^{}Laboratoire CEDRIC-CNAM, 292 rue St-Martin, 75003 Paris, France,
Gabes University of Sciences, 6072 Gabes, Tunisia

Abstract

We study the problem of reconstructing binary images from four projections data in a fuzzy environment. Given the uncertainly projections,w e want to find a binary image that respects as best as possible these projections. We provide an iterative algorithm based on fuzzy integer programming and linear membership functions.

[1] T. Allahviranloo,K. Shamsolkotabi, N. A. Kiani and L. Alizadeh, Fuzzy integer linear programming problems,In t. J. Contemp. Math. Sciences, 2(4) (2007),167- 181. [2] M. G. Bailey and B. E. Gillett, Parametric integer programming analysis: A contraction approach,Jour nal of the Operational Research Society, 31 (1980),253- 262. [3] K. J. Batenburg, Network flow algorithms for discrete tomography,A dvances in Discrete Tomography and its Applications,Bi rkh¨auser,Bos ton, (2007), 175-205. [4] J. J. Buckley and L. J. Jowers, Monte carlo methods in fuzzy optimization,Studi es in Fuzziness and Soft Computig, 222 (2008),223- 226. [5] J. M. Cadenas and J. L. Verdegay, A primer on fuzzy optimization models and methods, Iranian Journal of Fuzzy Systems, 3(1) (2006),1- 21. [6] R. J. Gardner,P . Gritzmann and D. Prangenberg, The computational complexity of reconstructing lattice sets from their X-rays,D iscrete Math., 202 (1999),45- 71. [7] F. Herrera and J. L. Verdegay, Three models of fuzzy integer linear programming,Eur opean Journal of Operational Research, 83 (1995),581- 593. [8] F. Jarray, Solving problems of discrete tomography: applications in workforce scheduling, Ph.D. Thesis,U niversity of CNAM,P aris, 2004. [9] F. Jarray,M . C. Costa and C. Picouleau, Complexity results for the horizontal bar packing problem,I nformation Processing Letters, 108(6) (2008),356- 359. [10] N. Javadian,Y . Maali and N. Mahdavi-Amiri, Fuzzy linear programming with grades of satisfaction in constraints,Ir anian Journal of Fuzzy Systems, 6(3) (2009),17- 35. [11] A. Mitsos and P. I. Barton, Parametric mixed-integer 0-1 linear programming: the general case for a single parameter,Eur opean Journal of Operational Research, 194 (2009),663- 686. [12] S. A. Orlovski, On programming with fuzzy constraint sets,K ybernetes, 6 (1977),197- 201. [13] M. S. Osman,O . M. Saad and A. G. Hasan, Solving a special class of Large-Scale fuzzy multiobjective integer linear programming problems,F uzzy sets and systems, 107 (1999), 289-297. [14] H. J. Ryser, Combinatorial properties of matrices of zeros and ones,Canad. J. Math, 9 (1957),371- 377. [15] E. Shivantian,E. Khorram and A. Ghodousian, Optimization of linear objective function subject to fuzzy relation inequalities constraints with max-average composition,Ir anian Journal of Fuzzy Systems, 4(2) (2007),15- 29. [16] J. L. Verdegay, Fuzzy mathematical programming,In M. M. Gupta and E. Sanchez,Eds ., Fuzzy Information and Decision Processes,N orth-Holland,(1982), 231-236. [17] S. Weber,T. Schule,J. Hornegger and C. Schnorr, Binary tomography by iterating linear programs from noisy projections,LNCS, 233 (2004),38- 51. [18] H. J. Zimmermann, Description and optimization of fuzzy systems,In ternational Journal General Systems, 2 (1976),209- 215. [19] H. J. Zimmermann, Fuzzy programming and linear programming with several objective functions, F uzzy Sets and Systems, 1 (1978),45- 55.