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.
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References
Alpers, A.: Instability and Stability in Discrete Tomography. Ph.D. Thesis, Technical Univ. of Munich, Shaker Verlag, Germany (2004).
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).
Alpers, A., Gritzmann, P.: On stability, error correction, and noise compensation in discrete tomography. SIAM J. Discr. Math., 20, 227–239 (2006).
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).
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).
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).
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).
Baake, M., Gritzmann, P., Huck, C., Langfeld, B., Lord, K.: Discrete tomography of planar model sets. Acta Crystallographica Section A., 62, 419–433 (2006).
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).
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).
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).
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).
Balázs, P.: Reconstruction of discrete sets from four projections: Strong decomposability.Electr. Notes Discr. Math., 20, 329–345 (2005)
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).
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).
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).
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).
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).
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).
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).
Barucci, E., Del Lungo, A., Nivat, M., Pinzani, R.: X-rays characterizing some classes of discrete sets. J. Linear Algebra, 339, 3–21 (2001).
Barcucci, E., Frosini, A., Rinaldi, S.: Reconstruction of discrete sets from two absorbed projections: An algorithm. Electr. Notes Discr. Math., 12 (2003)
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)
Batenburg, K.J.: Analysis and optimization of an algorithm for discrete tomography.Electr. Notes Discr. Math., 12 (2003)
Batenburg, K.J.: A new algorithm for 3D binary tomography. Electr. Notes Discr. Math., 20, 247–261 (2005)
Batenburg, K.J.: An evolutionary algorithm for discrete tomography. Discr.Appl. Math., 151, 36–54 (2005)
Batenburg, K.J.: Network Flow Algorithms for Discrete Tomography. Ph.D.Thesis, Leiden Univ., The Netherlands (2006)
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)
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)
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)
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)
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)
Battle, X.L., Bizais, Y., Le Rest, C., Turzo, A.: Tomographic reconstruction using free-form deformation models. Proc. SPIE, 3661, 356–367 (1999)
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)
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)
Bebeacua, C., Mansour, T., Postnikov, A., Severini, S.: On the X-rays of permutations.Electr. Notes Discr. Math., 20, 193–203 (2005)
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)
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)
Brewbaker, C.R.: Lonesum (0,1)-Matrices and Poly-Bernoulli Numbers of Negative Index. MSc. Thesis, Iowa State Univ. (2005)
Brimkov, V.E., Barneva, R.P.: Exact image reconstruction from a single projection through real computation. Electr. Notes Discr. Math., 20, 233–246 (2005)
Brualdi, R.A.: Minimal nonnegative integral matrices and uniquely determined (0,1)-matrices. Lin. Algebra Appl., 341, 351–356 (2002)
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)
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)
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)
Brunetti, S.: Convexity and Complexity in Discrete Tomography. Ph.D. Thesis, Univ. of Florence, Italy (2001)
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)
Brunetti, S., Daurat, A.: An algorithm reconstructing lattice convex sets. Theoret.Comput. Sci., 304, 35–57 (2003)
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)
Brunetti, S., Daurat, A.: Determination of Q-convex bodies by X-rays. Electr.Notes Discr. Math., 20, pp. 67–81 (2005)
Brunetti, S., Daurat, A.: Stability in discrete tomography: Some positive results.Discr. Appl. Math., 147, pp. 207–226 (2005)
Brunetti, S., Daurat, A., Del Lungo, A.: Approximate X-rays reconstruction of special lattice sets. Pure Math. Appl., 11, 409–425 (2000)
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)
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)
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)
Capricelli, T.D., Combettes, P.L.: Parallel block-iterative reconstruction algorithms for binary tomography. Electr. Notes Discr. Math., 20, 263–280 (2005)
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)
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)
Castiglione, G., Restivo, A.: Reconstruction of L-convex polyominoes. Electr. Notes Discr. Math., 12 (2003).
Censor, Y.: Binary steering in discrete tomography reconstruction with sequential and simultaneous iterative algorithms. Lin. Algebra Appl., 339 111–124 (2001).
Chrobak, M., Couperus, P., Dürr, C., Woeginger, G.: On tiling under tomographic constraints. Theor. Comp. Sci. 290, 2125–2136 (2003).
Chrobak, M., Dürr, C.: Reconstructing hv-convex polyominoes from orthogonal projections. Inform. Process. Lett., 69, 283–289 (1999).
Chrobak, M., Dürr, C.: Reconstructing polyatomic structures from X-rays: NPcompleteness proof for three atoms. Theor. Comp. Sci., 259, 81–98 (2001).
Costa, M. C., Jarray, F., Picouleau, C.: Reconstruction of binary matrices under adjacency constraints. Electr. Notes Discr. Math. 20, 281–297 (2005).
Costa, M. C., de Werra, D., Picouleau, C.: Using graphs for some discrete tomography problems. Discr. Appl. Math., 154, 35–46 (2006).
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).
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).
Dahl, G., Flatberg, T.: Optimization and reconstruction of hv-convex (0, 1)matrices. Electr. Notes Discr. Math., 12 (2003).
Dahl, G., Flatberg, T.: Optimization and reconstruction of hv-convex (0, 1)matrices. Discr. Appl. Math., 151, 93–105 (2005).
Daurat, A.: Convexité dans le Plan Discret. Application à la Tomographie, Ph.D. Thesis, LLAIC1, and LIAFA, Université Paris 7, France (2000).
Daurat, A.: Determination of Q-convex sets by X-rays. Theoret. Comput. Sci.,332 19–45 (2005).
Daurat, A., Del Lungo, A., Nivat, M.: Medians of discrete sets according to a linear distance. Discrete Comput. Geom., 23, 465–483 (2000).
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).
Debled-Rennesson, I., Remy, J.-L., Rouyer-Degli, J.: Detection of the discrete convexity of polyominoes. Discr. Appl. Math., 125, 115–133 (2003).
Del Lungo, A.: Reconstructing permutation matrices from diagonal sums. Theor. Comput. Sci., 281, 235–249 (2002).
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).
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).
Di Gesu, V.D., Valenti, C.: The stability problem and noisy projections in discrete tomography. J. Visual Languages and Computing, 15 361–371 (2004).
Dulio, P., Gardner, R.J., Peri, C.: Discrete point X-rays of convex lattice sets. Electronic Notes in Discrete Math., 20, 1–13 (2005).
Dulio, P., Gardner, R.J., Peri, C.: Discrete point X-rays. SIAM J. Discrete Math., 20, 171–188 (2006).
Dürr, C., Goles, E., Rapaport, I., Remila, E.: Tiling with bars under tomographic constraints. Theor. Comput. Sci., 290, 1317–1329 (2003).
Frosini, A.: Complexity Results and Reconstruction Algorithms for Discrete Tomography, Ph.D. Thesis, Univ. of Siena, Italy (2003).
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).
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).
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).
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).
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).
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).
Frosini, A., Simi, G.: Reconstruction of low degree domino tilings. Electr. Notes Discr. Math., 12 (2003).
Frosini, A., Simi, G.: The NP-completeness of a tomographical problem on bicolored domino tilings. Theor. Comput. Sci., 319, 447–454 (2004).
Gardner, R.J.: Geometric Tomography 2nd edition, Cambridge University Press, New York, NY (2006).
Gardner, R.J., Gritzmann, P.: Discrete tomography: Determination of finite sets by X-rays. J. Linear Algebra, 339, 3–21 (2001).
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).
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).
Gerard, Y.: Reduction from three-dimensional discrete tomography to multicommodity flow problem. Theor. Comput. Sci., 346, 300–306 (2005).
Gerard, Y., Feschet, F.: Application of a discrete tomography algorithm to computerized tomography. Electr. Notes Discr. Math., 20, 501–517 (2005).
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).
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).
Gritzmann, P., de Vries, S.: On the algorithmic inversion of the discrete Radon transform. Theor. Comp. Science, 281, 455–469 (2001).
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).
Gritzmann, P, de Vries, S., Wiegelmann, M.: Approximating binary images from discrete X-rays. SIAM J. Optim., 11, 522–546 (2000).
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).
Hajdu, L.: Unique reconstruction of bounded sets in discrete tomography. Electr. Notes Discr. Math., 20, 15–25 (2005).
