Document Type : Research Paper
Author
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.
Keywords
[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.
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.
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