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| [[File:Discrete tomography.png|thumb|A discrete tomography reconstruction problem for two vertical and horizontal directions (left), together with its (non-unique) solution (right). The task is to color some of the white points black so that the number of black points in the rows and columns match the blue numbers.]]'''Discrete Tomography'''<ref name="ref4">
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| Herman, G. T. and Kuba, A., Discrete Tomography: Foundations, Algorithms, and Applications, Birkhäuser Boston, 1999
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| </ref><ref name="ref5">
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| Herman, G. T. and Kuba, A., Advances in Discrete Tomography and Its Applications, Birkhäuser Boston, 2007
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| </ref> focuses on the problem of reconstruction of [[binary image]]s (or finite subsets of the [[integer lattice]]) from a small number of their [[projection (mathematics)|projection]]s.
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| In general, [[tomography]] deals with the problem of determining shape and dimensional information of an object from a set of projections. From the mathematical point of view, the object corresponds to a [[function (mathematics)|function]] and the problem posed is to reconstruct this function from its [[integral]]s or sums over subsets of its [[Domain of a function|domain]]. In general, the tomographic inversion problem may be continuous or discrete. In continuous tomography both the
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| domain and the range of the function are continuous and line integrals are used. In discrete tomography the domain of the function may be either discrete or continuous, and the range of the function is a finite set of real, usually nonnegative numbers. In continuous tomography when a large number of projections is available, accurate reconstructions can be made by many different algorithms.
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| It is typical for discrete tomography that only a few projections (line sums) are used. In this case, conventional techniques all fail. A special case of discrete tomography deals with the problem of the reconstruction of
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| a binary image from a small number of projections. The name ''discrete tomography'' is due to [[Larry Shepp]], who organized the first meeting devoted to this topic ([[DIMACS]] Mini-Symposium on Discrete Tomography, September 19, 1994, [[Rutgers University]]).
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| ==Theory==
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| Discrete tomography has strong connections with other mathematical fields, such as [[number theory]],<ref name="ref11">R.J. Gardner, P. Gritzmann, Discrete tomography: determination of finite sets by X-rays, Trans. Amer. Math. Soc. 349 (1997), no. 6, 2271-2295.</ref><ref name="ref9">L. Hajdu, R. Tijdeman, Algebraic aspects of discrete tomography, J. reine angew. Math. 534 (2001), 119-128.</ref><ref name="ref10">A. Alpers, R. Tijdeman, The Two-Dimensional Prouhet-Tarry-Escott Problem, Journal of Number Theory, 123 (2), pp. 403-412, 2007 [http://www-m9.ma.tum.de/Allgemeines/AndreasAlpersPublications].</ref> [[discrete mathematics]],<ref name="ref14">S. Brunetti, A. Del Lungo, P. Gritzmann, S. de Vries, On the reconstruction of binary and permutation matrices under (binary) tomographic constraints. Theoret. Comput. Sci. 406 (2008), no. 1-2, 63-71.</ref><ref name="ref13">A. Alpers, P. Gritzmann, On Stability, Error Correction, and Noise Compensation in Discrete Tomography, SIAM Journal on Discrete Mathematics 20 (1), pp. 227-239, 2006 [http://www-m9.ma.tum.de/Allgemeines/AndreasAlpersPublications]</ref><ref name="ref12">P. Gritzmann, B. Langfeld, On the index of Siegel grids and its application to the tomography of quasicrystals. European J. Combin. 29 (2008), no. 8, 1894-1909.</ref> [[Complex systems|complexity theory]]<ref name="ref15">R.J. Gardner, P. Gritzmann, D. Prangenberg, On the computational complexity of reconstructing lattice sets from their X-rays. Discrete Math. 202 (1999), no. 1-3, 45-71.</ref><ref name="ref16">C. Dürr, F. Guiñez, M. Matamala, Reconstructing 3-Colored Grids from Horizontal and Vertical Projections Is NP-hard. ESA 2009: 776-787.</ref> and [[combinatorics]].