Abstract

New methods of optical fuzzy-logic operations and optical fuzzy-controller synthesis are proposed and experimentally demonstrated by use of optical fan-out elements to achieve multiple imaging and polarization-space/aperture data encoding to represent fuzzy variables in optics. Sixteen fuzzy-logic operations between two inputs are achieved by use of a simple polarization-space data-encoding and kernel-operation scheme. In addition, a max–min composition-based fuzzy controller is implemented by use of an aperture-data-encoding and a double-multi-imaging approach. Our systems exhibit a high operation speed, a large information throughput, and a high signal-to-noise ratio.

© 1994 Optical Society of America

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  1. L. A. Zadeh, “Fuzzy sets,” Inf. Control 8, 338–353 (1965).
    [CrossRef]
  2. P. Siy, C. S. Chen, “Fuzzy logic for handwritten numerical character recognition,” IEEE Trans. Syst. Man Cybern. SMC-4, 570–575 (1974).
  3. J. C. Bezdek, “Numerical taxonomy with fuzzy sets,” J. Math. Biol. 1, 57–71 (1974).
    [CrossRef]
  4. S. Gale, “Conjectures on many-valued logic, regions and criteria for conflict resolution,” Proc. IEEE 75, 212–225 (1975).
  5. W. G. Wee, S. K. Fu, “A formulation of fuzzy automata and its application as a model of learning systems,” IEEE Trans. Syst. Sci. Cybern. SSC-5, 215–223 (1969).
    [CrossRef]
  6. R. Kling, “Fuzzy PLANNER: reasoning with inexact concepts in a procedural problem-solving language,” J. Cybern. 4, 105–122 (1974).
    [CrossRef]
  7. T. E. Lee, “Shape-oriented chromosome classification,” IEEE Trans. Syst. Man Cybern. SMC-5, 629–682 (1975).
  8. M. Albn, “Fuzzy sets and their application to medical diagnosis,” Ph.D. dissertation (University of California, Berkeley, Berkeley, Calif., 1975).
  9. J. M. Becker, “A structural design process: philosophy and methodology,” Ph.D. dissertation (University of California, Berkeley, Berkeley, Calif., 1973).
  10. W. L. Fellinger, “Specifications for a fuzzy system modeling language,” Ph.D. dissertation (Oregon State University, Corvallis, Ore., 1974).
  11. E. H. Mamdani, S. Assilian, “An experiment in linguistic synthesis with a fuzzy logic controller,” Int. J. Man-Mach. Stud. 7, 1–13 (1975).
    [CrossRef]
  12. F. Wenstop, “Deductive verbal models of organizations,” Int. J. Man-Mach. Stud. 8, 293–311 (1976).
    [CrossRef]
  13. A. Kandel, S. C. Lee, Fuzzy Switching and Automata: Theory and Applications (Crane, Russak, N.Y., 1979).
  14. L. Liu, “Optical implementation of parallel fuzzy logic,” Opt. Commun. 73, 183–187 (1989).
    [CrossRef]
  15. G. C. Marsden, B. Olson, S. Esener, S. H. Lee, “Optoelectronic fuzzy logic system,” in Optical Computing, Vol. 6 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), pp. 212–215.
  16. S. Zhang, S. Lin, C. Cheng, “Optical fuzzy vector-matrix composition operation using shadow-casting,” Opt. Commun. 94, 497–500 (1992).
    [CrossRef]
  17. S. Lin, I. Kumazawa, “Optical fuzzy image processing based-on shadow casting,” Opt. Commun. 94, 397–405 (1992).
    [CrossRef]
  18. S. Lin, S. Zhang, C. Chen, R. Liu, J. Wu, “Optical multiple-variable fuzzy logic array using shadow casting,” Microwave Opt. Tech. Lett. 6, 106–109 (1993).
    [CrossRef]
  19. H. Itoh, M. Watanabe, S. Mukai, H. Yajima, “Optoelectronic fuzzy logic inference system using beam scanning laser diodes,” in Optical Computing, Vol. 7 of 1993 OSA Technical Digtest Series (Optical Society of America, Washington, D.C., 1993), p. 123.
  20. M. M. Gupta, G. N. Saridis, B. R. Gaines, Fuzzy Automata and Decision Processes (North-Holland, Amsterdam, 1977), Chap. 4, pp. 83–97.
  21. T. Kohonen, Self-Organization and Associative Memory (Springer-Verlag, New York, (1984), pp. 1–176.
  22. S. Zhou, S. Campbell, P. Yeh, H. K. Liu, “Modified-signed-digit optical computing by using optical fan-out elements,” Opt. Lett. 17, 1697–1699 (1992).
    [CrossRef] [PubMed]