Hajdu, L., Tijdeman, R.: Algebraic aspects of discrete tomography. J. Reine Angew. Math., 534, 119–128 (2001).
Hajdu, L., Tijdeman, R.: An algorithm for discrete tomography. J. Linear Algebra, 339, 147–169 (2001).
Hajdu, L., Tijdeman, R.: Algebraic aspects of emission tomography with absorption. Theoret. Comput. Sci., 290, 2169–2181 (2003).
Herman, G.T., Kuba, A. (eds.): Discrete Tomography: Foundations, Algorithms, and Applications. Birkhäuser, Boston, MA (1999).
Herman, G.T., Kuba, A.: Discrete tomography in medical imaging. Proc. IEEE, 91, 380–385 (2003).
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).
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).
Imiya, A., Tirii, A., Sato, K.: Tomography on finite graphs. Electr. Notes Discr. Math., 20, 217–232 (2005).
Jarray F.: R’esolution de Problemes de Domographie Discrète. Applications à la Planification de Ppersonnel. Ph.D. Thesis, CNAM, Paris, France (2004).
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).
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).
Kaneko, A., Nagahama, R: Structure of total reconstructed sets from given two projection data. Electr. Notes Discr. Math., 20, 27–46 (2005).
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).
Kingston, A.M.: Extension and Application of Finite Discrete Radon Projection Theory. Ph.D. Thesis, Monash Univ., Australia (2005).
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).
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).
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).
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).
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).
Kuba, A., Balogh, E.: Reconstruction of convex 2D discrete sets in polynomial time. Theor. Comput. Sci., 283, 223–242 (2002).
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).
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).
Kuba, A., Nagy, A.: Reconstruction of hv-convex binary matrices from their absorbed projections. Electr. Notes Theor. Comput. Sci., 46 (2001).
Kuba, A., Nagy, A., Balogh, E.: Reconstruction of hv-convex binary matrices from their absorbed projections. Discr. Appl. Math., 139, 137–148 (2004).
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).
Kuba, A., Nivat, M.: Reconstruction of discrete sets with absorption. Lin. Algebra Appl., 339, 171–194 (2001).
Kuba, A., Nivat, M.: A sufficient condition for non-uniqueness in binary tomography with absorption. Discr. Appl. Math., 346, 335–357 (2005).
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).
Kuba, A., Ruskó, L., Kiss, Z., Nagy, A.: Discrete reconstruction techniques, Electr. Notes Discr. Math., 20, 385–398 (2005).
Kuba, A., Ruskó, L., Rodek, L., Kiss, Z.: Preliminary studies of discrete tomography in neutron imaging, IEEE Trans. Nuclear Sci., 52, 380–385 (2005).
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).
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).
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).
Liao, H.Y.: Reconstruction of Label Images Using Gibbs Priors. Ph.D. Thesis, City University of New York, New York, USA (2005).
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).
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).
Liao, H.Y., Herman, G.T.: Tomographic reconstruction of label images from a new projections. Electr. Notes Discr. Math., 12 (2003).
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).
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).
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).
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).
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).
Masilamani, V., Dersanambika, K.S., Krithivasan, K.: Binary 3D matrices under the microscope: A tomographical problem. Electr. Notes Discr. Math., 20, 573–586 (2005).
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).
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).
Nagy, A., Kuba, A.: Reconstruction of binary matrices from fan-beam projections. Acta Cybernetica, 17, 359–385 (2005).
Nagy, A., Kuba, A.: Parameter settings for reconstructing binary matrices from fan-beam projections. J. Comput. Info. Tech., 14, 101–110 (2006).
Nagy, A., Kuba, A., Samal, M.: Reconstruction of factor structures using discrete tomography method. Electr. Notes Comput. Sci., 20, 519–534 (2005).
Nam, Y.: Integral matrices with given row and column sums. Ars Combinatorica, 52, 141–151 (1999).
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).
Normand, N., Guédon, J.: Spline mojette transform. Application in tomography and communication. Proc. EUSIPCO, 2, 407–410 (2002).