<ref name="ref17">H.J. Ryser, Matrices of zeros and ones, Bull. Amer. Math. Soc. 66 1960 442-464.</ref><ref name="ref18">D. Gale, A theorem on flows in networks, Pacific J. Math. 7 (1957), 1073-1082.</ref><ref name="ref19">E. Barcucci, S. Brunetti, A. Del Lungo, M. Nivat, Reconstruction of lattice sets from their horizontal, vertical and diagonal X-rays, Discrete Mathematics 241(1-3): 65-78 (2001).</ref> In fact, a number of discrete tomography problems were first discussed as combinatorial problems. In 1957, [[Herbert John Ryser]] found a necessary and sufficient condition for a pair of vectors being the two orthogonal projections of a discrete set. In the proof of his theorem, Ryser also described a reconstruction algorithm, the very first reconstruction algorithm for a general discrete set from two orthogonal projections. In the same year, [[David Gale]] found the same consistency conditions, but in connection with the [[flow network|network flow]] problem.<ref name=CMC27>{{cite book | last=Brualdi | first=Richard A. | title=Combinatorial matrix classes | series=Encyclopedia of Mathematics and Its Applications | volume=108 | location=Cambridge | publisher=[[Cambridge University Press]] | year=2006 | isbn=0-521-86565-4 | zbl=1106.05001 | page=27 }}</ref> Another result of Ryser is the definition of the switching operation by which discrete sets having the same projections can be transformed into each other.
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| The problem of reconstructing a [[binary image]] from a small number of projections generally leads to a large number of solutions. It is desirable to limit the class of possible solutions to only those that are typical of the class of the images which contains the image being reconstructed by using a priori information, such as convexity or connectedness.
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| ===Theorems===
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| * Reconstructing (finite) planar lattice sets from their 1-dimensional X-rays is an [[NP-hard]] problem if the X-rays are taken from <math> m\geq 3 </math> lattice directions (for <math> m=2 </math> the problem is in P).<ref name="ref15" />
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| *The reconstruction problem is highly unstable for <math> m\geq 3 </math> (meaning that a small perturbation of the X-rays may lead to completely different reconstructions)<ref name="ref23" /> and stable for <math> m=2 </math>, see.<ref name="ref23">A. Alpers, P. Gritzmann, L. Thorens, Stability and Instability in Discrete Tomography, Lecture Notes in Computer Science 2243; Digital and Image Geometry (Advanced Lectures), G. Bertrand, A. Imiya, R. Klette (Eds.), pp. 175-186, Springer-Verlag, 2001.</ref><ref name="ref24">A. Alpers, S. Brunetti, On the Stability of Reconstructing Lattice Sets from X-rays Along Two Directions, Lecture Notes in Computer Science 3429; Digital Geometry for Computer Imagery, E. Andres, G. Damiand, P. Lienhardt (Eds.) , pp. 92-103, Springer-Verlag, 2005.</ref><ref name="ref25">B. van Dalen, Stability results for uniquely determined sets from two directions in discrete tomography, Discrete Mathematics 309(12): 3905-3916 (2009).</ref>
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| *Coloring a grid using <math> k </math> colors with the restriction that each row and each column has a specific number of cells of each color is known as the <math>(k-1)</math>−atom problem in the discrete tomography community. The problem is NP-hard for <math> k \geq 3 </math>, see.<ref name="ref16" />
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| For further results see.<ref name="ref4" /><ref name="ref5" />
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| ==Algorithms==
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| Among the reconstruction methods one can find [[algebraic reconstruction technique]]s (e.g., DART<ref name="ref8" />
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| <ref>W. van Aarle, K J. Batenburg, and J. Sijbers, Automatic parameter estimation for the Discrete Algebraic Reconstruction Technique (DART), IEEE Transactions on Image Processing, 2012 [http://dx.doi.org/10.1109/TIP.2012.2206042]</ref> or <ref name="ref8b">K. J. Batenburg, and J. Sijbers, "Generic iterative subset algorithms for discrete tomography", Discrete Applied Mathematics, vol. 157, no. 3, pp. 438-451, 2009</ref>), [[greedy algorithm]]s (see <ref name="ref22">P. Gritzmann, S. de Vries, M. Wiegelmann, Approximating binary images from discrete X-rays, SIAM J. Optim. 11 (2000), no. 2, 522-546.</ref> for approximation guarantees), and [[Monte Carlo algorithm]]s.<ref name="ref20" /><ref name="ref21" />