1993 (1)

S. Lin, S. Zhang, C. Chen, R. Liu, J. Wu, “Optical multiple-variable fuzzy logic array using shadow casting,” Microwave Opt. Tech. Lett. 6, 106–109 (1993).
[CrossRef]

1992 (3)

S. Zhou, S. Campbell, P. Yeh, H. K. Liu, “Modified-signed-digit optical computing by using optical fan-out elements,” Opt. Lett. 17, 1697–1699 (1992).
[CrossRef] [PubMed]

S. Zhang, S. Lin, C. Cheng, “Optical fuzzy vector-matrix composition operation using shadow-casting,” Opt. Commun. 94, 497–500 (1992).
[CrossRef]

S. Lin, I. Kumazawa, “Optical fuzzy image processing based-on shadow casting,” Opt. Commun. 94, 397–405 (1992).
[CrossRef]

1989 (1)

L. Liu, “Optical implementation of parallel fuzzy logic,” Opt. Commun. 73, 183–187 (1989).
[CrossRef]

1976 (1)

F. Wenstop, “Deductive verbal models of organizations,” Int. J. Man-Mach. Stud. 8, 293–311 (1976).
[CrossRef]

1975 (3)

S. Gale, “Conjectures on many-valued logic, regions and criteria for conflict resolution,” Proc. IEEE 75, 212–225 (1975).

T. E. Lee, “Shape-oriented chromosome classification,” IEEE Trans. Syst. Man Cybern. SMC-5, 629–682 (1975).

E. H. Mamdani, S. Assilian, “An experiment in linguistic synthesis with a fuzzy logic controller,” Int. J. Man-Mach. Stud. 7, 1–13 (1975).
[CrossRef]

1974 (3)

R. Kling, “Fuzzy PLANNER: reasoning with inexact concepts in a procedural problem-solving language,” J. Cybern. 4, 105–122 (1974).
[CrossRef]

P. Siy, C. S. Chen, “Fuzzy logic for handwritten numerical character recognition,” IEEE Trans. Syst. Man Cybern. SMC-4, 570–575 (1974).

J. C. Bezdek, “Numerical taxonomy with fuzzy sets,” J. Math. Biol. 1, 57–71 (1974).
[CrossRef]

1969 (1)

W. G. Wee, S. K. Fu, “A formulation of fuzzy automata and its application as a model of learning systems,” IEEE Trans. Syst. Sci. Cybern. SSC-5, 215–223 (1969).
[CrossRef]

1965 (1)

L. A. Zadeh, “Fuzzy sets,” Inf. Control 8, 338–353 (1965).
[CrossRef]

Albn, M.

M. Albn, “Fuzzy sets and their application to medical diagnosis,” Ph.D. dissertation (University of California, Berkeley, Berkeley, Calif., 1975).

Assilian, S.

E. H. Mamdani, S. Assilian, “An experiment in linguistic synthesis with a fuzzy logic controller,” Int. J. Man-Mach. Stud. 7, 1–13 (1975).
[CrossRef]

Becker, J. M.

J. M. Becker, “A structural design process: philosophy and methodology,” Ph.D. dissertation (University of California, Berkeley, Berkeley, Calif., 1973).

Bezdek, J. C.

J. C. Bezdek, “Numerical taxonomy with fuzzy sets,” J. Math. Biol. 1, 57–71 (1974).
[CrossRef]

Campbell, S.

Chen, C.

S. Lin, S. Zhang, C. Chen, R. Liu, J. Wu, “Optical multiple-variable fuzzy logic array using shadow casting,” Microwave Opt. Tech. Lett. 6, 106–109 (1993).
[CrossRef]

Chen, C. S.

P. Siy, C. S. Chen, “Fuzzy logic for handwritten numerical character recognition,” IEEE Trans. Syst. Man Cybern. SMC-4, 570–575 (1974).

