Brest, France (2003).

    Google Scholar 

  165. Sharif, B., Sharif, B.: Discrete tomography in discrete deconvolution: Deconvolution of binary images using Ryser’s algorithm. Electr. Notes in Discr. Math.,20, 555–571 (2005).

    Article  MathSciNet  Google Scholar 

  166. Soussen, C.: Reconstruction 3D d’un Objet Compact en Tomographie. Ph.D. Thesis, Université de Paris-Sud, France, (2000).

    Google Scholar 

  167. Soussen, C., Muhammad-Djafari, A.: Contour-based models for 3D binary reconstruction in X-ray tomography. In: AIP Conference Proceedings, Gif-sur-Yvette, France, 568, pp. 543–554 (2001).

    Chapter  Google Scholar 

  168. Soussen, C., Muhammad-Djafari, A.: Polygonal and polyhedral contour reconstruction in computed tomography. IEEE Trans. Image Processing, 13, 1507–1523 (2004).

    Article  Google Scholar 

  169. Takiguchi, T.: Reconstruction of measurable plane sets from their orthogonal projections. Contemporary Math., 348, 199–208 (2004).

    MathSciNet  Google Scholar 

  170. Valenti, C.: An Experimental Study of the Stability Problem in Discrete Tomography. Ph.D. Thesis, Univ. of Palermo, Italy (2002).

    Google Scholar 

  171. Valenti, C.: Discrete tomography from noisy projections. In: Series on Software Engineering and Knowledge Engineering, World Scientific Publishing, 15, 38–45 (2003).

    Google Scholar 

  172. Valenti, C.: An experimental study of the stability problem in discrete tomography. Electr. Notes Discr. Math., 20, 113–132 (2005).

    Article  Google Scholar 

  173. Vallejo, E.: Plane partitions and characters of the symmetric group. J. Algebraic Comb.,11, 79–88 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  174. Vallejo, E.: The classification of minimal matrices of size 2×q. Lin. Algebra Appl., 340, 169–181 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  175. Vallejo, E.: A characterization of additive sets. Discr. Math., 259, 201–210 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  176. Vallejo, E.: Minimal matrices and discrete tomography. Electr. Notes in Discr. Math., 20, 113–132 (2005).

    Article  MathSciNet  Google Scholar 

  177. Vardi, E., Herman, G.T., Kong, T.Y.: Speeding up stochastic reconstructions of binary images from limited projection directions. Lin. Algebra Appl., 339,75–89 (2001).

    Article  MATH  MathSciNet  Google Scholar 

  178. Venere, M., Liao, H., Clausse, A.: A genetic algorithm for adaptive tomography of elliptical objects. IEEE Signal Processing Letters, 7, 176–178 (2000).

    Article  Google Scholar 

  179. de Vries, S.: Discrete Tomography, Packing and Covering, and Stable Set Problems: Polytopes and Algorithms. Ph.D. Thesis, Technical Univ. of Munich, Germany (1999).

    Google Scholar 

  180. Weber, S., Schnörr, C., Hornegger, J.: A linear programming relaxation for binary tomography with smoothness priors. Electr. Notes Discr. Math., 12 (2003).

    Google Scholar 

  181. Weber, S., Schüle, T., Hornegger, J., Schnörr, C.: Binary tomography by iterating linear programs from noisy projections. In: Klette, R, Zunic, J.D. (eds.), Combinatorial Image Analysis, Springer, Berlin, Germany, pp. 38–51 (2004).

    Google Scholar 

  182. Weber, S., Schüle, T., Kuba, A., Schnörr, C.: Binary tomography with deblurring. In: Reulke, R., Eckardt, U., Flach, B., Knauer, U., Polthier, K. (eds.), Combinatorial Image Analysis, Springer, Berlin, Germany, pp. 375–388 (2006).

    Chapter  Google Scholar 

  183. Weber, S., Schüle, T., Schnörr, C.: Prior learning and convex-concave regularization of binary tomography. Electr. Notes Discr. Math., 20, 313–327 (2005).

    Article  Google Scholar 

  184. Weber, S., Schüle, T., Schnörr, C., Hornegger, J.: A linear programming approach to limited angle 3D reconstruction from DSA projections. Methods of Information in Medicine, 43, 320–326 (2003).

    Google Scholar 

  185. Weber, G.-W., Yasar, Ô.: Discrete tomography: A modern inverse problem reconsidered by optimization. J. Comp. Tech., 9, 115–121 (2004).

    Google Scholar 

  186. Woeginger, G.J.: The reconstruction of polyominoes from their orthogonal projections. Inform. Process. Lett., 77, 225–229 (2001).

    Article  MATH  MathSciNet  Google Scholar 

  187. Yagle, A.: A convergent composite mapping Fourier domain iterative algorithm for 3-D discrete tomography. Lin. Algebra Appl., 339, 91–109 (2001).

    Article  MATH  MathSciNet  Google Scholar 

  188. Yasar, Ô., Diner, C., Dogan, A., Weber, G.-W., Özbudak, F., Tiefenbach, A.: On the applied mathematics of discrete tomography. J. Comp. Tech., 9, 14–32 (2004).

    MATH  Google Scholar 

  189. Ye, Y., Wang, G., Zhu, J.: Linear diophantine equations for discrete tomography. J. X-Ray Sci. Tech., 10, 59–66 (2001).

    Google Scholar 

  190. Zdunek, R., Pralat, A.: Detection of subsurface bubbles with discrete electromagnetic geotomography. Electr. Notes Discr. Math., 20, 535–553 (2005).

    Article  MathSciNet  Google Scholar 

  191. Zopf, S., Kuba, A.: Reconstruction of measurable sets from two generalized projections. Electr. Notes Discr. Math., 20, 47–66 (2005).

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dept. of Image Processing and Computer Graphics, University of Szeged, Árpád tér 2., Szeged, H-6720, Hungary

    A. Kuba

  2. Dept. of Computer Science, The Graduate Center, City Univ. of New York, 365 Fifth Avenue, New York, NY, 10016, USA

    G.T. Herman

Authors
  1. A. Kuba
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. G.T. Herman
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Program in Computer Science The Graduate Center, The City University of New York, 365 Fifth Avenue, New York, NY, 10016, USA

    Gabor T. Herman

  2. Department of Image Processing and Computer Graphics, University of Szeged, Árpád tér 2 H-6720 , Szeged, Hungary

    Attila Kuba

Rights and permissions

Reprints and permissions

Copyright information

© 2007 Birkhäuser Boston

About this chapter

Cite this chapter

Kuba, A., Herman, G. (2007). Introduction. 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_1

Download citation

  • DOI: https://doi.org/10.1007/978-0-8176-4543-4_1

  • Publisher Name: Birkhäuser Boston

  • Print ISBN: 978-0-8176-3614-2

  • Online ISBN: 978-0-8176-4543-4

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation