Discrete Tomography pp 3-34 | Cite as

# Discrete Tomography: A Historical Overview

Chapter

## Abstract

In this chapter we introduce the topic of discrete tomography and give a brief historical survey of the relevant contributions. After discussing the nature of the basic theoretical problems (those of consistency, uniqueness, and reconstruction) that arise in discrete tomography, we give the details of the classical special case (namely, two-dimensional discrete sets — i.e.,binary matrices — and two orthogonal projections) including a polynomial time reconstruction algorithm. We conclude the chapter with a summary of some of the applications of discrete tomography.

## Keywords

Lattice Line Variant Position Binary Matrix Historical Overview Algebraic Reconstruction Technique
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## References

- [1]G. T. Herman,
*Image Reconstruction from Projections: The Fundamentals of Computerized Tomography*, (Academic Press, New York), 1980.Google Scholar - [2]G. G. Lorentz, “A problem of plane measure,”
*Amer. J. Math.*,**71**, 417–426 (1949).CrossRefGoogle Scholar - [3]A. Rényi, “On projections of probability distributions,”
*Acta Math. Acad. Sci. Hung.*,**3**, 131–142 (1952).CrossRefGoogle Scholar - [4]A. Heppes, “On the determination of probability distributions of more dimensions by their projections,”
*Acta Math. Acad. Sci. Hung.*,**7**, 403–410 (1956).CrossRefGoogle Scholar - [5]H. Kellerer, “Funktionen auf Produkträumen mit vorgegebenen Marginal-Funktionen,”
*Math. Ann.*,**144**, 323–344 (1961).CrossRefGoogle Scholar - [6]H. Kellerer, “Masstheoretische Marginalprobleme,”
*Math. Ann. 153*, 168–198 (1964).CrossRefGoogle Scholar - [7]H. Kellerer, “Marginalprobleme für Funktionen,”
*Math. Ann.*,**154**, 147–150 (1964).CrossRefGoogle Scholar - [8]H. J. Ryser, “Combinatorial properties of matrices of zeros and ones,”
*Ganad. J. Math.*,**9**, 371–377 (1957).CrossRefGoogle Scholar - [9]D. Gale, “A theorem on flows in networks,”
*Pacific J. Math.*,**7**, 1073–1082 (1957).CrossRefGoogle Scholar - [10]H. J. Ryser, “Traces of matrices of zeros and ones,”
*Canad. J. Math.*,**12**, 463–476 (1960).CrossRefGoogle Scholar - [11]L. R. Ford, Jr. and D. R. Fulkerson,
*Flows in Networks*, (Princeton University Press, Princeton, NJ), 1962.Google Scholar - [12]J. Steiner “Einfache Beweis der isoperimetrischen Hauptsätze,”
*J. reine angew. Math.*,**18**, 289–296 (1838).Google Scholar - [13]P. C. Hammer “Problem 2,” In
*Proc. Symp. Pure Math.*,*vol. VII: Convexity*, (Amer. Math. Soc., Providence, RI), pp. 498–499, 1963.Google Scholar - [14]O. Giering, “Bestimmung von Eibereichen und Eikörpern durch Steiner-Symmetrisierungen,”
*Sitzungsberichten Bayer. Akad. Wiss. München, Math.-Nat. Kl.*, pp. 225–253 (1962).Google Scholar - [15]R. J. Gardner and P. McMullen, “On Hammer’s X-ray problem,”
*J. London Math. Soc.*,**21**, 171–175 (1980).CrossRefGoogle Scholar - [16]R. J. Gardner,
*Geometric Tomography*, (Cambridge University Press, Cambridge, UK), 1995.Google Scholar - [17]G. Bianchi and M. Longinetti, “Reconstructing plane sets from projections,”
*Discrete Comp. Geom.*,**5**, 223–242 (1990).CrossRefGoogle Scholar - [18]R. J. Gardner and P. Gritzmann, “Discrete tomography: Determination of finite sets by X-rays,”