Cheng, C.

S. Zhang, S. Lin, C. Cheng, “Optical fuzzy vector-matrix composition operation using shadow-casting,” Opt. Commun. 94, 497–500 (1992).
[CrossRef]

Esener, S.

G. C. Marsden, B. Olson, S. Esener, S. H. Lee, “Optoelectronic fuzzy logic system,” in Optical Computing, Vol. 6 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), pp. 212–215.

Fellinger, W. L.

W. L. Fellinger, “Specifications for a fuzzy system modeling language,” Ph.D. dissertation (Oregon State University, Corvallis, Ore., 1974).

Fu, S. K.

W. G. Wee, S. K. Fu, “A formulation of fuzzy automata and its application as a model of learning systems,” IEEE Trans. Syst. Sci. Cybern. SSC-5, 215–223 (1969).
[CrossRef]

Gaines, B. R.

M. M. Gupta, G. N. Saridis, B. R. Gaines, Fuzzy Automata and Decision Processes (North-Holland, Amsterdam, 1977), Chap. 4, pp. 83–97.

Gale, S.

S. Gale, “Conjectures on many-valued logic, regions and criteria for conflict resolution,” Proc. IEEE 75, 212–225 (1975).

Gupta, M. M.

M. M. Gupta, G. N. Saridis, B. R. Gaines, Fuzzy Automata and Decision Processes (North-Holland, Amsterdam, 1977), Chap. 4, pp. 83–97.

Itoh, H.

H. Itoh, M. Watanabe, S. Mukai, H. Yajima, “Optoelectronic fuzzy logic inference system using beam scanning laser diodes,” in Optical Computing, Vol. 7 of 1993 OSA Technical Digtest Series (Optical Society of America, Washington, D.C., 1993), p. 123.

Kandel, A.

A. Kandel, S. C. Lee, Fuzzy Switching and Automata: Theory and Applications (Crane, Russak, N.Y., 1979).

Kling, R.

R. Kling, “Fuzzy PLANNER: reasoning with inexact concepts in a procedural problem-solving language,” J. Cybern. 4, 105–122 (1974).
[CrossRef]

Kohonen, T.

T. Kohonen, Self-Organization and Associative Memory (Springer-Verlag, New York, (1984), pp. 1–176.

Kumazawa, I.

S. Lin, I. Kumazawa, “Optical fuzzy image processing based-on shadow casting,” Opt. Commun. 94, 397–405 (1992).
[CrossRef]

Lee, S. C.

A. Kandel, S. C. Lee, Fuzzy Switching and Automata: Theory and Applications (Crane, Russak, N.Y., 1979).

Lee, S. H.

G. C. Marsden, B. Olson, S. Esener, S. H. Lee, “Optoelectronic fuzzy logic system,” in Optical Computing, Vol. 6 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), pp. 212–215.

Lee, T. E.

T. E. Lee, “Shape-oriented chromosome classification,” IEEE Trans. Syst. Man Cybern. SMC-5, 629–682 (1975).

Lin, S.

S. Lin, S. Zhang, C. Chen, R. Liu, J. Wu, “Optical multiple-variable fuzzy logic array using shadow casting,” Microwave Opt. Tech. Lett. 6, 106–109 (1993).
[CrossRef]

S. Lin, I. Kumazawa, “Optical fuzzy image processing based-on shadow casting,” Opt. Commun. 94, 397–405 (1992).
[CrossRef]

S. Zhang, S. Lin, C. Cheng, “Optical fuzzy vector-matrix composition operation using shadow-casting,” Opt. Commun. 94, 497–500 (1992).
[CrossRef]

Liu, H. K.

Liu, L.

L. Liu, “Optical implementation of parallel fuzzy logic,” Opt. Commun. 73, 183–187 (1989).
[CrossRef]

Liu, R.

S. Lin, S. Zhang, C. Chen, R. Liu, J. Wu, “Optical multiple-variable fuzzy logic array using shadow casting,” Microwave Opt. Tech. Lett. 6, 106–109 (1993).
[CrossRef]

Mamdani, E. H.

E. H. Mamdani, S. Assilian, “An experiment in linguistic synthesis with a fuzzy logic controller,” Int. J. Man-Mach. Stud. 7, 1–13 (1975).
[CrossRef]

Marsden, G. C.

