Discrete Tomography

Discrete tomography
Herman, G. T. and Kuba, A., Discrete Tomography: Foundations, Algorithms, and Applications, Birkhäuser Boston, 1999

Herman, G. T. and Kuba, A., Advances in Discrete Tomography and Its Applications, Birkhäuser Boston, 2007
focuses on the problem of reconstruction of binary images (or finite subsets of the integer lattice) from a small number of their projections.

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 and the problem posed is to reconstruct this function from its integrals or sums over subsets of its domain. In general, the tomographic inversion problem may be continuous or discrete. In continuous tomography both the
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.
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
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).

Theory

Discrete tomography has strong connections with other mathematical fields, such as number theory,R.J. Gardner, P. Gritzmann, Discrete tomography: determination of finite sets by X-rays, Trans. Amer. Math. Soc. 349 (1997), no. 6, 2271-2295.L. Hajdu, R. Tijdeman
Algebraic aspects of discrete tomography
J. reine angew. Math. 534 (2001), 119-128.A. Alpers, R. Tijdeman, The Two-Dimensional Prouhet-Tarry-Escott Problem, Journal of Number Theory, 123 (2), pp. 403-412, 200

discrete mathematics,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.A. Alpers, P. Gritzmann, On Stability, Error Correction, and Noise Compensation in Discrete Tomography, SIAM Journal on Discrete Mathematics 20 (1), pp. 227-239, 200

/ref>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. computational complexity theoryR.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.C. Dürr, F. Guiñez, M. Matamala
Reconstructing 3-Colored Grids from Horizontal and Vertical Projections Is NP-hard
ESA 2009: 776-787. and combinatorics.H.J. Ryser, Matrices of zeros and ones, Bull. Amer. Math. Soc. 66 1960 442-464.D. Gale
A theorem on flows in networks
Pacific J. Math. 7 (1957), 1073-1082.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). In fact, a number of discrete tomography problems were first discussed as combinatorial problems. In 1957, H. J. Ryser found the same consistency conditions, but in connection with the network flow problem. Another result of Ryser's is the definition of the switching operation by which discrete sets having the same projections can be transformed into each other.

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.

Theorems

* Reconstructing (finite) planar lattice sets from their 1-dimensional X-rays is an NP-hard problem if the X-rays are taken from $m\geq 3$ lattice directions (for $m=2$ the problem is in P).
*The reconstruction problem is highly unstable for $m\geq 3$ (meaning that a small perturbation of the X-rays may lead to completely different reconstructions) and stable for $m=2$, see.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.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.B. van Dalen
Stability results for uniquely determined sets from two directions in discrete tomography
Discrete Mathematics 309(12): 3905-3916 (2009).
*Coloring a grid using $k$ colors with the restriction that each row and each column has a specific number of cells of each color is known as the $(k-1)$−atom problem in the discrete tomography community. The problem is NP-hard for $k \geq 3$, see.

For further results see.

Algorithms

Among the reconstruction methods one can find algebraic reconstruction techniques (e.g., DART
or K. J. Batenburg, and J. Sijbers, "Generic iterative subset algorithms for discrete tomography", Discrete Applied Mathematics, vol. 157, no. 3, pp. 438-451, 2009), greedy algorithms (see P. Gritzmann, S. de Vries, M. Wiegelmann
Approximating binary images from discrete X-rays
SIAM J. Optim. 11 (2000), no. 2, 522-546. for approximation guarantees), and Monte Carlo algorithms.

Applications

Various algorithms have been applied in image processing
,
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).
medicine,
three-dimensional statistical data security problems, computer
tomograph assisted engineering and design, electron microscopy

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)

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)
and materials science, including the 3DXRD microscope.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, 200

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.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

A form of discrete tomography also forms the basis of nonograms, a type of logic puzzle in which information about the rows and columns of a digital image is used to reconstruct the image.

See also

* Geometric tomography

References

External links

Euro DT (a Discrete Tomography Wiki site for researchers)Tomography applet by Christoph DürrPhD thesis on discrete tomography (2012): Tomographic segmentation and discrete tomography for quantitative analysis of transmission tomography data

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Applied mathematics
Digital geometry