*Trans. Amer. Math. Soc.*,**349**, 2271–2295 (1997).CrossRefGoogle Scholar - [19]P. Gritzmann and M. Nivat (Eds.),
*Discrete Tomography: Algorithms and Complexity*, (Dagstuhl-Seminar-Report 165, Dagstuhl, Germany), 1997.Google Scholar - [20]G. T. Herman and A. Kuba (Editors),
*Discrete Tomography*,. Special Issue of Intern. J. of Imaging Systems and Techn., Vol. 9, No. 2/3, 1998.Google Scholar - [21]R. Gordon, R. Bender, and G. T. Herman, “Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and X-ray photography,”
*J. Theor. Biol.*,**29**, 471–481 (1970).CrossRefPubMedGoogle Scholar - [22]G. T. Herman, A. Lent, and S. W. Rowland, “ART: Mathematics and applications,”
*J. Theor. Biol.*,**42**, 1–32 (1973).CrossRefPubMedGoogle Scholar - [23]G. T. Herman, “Reconstruction of binary patterns from a few projections,” In A. Günther, B. Levrat and H. Lipps,
*Intern*,*Computing Symposium 1973*, (North-Holland Publ. Co., Amsterdam), pp. 371–378, 1974.Google Scholar - [24]P. Gritzmann, D. Prangenberg, S. de Vries, and M. Wiegelmann, “Success and failure of certain reconstruction and uniqueness algorithms in discrete tomography,”
*Intern. J. of Imaging Systems and Techn.*,**9**, 101–109 (1998).CrossRefGoogle Scholar - [25]Y. Vardi and D. Lee, “The discrete Radon transform and its approximate inversion via the EM algorithm,”
*Intern. J. Imaging Systems and Techn.*,**9**, 155–173 (1998).CrossRefGoogle Scholar - [26]J. H. B. Kemperman and A. Kuba, “Reconstruction of two-valued matrices from their two projections,”
*Intern. J. of Imaging Systems and Techn.*,**9**, 110–117 (1998).CrossRefGoogle Scholar - [27]D. Rossi and A. Willsky, “Reconstruction from projections based on detection and estimation of objects: Performance analysis and robustness analysis,”
*IEEE Trans. Acoust. Speech Signal Process.*,**32**, 886–906 (1984).CrossRefGoogle Scholar - [28]J. Singer, F. A. Grünbaum, P. Kohn, and J. Zubelli, “Image reconstruction of the interior of bodies that diffuse radiation,”
*Science*,**248**, 990–993 (1990).CrossRefPubMedGoogle Scholar - [29]D. R. Fulkerson and H. J. Ryser, “Width sequences for special classes of (0,1)-matrices,”
*Canad. J. Math.*,**15**, 371–396 (1963).CrossRefGoogle Scholar - [30]S.-K. Chang and Y. R. Wang, “Three-dimensional reconstruction from orthogonal projections,”
*Pattern Recognition*,**7**, 167–176 (1975).CrossRefGoogle Scholar - [31]S.-K. Chang, “The reconstruction of binary patterns from their projections,”
*Commun. ACM*,**14**, 21–25 (1971).CrossRefGoogle Scholar - [32]S.-K. Chang, “Algorithm 445. Binary pattern reconstruction from projections,”
*Commun. ACM*,**16**, 185–186 (1973).CrossRefGoogle Scholar - [33]Y. R. Wang, “Characterization of binary patterns and their projections,”
*IEEE Trans. Computers*,**C24**, 1032–1035 (1975).CrossRefGoogle Scholar - [34]W. Wandi, “The class
*2t(R*,*S)*, of (0,1)-matrices,”*Discrete Math.*,**39**, 301–305 (1982).CrossRefGoogle Scholar - [35]W. Honghui, “Structure and cardinality of the class
*2t(R*,*S)*, of (0,1)-matrices,”*J. Math. Research and Exposition*,**4**, 87–93 (1984).Google Scholar - [36]S. Jiayu, “On a guess about the cardinality of the class
*2t(R*,*S)*, of (0,1)-matrices,”*J. Tongji Univ.*,**14**, 52–55 (1986).Google Scholar - [37]W. Xiaohong, “The cardinality of the class