G. C. Marsden, B. Olson, S. Esener, S. H. Lee, “Optoelectronic fuzzy logic system,” in Optical Computing, Vol. 6 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), pp. 212–215.

Mukai, S.

H. Itoh, M. Watanabe, S. Mukai, H. Yajima, “Optoelectronic fuzzy logic inference system using beam scanning laser diodes,” in Optical Computing, Vol. 7 of 1993 OSA Technical Digtest Series (Optical Society of America, Washington, D.C., 1993), p. 123.

Olson, B.

G. C. Marsden, B. Olson, S. Esener, S. H. Lee, “Optoelectronic fuzzy logic system,” in Optical Computing, Vol. 6 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), pp. 212–215.

Saridis, G. N.

M. M. Gupta, G. N. Saridis, B. R. Gaines, Fuzzy Automata and Decision Processes (North-Holland, Amsterdam, 1977), Chap. 4, pp. 83–97.

Siy, P.

P. Siy, C. S. Chen, “Fuzzy logic for handwritten numerical character recognition,” IEEE Trans. Syst. Man Cybern. SMC-4, 570–575 (1974).

Watanabe, M.

H. Itoh, M. Watanabe, S. Mukai, H. Yajima, “Optoelectronic fuzzy logic inference system using beam scanning laser diodes,” in Optical Computing, Vol. 7 of 1993 OSA Technical Digtest Series (Optical Society of America, Washington, D.C., 1993), p. 123.

Wee, W. G.

W. G. Wee, S. K. Fu, “A formulation of fuzzy automata and its application as a model of learning systems,” IEEE Trans. Syst. Sci. Cybern. SSC-5, 215–223 (1969).
[CrossRef]

Wenstop, F.

F. Wenstop, “Deductive verbal models of organizations,” Int. J. Man-Mach. Stud. 8, 293–311 (1976).
[CrossRef]

Wu, J.

S. Lin, S. Zhang, C. Chen, R. Liu, J. Wu, “Optical multiple-variable fuzzy logic array using shadow casting,” Microwave Opt. Tech. Lett. 6, 106–109 (1993).
[CrossRef]

Yajima, H.

H. Itoh, M. Watanabe, S. Mukai, H. Yajima, “Optoelectronic fuzzy logic inference system using beam scanning laser diodes,” in Optical Computing, Vol. 7 of 1993 OSA Technical Digtest Series (Optical Society of America, Washington, D.C., 1993), p. 123.

Yeh, P.

Zadeh, L. A.

L. A. Zadeh, “Fuzzy sets,” Inf. Control 8, 338–353 (1965).
[CrossRef]

Zhang, S.

S. Lin, S. Zhang, C. Chen, R. Liu, J. Wu, “Optical multiple-variable fuzzy logic array using shadow casting,” Microwave Opt. Tech. Lett. 6, 106–109 (1993).
[CrossRef]

S. Zhang, S. Lin, C. Cheng, “Optical fuzzy vector-matrix composition operation using shadow-casting,” Opt. Commun. 94, 497–500 (1992).
[CrossRef]

Zhou, S.

IEEE Trans. Syst. Man Cybern. (2)

P. Siy, C. S. Chen, “Fuzzy logic for handwritten numerical character recognition,” IEEE Trans. Syst. Man Cybern. SMC-4, 570–575 (1974).

T. E. Lee, “Shape-oriented chromosome classification,” IEEE Trans. Syst. Man Cybern. SMC-5, 629–682 (1975).

IEEE Trans. Syst. Sci. Cybern. (1)

W. G. Wee, S. K. Fu, “A formulation of fuzzy automata and its application as a model of learning systems,” IEEE Trans. Syst. Sci. Cybern. SSC-5, 215–223 (1969).
[CrossRef]

Inf. Control (1)

L. A. Zadeh, “Fuzzy sets,” Inf. Control 8, 338–353 (1965).
[CrossRef]

Int. J. Man-Mach. Stud. (2)

E. H. Mamdani, S. Assilian, “An experiment in linguistic synthesis with a fuzzy logic controller,” Int. J. Man-Mach. Stud. 7, 1–13 (1975).
[CrossRef]

