BRST quantization

From formulasearchengine
Revision as of 21:16, 20 November 2013 by en>BD2412 (→‎Quantum version: clean up obsolete tags, removed: {{Disambiguation needed|date=June 2011}} using AWB)
Jump to navigation Jump to search
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[1][2] 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,[3][4][5] discrete mathematics,[6][7][8] complexity theory[9][10] and combinatorics.[11][12][13] 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 network flow problem.[14] 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.

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

For further results see.[1][2]

Algorithms

Among the reconstruction methods one can find algebraic reconstruction techniques (e.g., DART[18] [19] or [20]), greedy algorithms (see [21] for approximation guarantees), and Monte Carlo algorithms.[22][23]

Applications

Various algorithms have been applied in image processing ,[18] medicine, three-dimensional statistical data security problems, computer tomograph assisted engineering and design, electron microscopy [24] ,[25] and materials science.,[22][23] [26]

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.[27]

See also

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

External links

  1. ↑ 1.0 1.1 Herman, G. T. and Kuba, A., Discrete Tomography: Foundations, Algorithms, and Applications, Birkhäuser Boston, 1999
  2. ↑ 2.0 2.1 Herman, G. T. and Kuba, A., Advances in Discrete Tomography and Its Applications, Birkhäuser Boston, 2007
  3. ↑ R.J. Gardner, P. Gritzmann, Discrete tomography: determination of finite sets by X-rays, Trans. Amer. Math. Soc. 349 (1997), no. 6, 2271-2295.
  4. ↑ L. Hajdu, R. Tijdeman, Algebraic aspects of discrete tomography, J. reine angew. Math. 534 (2001), 119-128.
  5. ↑ A. Alpers, R. Tijdeman, The Two-Dimensional Prouhet-Tarry-Escott Problem, Journal of Number Theory, 123 (2), pp. 403-412, 2007 [1].
  6. ↑ 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.
  7. ↑ 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 [2]
  8. ↑ 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.
  9. ↑ 9.0 9.1 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.
  10. ↑ 10.0 10.1 C. Dürr, F. Guiñez, M. Matamala, Reconstructing 3-Colored Grids from Horizontal and Vertical Projections Is NP-hard. ESA 2009: 776-787.
  11. ↑ H.J. Ryser, Matrices of zeros and ones, Bull. Amer. Math. Soc. 66 1960 442-464.
  12. ↑ D. Gale, A theorem on flows in networks, Pacific J. Math. 7 (1957), 1073-1082.
  13. ↑ 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).
  14. ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  15. ↑ 15.0 15.1 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.
  16. ↑ 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.
  17. ↑ B. van Dalen, Stability results for uniquely determined sets from two directions in discrete tomography, Discrete Mathematics 309(12): 3905-3916 (2009).
  18. ↑ 18.0 18.1 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). 21 year-old Glazier James Grippo from Edam, enjoys hang gliding, industrial property developers in singapore developers in singapore and camping. Finds the entire world an motivating place we have spent 4 months at Alejandro de Humboldt National Park.
  19. ↑ 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 [3]
  20. ↑ K. J. Batenburg, and J. Sijbers, "Generic iterative subset algorithms for discrete tomography", Discrete Applied Mathematics, vol. 157, no. 3, pp. 438-451, 2009
  21. ↑ P. Gritzmann, S. de Vries, M. Wiegelmann, Approximating binary images from discrete X-rays, SIAM J. Optim. 11 (2000), no. 2, 522-546.
  22. ↑ 22.0 22.1 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 [4].
  23. ↑ 23.0 23.1 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.
  24. ↑ 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) 21 year-old Glazier James Grippo from Edam, enjoys hang gliding, industrial property developers in singapore developers in singapore and camping. Finds the entire world an motivating place we have spent 4 months at Alejandro de Humboldt National Park.
  25. ↑ 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) 21 year-old Glazier James Grippo from Edam, enjoys hang gliding, industrial property developers in singapore developers in singapore and camping. Finds the entire world an motivating place we have spent 4 months at Alejandro de Humboldt National Park.
  26. ↑ 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
  27. ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
Retrieved from "https://en.formulasearchengine.com/index.php?title=BRST_quantization&oldid=14889"