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| ==Applications==
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| Various algorithms have been applied in [[image processing]]
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| ,<ref name="ref8">
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| Batenburg, Joost; Sijbers, Jan - DART: A practical reconstruction algorithm for discrete tomography - In: IEEE transactions on image processing, Vol. 20, Nr. 9, p. 2542-2553, (2011). {{doi|10.1109/TIP.2011.2131661}}
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| </ref> [[medicine]],
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| three-dimensional statistical data security problems, computer
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| tomograph assisted engineering and design, [[electron microscopy]]
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| <ref name="ref6">
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| S. Bals, K. J. Batenburg, J. Verbeeck, J. Sijbers and G. Van Tendeloo, "Quantitative 3D reconstruction of catalyst particles for bamboo-like carbon-nanotubes", Nano Letters, Vol. 7, Nr. 12, p. 3669-3674, (2007) {{doi|10.1021/nl071899m}}
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| </ref>
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| ,<ref name="ref7">
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| Batenburg J., S. Bals, Sijbers J., C. Kubel, P.A. Midgley, J.C. Hernandez, U. Kaiser, E.R. Encina, E.A. Coronado and G. Van Tendeloo, "3D imaging of nanomaterials by discrete tomography", Ultramicroscopy, Vol. 109, p. 730-740, (2009) {{doi|10.1016/j.ultramic.2009.01.009}}
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| </ref> and [[materials science]].,<ref name="ref20">A. Alpers, H.F. Poulsen, E. Knudsen, G.T. Herman, A Discrete Tomography Algorithm for Improving the Quality of 3DXRD Grain Maps, Journal of Applied Crystallography 39, pp. 582-588, 2006 [http://www-m9.ma.tum.de/Allgemeines/AndreasAlpersPublications].</ref><ref name="ref21">L. Rodek, H.F. Poulsen, E. Knudsen, G.T. Herman, A stochastic algorithm for reconstruction of grain maps of moderately deformed specimens based on X-ray diffraction, Journal of Applied Crystallography 40:313-321, 2007.</ref>
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| <ref name="ref21b">K. J. Batenburg, J. Sijbers, H. F. Poulsen, and E. Knudsen, "DART: A Robust Algorithm for Fast Reconstruction of 3D Grain Maps", Journal of Applied Crystallography, vol. 43, pp. 1464-1473, 2010</ref>
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| A form of discrete tomography also forms the basis of [[nonogram]]s, a type of [[logic puzzle]] in which information about the rows and columns of a digital image is used to reconstruct the image.<ref>{{Cite book| url=http://books.google.ca/books?id=K98BAAAACAAJ | title=Games Magazine Presents Paint by Numbers | publisher=[[Random House]] | year=1994 | ISBN=0-8129-2384-7}}</ref>
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| ==See also==
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| *[[Geometric tomography]]
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| ==References==
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| {{reflist}}
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| == External links ==
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| *[http://astra.ua.ac.be/euroDT/index.php/Main_Page Euro DT (a Discrete Tomography Wiki site for researchers)]
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| *[http://www-desir.lip6.fr/~durrc/Xray/Complexity/ Tomography applet by Christoph Dürr]
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| *[http://www.visielab.ua.ac.be/publications/tomographic-segmentation-and-discrete-tomography-quantitative-analysis-transmission PhD thesis on discrete tomography (2012): Tomographic segmentation and discrete tomography for quantitative analysis of transmission tomography data]
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| [[Category:Applied mathematics]]
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| [[Category:Digital geometry]]
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