F. Wenstop, “Deductive verbal models of organizations,” Int. J. Man-Mach. Stud. 8, 293–311 (1976).
[CrossRef]

J. Cybern. (1)

R. Kling, “Fuzzy PLANNER: reasoning with inexact concepts in a procedural problem-solving language,” J. Cybern. 4, 105–122 (1974).
[CrossRef]

J. Math. Biol. (1)

J. C. Bezdek, “Numerical taxonomy with fuzzy sets,” J. Math. Biol. 1, 57–71 (1974).
[CrossRef]

Microwave Opt. Tech. Lett. (1)

S. Lin, S. Zhang, C. Chen, R. Liu, J. Wu, “Optical multiple-variable fuzzy logic array using shadow casting,” Microwave Opt. Tech. Lett. 6, 106–109 (1993).
[CrossRef]

Opt. Commun. (3)

L. Liu, “Optical implementation of parallel fuzzy logic,” Opt. Commun. 73, 183–187 (1989).
[CrossRef]

S. Zhang, S. Lin, C. Cheng, “Optical fuzzy vector-matrix composition operation using shadow-casting,” Opt. Commun. 94, 497–500 (1992).
[CrossRef]

S. Lin, I. Kumazawa, “Optical fuzzy image processing based-on shadow casting,” Opt. Commun. 94, 397–405 (1992).
[CrossRef]

Opt. Lett. (1)

Proc. IEEE (1)

S. Gale, “Conjectures on many-valued logic, regions and criteria for conflict resolution,” Proc. IEEE 75, 212–225 (1975).

Other (8)

M. Albn, “Fuzzy sets and their application to medical diagnosis,” Ph.D. dissertation (University of California, Berkeley, Berkeley, Calif., 1975).

J. M. Becker, “A structural design process: philosophy and methodology,” Ph.D. dissertation (University of California, Berkeley, Berkeley, Calif., 1973).

W. L. Fellinger, “Specifications for a fuzzy system modeling language,” Ph.D. dissertation (Oregon State University, Corvallis, Ore., 1974).

G. C. Marsden, B. Olson, S. Esener, S. H. Lee, “Optoelectronic fuzzy logic system,” in Optical Computing, Vol. 6 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), pp. 212–215.

A. Kandel, S. C. Lee, Fuzzy Switching and Automata: Theory and Applications (Crane, Russak, N.Y., 1979).

H. Itoh, M. Watanabe, S. Mukai, H. Yajima, “Optoelectronic fuzzy logic inference system using beam scanning laser diodes,” in Optical Computing, Vol. 7 of 1993 OSA Technical Digtest Series (Optical Society of America, Washington, D.C., 1993), p. 123.

M. M. Gupta, G. N. Saridis, B. R. Gaines, Fuzzy Automata and Decision Processes (North-Holland, Amsterdam, 1977), Chap. 4, pp. 83–97.

T. Kohonen, Self-Organization and Associative Memory (Springer-Verlag, New York, (1984), pp. 1–176.

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Figures (11)

Fig. 1
Fig. 1

Venn diagrams for ordinary-set theory.

Fig. 2
Fig. 2

Profile diagrams for fuzzy-set theory.

Fig. 3
Fig. 3

Density-graphic representations of nine fuzzy-operation results.

Fig. 4
Fig. 4

Weighted bipartite graph to describe the relationship between two fuzzy sets.

Fig. 5
Fig. 5

Optical setup to achieve polarization-space-based parallel fuzzy-logic operaticons: OFE, optical fan-out element; PMOK’s, polarization-multiplexed operation kernels; PBS, polarizing beam splitter. The symbols ● and ↕ represent p- and s-polarization components, respectively.

Fig. 6
Fig. 6

Polarization-space-based data-encoding scheme: (a) AB and (b) A < B. A and B are the two input fuzzy variables with values that vary continuously through the interval [0, 1]. a and 2b are the widths of each encoded pixel in the horizontal and the vertical directions, respectively.

Fig. 7
Fig. 7

Sixteen different fuzzy-logic operations of two fuzzy inputs and the corresponding kernel configurations.

Fig. 8
Fig. 8

Patterns corresponding to the various stages of our optical system for the case of the fuzzy-logic operation max(A, B): (a) the input pattern set, (b) the polarized multiple images before the decoding mask, and (c) the decoded output pattern.

