Abstract
The 2-color problem in discrete tomography requires to construct a 2-colored matrix consistent with a given set of projections representing the number of elements of each color in each one of its rows and columns.
In this paper, we describe an heuristic algorithm to find a solution of the 2-color problem, that has been recently proved to be NP-complete. The algorithm starts by computing a solution where elements of different colors may overlap, and then it proceeds in searching for switches that leave unaltered the projections but remove the overlaps. Experimental results show that this heuristic approach finds a solution in a short computational time to almost all the randomly generated 2-color instances, and it provides for the remaining ones a high quality approximation.
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References
Anstee, R.P.: Invariant sets of arcs in network flow problems. Discrete Applied Mathematics 13, 1–7 (1986)
Balogh, E., Sorantin, E., Nyúl, L.G., Palágyi, K., Kuba, A., Werkgartner, G., Spuller, E.: Virtual dissection of the colon: technique and first experiments with artificial and cadaveric phantoms. In: Proceedings of SPIE Medical Imaging 2002: Image Processing, San Diego, USA, vol. 4681, pp. 713–721 (2002)
Batenburg, K.J., Bals, S., Sijbers, J., Kuebel, C., Midgley, P.A., Hernandez, J.C., Kaiser, U., Encina, E.R., Coronado, E.A., Van Tendeloo, G.: 3D imaging of nanomaterials by discrete tomography. Ultramicroscopy 109(6), 730–740 (2009)
Bentz, C., Costa, M.C., Picouleau, C., Ries, B., de Werra, D.: Degree-constrained edge partitioning in graphs arising from discrete tomography. Journal of Graph Algorithms and Applications 13(2), 99–118 (2009)
Brocchi, S., Frosini, A., Picouleau, C.: Reconstruction of binary matrices under fixed size neighborhood constraints. Theoretical Computer Science 406(1-2), 43–54 (2008)
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, M.: Reconstructing polyatomic structures from X-rays: NP completness proof for three atoms. Theoretical Computer Science 259, 81–98 (2001)
Costa, M.-C., de Werra, D., Picouleau, C., Schindl, D.: A solvable case of image reconstruction in discrete tomography. Discrete Applied Mathematics 148(3), 240–245 (2005)
Dürr, C., Guiñez, F., Matamala, M.: Reconstructing 3-colored grids from horizontal and vertical projections is NP-hard. In: Fiat, A., Sanders, P. (eds.) ESA 2009. LNCS, vol. 5757, pp. 776–787. Springer, Heidelberg (2009)
Gardner, R.J., Gritzmann, P.: Discrete tomography: determination of finite sets by X-rays. Trans. Amer. Math. Soc. 349, 2271–2295 (1997)
Gardner, R.J., Gritzmann, P., Pranenberg, D.: On the computational complexity of determining polyatomic structures by X-rays. Theoretical Computer Science 233, 91–106 (2000)
Herman, G.T., Kuba, A.: Discrete tomography: Foundations algorithms and applications. Birkhauser, Boston (1999)
Herman, G.T., Kuba, A.: Advances in Discrete Tomography and Its Applications. Birkhauser, Boston (2007)
Irving, R.W., Jerrum, M.R.: Three-dimensional statistical data security problems. SIAM Journal of Computing 23, 170–184 (1994)
Matej, S., Vardi, A., Hermann, G.T., Vardi, E.: Binary tomography using Gibbs priors. In: Herman, G.T., Kuba, A. (eds.) Discrete Tomography: Foundations, Algorithms and Applications, pp. 191–212. Birkhauser, Boston (1999)
Prause, G.P.M., Onnasch, D.G.W.: Binary reconstruction of the heart chambers from biplane angiographic image sequence. IEEE Transactions Medical Imaging 15, 532–559 (1996)
Ryser, H.J.: Combinatorial properties of matrices of zeros and ones. Canadian Journal of Mathematics 9, 371–377 (1957)
Shliferstein, A.R., Chien, Y.T.: Switching components and the ambiguity problem in the reconstruction of pictures from their projections. Pattern Recognition 10, 327–340 (1978)
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Barcucci, E., Brocchi, S., Frosini, A. (2011). Solving the Two Color Problem: An Heuristic Algorithm. 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_27
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DOI: https://doi.org/10.1007/978-3-642-21073-0_27
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