*2t(R*,*S)*, of (0,1)-matrices,”*J. Sichuan Univ. Nat. Sci. Edition*,**4**, 95–99 (1986).Google Scholar - [38]W. Xiaohong, “A necessary and sufficient condition for
*14(R*,*S)I*, to equal its lower bound,”*J. Sichuan Univ. Nat. Sci. Edition*,**24**, 136–143 (1987).Google Scholar - [39]L. Mirsky,
*Transversal Theory*, (Academic Press, New York), 1971.Google Scholar - [40]D. R. Fulkerson, “Zero-one matrices with zero trace,”
*Pacific J. Math.*,**10**, 831–836 (1960).CrossRefGoogle Scholar - [41]R. P. Anstee, “Properties of a class of (0,1)-matrices covering a given matrix,”
*Canad. J. Math.*,**34**, 438–453 (1982).CrossRefGoogle Scholar - [42]R. P. Anstee, “Triangular (0,1)-matrices with prescribed row and column sums,”
*Discrete Math.*,**40**, 1–10 (1982).CrossRefGoogle Scholar - [43]R. P. Anstee, “The network flows approach for matrices with given row and column sums,”
*Discrete Math.*,**44**, 125–138 (1983).CrossRefGoogle Scholar - [44]L. Mirsky, “Combinatorial theorems and integral matrices,”
*J. Combin. Theor.*,**5**, 30–44 (1968).CrossRefGoogle Scholar - [45]A. Kuba and A. Volcic, “Characterization of measurable plane sets which are reconstructable from their two projections,”
*Inverse Problems*,**4**, 513–527 (1988).CrossRefGoogle Scholar - [46]P. C. Fishburn, J. C. Lagarias, J. A. Reeds, and L. A. Shepp, “Sets uniquely determined by projections on axes II. Discrete case,”
*Discrete Math.*,**91**, 149–159 (1991).CrossRefGoogle Scholar - [47]H. J. Ryser,
*Combinatorial Mathematics*, (The Math. Assoc. Amer., Washington, DC), 1963.Google Scholar - [48]R. A. Brualdi, “Matrices of zeros and ones with fixed row and column sum vectors,”
*Lin. Algebra and Its Appl.*,**33**, 159–231 (1980).CrossRefGoogle Scholar - [49]A. Kuba, “Reconstruction of unique binary matrices with prescribed elements,”
*Acta Cybern.*,**12**, 57–70 (1995).Google Scholar - [50]R. Aharoni, G. T. Herman, and A. Kuba, “Binary vectors partially determined by linear equation systems,”
*Discrete Math.*,**171**,1–16 (1997).CrossRefGoogle Scholar - [51]
- [52]H. J. Ryser, “Matrices of zeros and ones,”
*Bull. Amer. Math. Soc.*,**66**, 442–464 (1960).CrossRefGoogle Scholar - [53]
- [54]A. Kuba, “Determination of the structure of the class
*21(R*,*S)*, of (0,1)-matrices,”*Acta Cybernetica*,**9**, 121–132 (1989).Google Scholar - [55]W. Y. C. Chen, “Integral matrices with given row and column sums,”
*J. Combin. Theory*,*Ser*,.**A 61**, 153–172 (1992).CrossRefGoogle Scholar - [56]S.-K. Chang and G. L. Shelton, “Two algorithms for multiple-view binary pattern reconstruction,”
*IEEE Trans. Systems*,*Man*,*Cybernetics SMC.*,**1**, 90–94 (1971).CrossRefGoogle Scholar - [57]L. Huang, “The reconstruction of uniquely determined plane sets from two projections in discrete case,”
*Preprint Series*,**UTMS 95–29**, Univ. of Tokyo (1995).Google Scholar - [58]A. Kuba, “The reconstruction of two-directionally connected binary patterns from their two orthogonal projections,”
*Comp. Vision*,*Graphics*,*Image Proc.*,**27**, 249–265 (1984).CrossRefGoogle Scholar - [59]A. Rosenfeld and A. C. Kak, “Digital Picture Processing,” (Academic Press, New York) 1976.Google Scholar
- [60]G. T. Herman, “Geometry of Digital Spaces,” (Birkhäuser, Boston) 1998.Google Scholar
- [61]A. Del Lungo, M. Nivat and R. Pinzani, “The number of convex polyominoes reconstructible from their orthogonal projections,”