Fig. 9
Fig. 9

Optical setup to implement the max–min composition-based fuzzy controller: OFE1, OFE2, optical fan-out elements; L1–L5, lenses.

Fig. 10
Fig. 10

Experimental results: (a), (b) the polarization-space-encoded patterns of the two input variables A and B as well as their resulting overlapping patterns; (c)–(h) the results for six typical fuzzy-logic operations, min (A, B), max(A, B), Ā, B ¯ , |AB|, and A B ¯ .

Fig. 11
Fig. 11

Experimental results: (a), (b) aperture-encoded patterns corresponding to fuzzy sets A and R, respectively; (c), and (d) experimental results of operations min{μ R (x, y), μ A (x)} and max{min{μ R (x, y), μ A (x)}}.

Equations (52)

Equations on this page are rendered with MathJax. Learn more.

x A x X
A X .
A = x X [ μ ( x ) x ] ,
A B = x X { max [ μ A ( x ) , μ B ( x ) x ] } ,
A B = x X { min [ μ A ( x ) , μ B ( x ) x ] } ,
A ˜ = x X [ 1 - μ A ( x ) x ] ,
A A ˜ X ,
A A ˜ .
CON ( A ) = x X [ μ A 2 ( x ) x ] ,
DIL ( A ) = x X [ μ A 0.5 ( x ) x ] ,
INT ( A ) = { CON ( A ) for all x such that μ A ( x ) < 0.5 DIL ( A ) for all x such that μ A ( x ) 0.5 ,
BLR ( A ) = { DIL ( A ) for all x such that μ A ( x ) < 0.5 CON ( A ) for all x such that μ A ( x ) 0.5 ,
X = [ x 11 x 12 x 13 x 14 x 15 x 21 x 22 x 23 x 24 x 25 x 31 x 32 x 33 x 34 x 35 x 41 x 42 x 43 x 44 x 45 x 51 x 52 x 53 x 54 x 55 ]
A = [ 1.0 x 11 1.0 x 12 1.0 x 13 1.0 x 14 1.0 x 15 1.0 x 21 0.8 x 22 0.6 x 23 0.8 x 24 1.0 x 25 1.0 x 31 1.0 x 32 1.0 x 33 1.0 x 34 1.0 x 35 1.0 x 41 0.6 x 42 0.4 x 43 0.4 x 44 0.4 x 45 1.0 x 51 0.2 x 52 0.2 x 53 0.0 x 54 0.0 x 55 ] ,
B = [ 1.0 x 11 0.2 x 12 0.2 x 13 0.0 x 14 0.0 x 15 1.0 x 21 0.6 x 22 0.4 x 23 0.4 x 24 0.4 x 25 1.0 x 31 1.0 x 32 1.0 x 33 1.0 x 34 1.0 x 35 1.0 x 41 0.8 x 42 0.6 x 43 0.8 x 44 1.0 x 45 1.0 x 51 1.0 x 52 1.0 x 53 1.0 x 54 1.0 x 55 ] ,
μ A B = max ( μ A , μ B ) = [ 1.0 1.0 1.0 1.0 1.0 1.0 0.8 0.6 0.8 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.8 0.6 0.8 1.0 1.0 1.0 1.0 1.0 1.0 ] ,
μ A B = max ( μ A , μ B ) = [ 1.0 0.2 0.2 0.0 0.0 1.0 0.6 0.4 0.4 0.4 1.0 1.0 1.0 1.0 1.0 1.0 0.6 0.4 0.4 0.4 1.0 0.2 0.2 0.0 0.0 ] ,
μ A ˜ = [ 0.0 0.0 0.0 0.0 0.0 0.0 0.2 0.4 0.2 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.4 0.6 0.6 0.6 0.0 0.8 0.8 1.0 1.0 ] ,
μ CON ( A ) = [ 1.0 1.0 1.0 1.0 1.0 1.0 0.64 0.36 0.64 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.36 0.16 0.16 0.16 1.0 0.04 0.04 0.0 0.0 ] ,
