Abstract
We study the problem of reconstructing bicolored images from their discrete projections that is the number of pixels of each color lying on each row and column. The problem is well known to be NP- complete so, we study a restricted case (with bounded projections) and present an approximating algorithm based on a max-flow technique for the general case.
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References
Anstee, R.P.: The network flows approach for matrices with given row and column sums. Discrete Math. 44, 125–138 (1983)
Barcucci, E., Del Lungo, A., Nivat, M., Pinzani, R.: X-rays characterizing some classes of discrete sets. Linear Algebra and its Applications 339, 3–21 (2001)
Batenburg, K.J.: Network flow algorithms for discrete tomography. In: Herman, G., Kuba, A. (eds.) Advances in Discrete Tomography and its Applications, pp. 175–205. Birkhäuser, Boston (2007)
Batenburg, K.J.: An evolutionary algorithm for discrete tomography. Discrete Applied Mathematics 151, 36–54 (2005)
Brunetti, S., Costa, M.C., Frosini, A., Jarray, F., Picouleau, C.: Reconstruction of binary matrices under adjacency constraints. In: Herman, G., Kuba, A. (eds.) Advances in Discrete Tomography and its Applications, Boston, pp. 125–150 (2007)
Baumann, J., Kiss, Z., Krimmel, S., Kauba, A., Nagy, A., Rodek, L., Schillinger, S., Stephan, J.: Discrete tomography methods for nondestructive testing. In: Herman, G., Kuba, A. (eds.) Advances in Discrete Tomography and its Applications, Boston, pp. 303–331 (2007)
Brocchi, S., Frosini, A., Rinaldi, S.: Solving some instances of the two color problem. In: Brlek, S., Reutenauer, C., Provençal, X. (eds.) DGCI 2009. LNCS, vol. 5810, pp. 505–516. Springer, Heidelberg (2009)
Chrobak, M., Dürr, C.: Reconstructing Polyatomic Structures from X-Rays: NP Completness proof for three Atoms. Theoretical computer Science 259(1), 1–98 (2001)
Chrobak, M., Couperus, P., Dürr, C., Woeginger, G.: A note on tiling under tomographic constraints. Theoretical computer Science 290, 2125–2136 (2003)
Costa, M.C., De Werra, D., Picouleau, C.: Using graphs for some discrete tomography problems. Discrete Applied Mathematics 154, 35–46 (2006)
Costa, M.C., de Werra, D., Picouleau, C., Schindld, D.: Discrete Applied Mathematics 148, 240–245 (2005)
Dürr, C., Guinez, F., Matamala, M.: Reconstructing 3-colored grids from horizontal and vertical projections is NP-hard, arXiv:0904.3169v1 (2009)
Gale, D.: A theorem on flows in networks. Pacific J. Math. 7, 1073–1082 (1957)
Gardner, R.J., Gritzmann, P., Prangenberg, D.: On the computionnal complexity of reconstructing lattice sets from their X-rays. Discrete Mathematics 202, 45–71 (1999)
Gardner, R.J., Gritzmann, P., Prangenberg, D.: On the computational complexity of determining polyatomic structures by X-rays. Theoretical Computer Science 233, 91–106 (2000)
Goldberg, A.V.: An efficient implementation of a scaling minimum-cost flow algorithm. Journal of Algorithms 22, 1–29 (1997)
Goldberg, A.V., Rao, S.: Beyond the flow decomposition barrier. Journal ACM 45, 783–797 (1998)
Hall, P.: A model for learning human vascular anatomy. DIMACS Serie in Discrete Mathematical Problems with Medical Applications 55, 11–27 (2000)
Herman, G.T., Kuba, A.: Advances in Discrete Tomography and its Applications. Birkhäuser, Boston (2007)
Jarray, F.: Solving problems of discrete tomography. Applications in workforce scheduling, Ph.D. Thesis, University of CNAM, Paris (2004)
Jarray, F.: Workforce scheduling and table coloring. In: Proceedings ROADEF 2005, Tours, pp. 92–100 (2004)
Jarray, F.: A lagrangean approach to reconstruct bicolored images from discrete orthogonal projections. Pure mathematics and applications (Linear algebra and computer science) 20(1), 17–25 (2010)
Kisielowski, C., Schwander, P., Baumann, F.H., Seibt, M., Kim, Y., Ourmazd, A.: An approach to quantitative high-resolution transmission electron microscopy of crystalline materials. Ultramicroscopy 58, 131–155 (1995)
Onnasch, D.G.W., Prause, G.P.M.: Heart Chamber Reconstruction from Biplane Angiography. In: Herman, G., Kuba, A. (eds.) Discrete Tomography: Foundations, Algorithms and Applications, pp. 385–403. Birkhäuser, Boston (1999)
Ryser, H.J.: Combinatorial properties of matrices of zeros and ones. Canad. J. Math. 9, 371–377 (1957)
Sachs, J.R.J., Sauer, K.: 3D Reconstruction from sparse Radiographic Data. In: Herman, G., Kuba, A. (eds.) Discrete Tomography: Foundations, Algorithms and Applications, pp. 363–383. Birkhäuser, Boston (1999)
Salzberg, P.M., Rivera-vega, P.I., Rodriguez, A.: Network flow model for binary tomography on lattices. Internal journal imaging system technology 9, 145–154 (1998)
Slump, C.H., et al.: A network flow approach to reconstruction of the left ventricle from two projections. Comput. Gr. Im. Proc. 18, 18–36 (1982)
Wang, B., Zhang, F.: On the precise Number of (0,1)-Matrices in \(\mathcal{U}(R,S)\). Discrete Mathematics 187, 211–220 (1998)
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Jarray, F., Tlig, G. (2011). Approximating Bicolored Images from Discrete Projections. In: Aggarwal, J.K., Barneva, R.P., Brimkov, V.E., Koroutchev, K.N., Korutcheva, E.R. (eds) Combinatorial Image Analysis. IWCIA 2011. Lecture Notes in Computer Science, vol 6636. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21073-0_28
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DOI: https://doi.org/10.1007/978-3-642-21073-0_28
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