*Discrete Math.*,**157**, 65–78 (1996).CrossRefGoogle Scholar - [62]E. Barcucci, A. Del Lungo, M. Nivat, and R. Pinzani, “Reconstructing convex polyominoes from horizontal and vertical projections,”
*Theor. Comput. Sci.*,**155**, 321–347 (1996).CrossRefGoogle Scholar - [63]E. Barcucci, A. Del Lungo, M. Nivat, and R. Pinzani, “Medians of polyominoes: A property for the reconstruction,”
*Int. J. Imaging Systems and Techn.*,**9**, 69–77 (1998).CrossRefGoogle Scholar - [64]M. Chrobak and C. Dürr, “Reconstructing hv-convex polyominoes from orthogonal projections,” (Preprint, Dept. of Computer Science, Univ. of California, Riverside, CA), 1998.Google Scholar
- [65]G. J. Woeginger, “The reconstruction of polyominoes from their orthogonal projections,” (Techn. Rep. F003, TU-Graz), 1996.Google Scholar
- [66]R. W. Irving and M. R. Jerrum, “Three-dimensional statistical data security problems,”
*SIAM J. Comput.*,**23**, 170–184 (1994).CrossRefGoogle Scholar - [67]A. R. Shliferstein and Y. T. Chien, “Some properties of image-processing operations on projection sets obtained from digital pictures,”
*IEEE Trans. Comput.*,**C-26**, 958–970 (1977).CrossRefGoogle Scholar - [68]Z. Mao and R. N. Strickland, “Image sequence processing for target estimation in forward-looking infrared imagery,”
*Optical Engrg.*,**27**, 541–549 (1988).Google Scholar - [69]A. Fazekas, G. T. Herman, and A. Matej, “On processing binary pictures via their projections,”
*Int. J. Imaging Systems and Techn.*,**9**, 99–100 (1998).CrossRefGoogle Scholar - [70]A. V. Crewe and D. A. Crewe, “Inexact reconstruction: Some improvements,”
*Ultramicroscopy*,**16**, 33–40 (1985).CrossRefGoogle Scholar - [71]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,”
*Phys. Rev. Letters*,**71**, 4150–4153 (1993).CrossRefGoogle Scholar - [72]C. Kisielowski, P. Schwander, F. H. Baumann, M. Seibt, Y. Kim, and A. Ourmazd, “An approach to quantitative high-resolution electron microscopy of crystalline materials,”
*Ultramicroscopy*,**58**, 131–155 (1995).CrossRefGoogle Scholar - [73]D. G. W. Onnasch, and P. H. Heintzen, “A new approach for the reconstruction of the right or left ventricular form from biplane angiocardiographic recordings,” In
*Conf. Comp. Card. 1976*, (IEEE Comp. Soc. Press, Washington), pp. 67–73, 1976.Google Scholar - [74]C. H. Slump and J. J. Gerbrands, “A network flow approach to reconstruction of the left ventricle from two projections,”
*Comp. Graphics and Image Proc.*,**18**, 18–36 (1982).CrossRefGoogle Scholar - [75]G. P. M. Prause and D. G. W. Onnasch, “Binary reconstruction of the heart chambers from biplane angiographic image sequences,”
*IEEE Trans. Medical Imaging*,**MI15**, 532–546 (1996).CrossRefGoogle Scholar - [76]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. Comput.*,**C22**, 18–28 (1973).CrossRefGoogle Scholar - [77]B. M. Carvalho, G. T. Herman, S. Matej, C. Salzberg, and E. Vardi, “Binary tomography for triplane cardiography,” In A. Kuba, M. Samal, A. Todd-Pokropek,
*Information Processing in Medical Imaging Conference*, 1999 (Springer-Verlag, Berlin), to be published.Google Scholar - [78]N. Robert, F. Peyrin, and M. J. Yaffe, “Binary vascular reconstruction from a limited number of cone beam projections,”
*Med. Phys.*,**21**, 1839–1851 (1994).CrossRefPubMedGoogle Scholar

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