Picouleau, C.: Reconstruction of domino tiling from its two orthogonal projections. Theor. Comp. Sci., 255, 437–447 (2001).
Picouleau, C., Brunetti, S., Frosini, A.: Reconstructing a binary matrix under timetabling constraints. Electr. Notes Discr. Math., 20, 99–112 (2005).
Popa, C., Zdunek, R.: Penalized least-squares image reconstruction for borehole tomography. Proc. ALGORITMY 2005, Podbanske, Slovakia, pp. 407–410 (2005).
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).
Ruskó, L., Kuba, A.: Multi-resolution method for binary tomography. Electr. Notes Discr. Math., 20, 299–311 (2005).
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).
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).
Schüle, T., Weber, S., Schnörr, C.: Adaptive reconstruction of discrete-valued objects from few projections. Electr. Notes in Discr. Math., 20, 365–384 (2005).
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).
Servières, M.: Reconstruction Tomographique Mojette. Thèse de doctorat, Universitéde Nantes, France (2005).
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).
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).
Soussen, C.: Reconstruction 3D d’un Objet Compact en Tomographie. Ph.D. Thesis, Université de Paris-Sud, France, (2000).
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).
Soussen, C., Muhammad-Djafari, A.: Polygonal and polyhedral contour reconstruction in computed tomography. IEEE Trans. Image Processing, 13, 1507–1523 (2004).
Takiguchi, T.: Reconstruction of measurable plane sets from their orthogonal projections. Contemporary Math., 348, 199–208 (2004).
Valenti, C.: An Experimental Study of the Stability Problem in Discrete Tomography. Ph.D. Thesis, Univ. of Palermo, Italy (2002).
Valenti, C.: Discrete tomography from noisy projections. In: Series on Software Engineering and Knowledge Engineering, World Scientific Publishing, 15, 38–45 (2003).
Valenti, C.: An experimental study of the stability problem in discrete tomography. Electr. Notes Discr. Math., 20, 113–132 (2005).
Vallejo, E.: Plane partitions and characters of the symmetric group. J. Algebraic Comb.,11, 79–88 (2000).
Vallejo, E.: The classification of minimal matrices of size 2×q. Lin. Algebra Appl., 340, 169–181 (2002).
Vallejo, E.: A characterization of additive sets. Discr. Math., 259, 201–210 (2002).
Vallejo, E.: Minimal matrices and discrete tomography. Electr. Notes in Discr. Math., 20, 113–132 (2005).
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).
Venere, M., Liao, H., Clausse, A.: A genetic algorithm for adaptive tomography of elliptical objects. IEEE Signal Processing Letters, 7, 176–178 (2000).
de Vries, S.: Discrete Tomography, Packing and Covering, and Stable Set Problems: Polytopes and Algorithms. Ph.D. Thesis, Technical Univ. of Munich, Germany (1999).
Weber, S., Schnörr, C., Hornegger, J.: A linear programming relaxation for binary tomography with smoothness priors. Electr. Notes Discr. Math., 12 (2003).
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).
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).
Weber, S., Schüle, T., Schnörr, C.: Prior learning and convex-concave regularization of binary tomography. Electr. Notes Discr. Math., 20, 313–327 (2005).
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).
Weber, G.-W., Yasar, Ô.: Discrete tomography: A modern inverse problem reconsidered by optimization. J. Comp. Tech., 9, 115–121 (2004).
Woeginger, G.J.: The reconstruction of polyominoes from their orthogonal projections. Inform. Process. Lett., 77, 225–229 (2001).
Yagle, A.: A convergent composite mapping Fourier domain iterative algorithm for 3-D discrete tomography. Lin. Algebra Appl., 339, 91–109 (2001).
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).
Ye, Y., Wang, G., Zhu, J.: Linear diophantine equations for discrete tomography. J. X-Ray Sci. Tech., 10, 59–66 (2001).
Zdunek, R., Pralat, A.: Detection of subsurface bubbles with discrete electromagnetic geotomography. Electr. Notes Discr. Math., 20, 535–553 (2005).
Zopf, S., Kuba, A.: Reconstruction of measurable sets from two generalized projections. Electr. Notes Discr. Math., 20, 47–66 (2005).
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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
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