μ DIL ( A ) = [ 1.0 1.0 1.0 1.0 1.0 1.0 0.89 0.77 0.89 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.77 0.63 0.63 0.63 1.0 0.45 0.45 0.0 0.0 ] ,
μ INT ( A ) = [ 1.0 1.0 1.0 1.0 1.0 1.0 0.89 0.77 0.89 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.77 0.16 0.14 0.14 1.0 0.04 0.04 0.0 0.0 ] ,
μ BLR ( A ) = [ 1.0 1.0 1.0 1.0 1.0 1.0 0.64 0.36 0.64 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.36 0.63 0.63 0.63 1.0 0.45 0.45 0.0 0.0 ] .
B = A R ,
μ B ( y ) = max x { min { μ R ( x , y ) , μ A ( x ) } } .
U = { 0.5 x 1 , 1.0 x 2 , 1.0 x 3 , 0.5 x 4 } ,
V = { 1.0 y 1 , 0.2 y 2 , 0.4 y 3 , 0.4 y 4 , 0.8 y 5 , 0.6 y 6 , 1.0 y 7 , 0.2 y 8 , 0.8 y 9 } ,
R = [ y 1 y 2 y 3 y 4 y 5 y 6 y 7 y 8 y 9 x 1 0.5 0.1 0.2 0.2 0.4 0.3 0.5 0.1 0.4 x 2 1.0 0.2 0.4 0.4 0.8 0.6 1.0 0.2 0.8 x 3 1.0 0.2 0.4 0.4 0.8 0.6 1.0 0.2 0.8 x 4 0.5 0.1 0.2 0.2 0.4 0.3 0.5 0.1 0.4 ] .
A = { 0.4 x 1 , 0.0 x 2 , 0.2 x 3 , 0.3 x 4 } ,
min { μ R ( x , y ) , μ A ( x ) } = [ 0.4 0.1 0.2 0.2 0.4 0.3 0.4 0.1 0.4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.3 0.1 0.2 0.2 0.3 0.3 0.3 0.1 0.3 ] ,
μ B = max x { min { μ R ( x , y ) , μ A ( x ) } } = [ 0.4 0.2 0.2 0.2 0.4 0.3 0.4 0.2 0.4 ] .
S A = p { rect ( x - a A / 2 a A , y - 0.5 b b ) + rect [ x - a ( 1 + A ) / 2 a ( 1 - A ) , y - 1.5 b b ] } + s { rect ( x - a A / 2 a A , y - 1.5 b b ) + rect [ x - a ( 1 + A ) / 2 a ( 1 - A ) , y - 0.5 b b ] } ,
S B = p { rect [ x - a ( 1 + B ) / 2 a ( 1 - B ) , y - b 2 b ] } + s [ rect ( x - a B / 2 a B , y - b 2 b ) ] ,
S A B = p { rect [ x - a ( A + B ) / 2 a ( A - B ) , y - 0.5 b b ] + rect [ x - a ( 1 + A ) / 2 a ( 1 - A ) , y - 1.5 b b ] } + s [ rect ( x - a B / 2 a B , y - 1.5 b b ) ]
S A B = p { rect [ x - a ( 1 + A ) / 2 a ( 1 - A ) , y - 1.5 b b ] } + s { rect [ x - a ( A + B ) / 2 a ( A - B ) , y - 0.5 b b ] + rect ( x - a B / 2 a B , y - 1.5 b b ) }
S A B * = p { rect [ x - a ( A + B ) / 2 a ( A - B ) , y - 0.5 b b ] + rect [ x - a ( 1 + A ) / 2 a ( 1 - A ) , y - 1.5 b b ] } + s [ rect ( x - a B / 2 a B , y - 0.5 b b ) + rect ( x - a B / 2 a B , y - 1.5 b b ) ]
S A B * = p { rect [ x - a ( 1 + B ) / 2 a ( 1 - B ) , y - 1.5 b b ] } + s { rect ( x - a A / 2 a A , y - 0.5 b b ) + rect ( x - a A / 2 a A , y - 1.5 b b ) + rect [ x - a ( B + A ) / 2 a ( B - A ) , y - 0.5 b b ] }
S DM = rect ( x - 0.5 a a , y - 0.5 b b ) .
S A B = p { rect [ x - a ( A + B ) / 2 a ( A - B ) , y - 0.5 b b ] } + s [ rect ( x - a B / 2 a B , y - 0.5 b b ) ] ,
S A B = s { rect ( x - a A / 2 a A , y - 0.5 b b ) + rect [ x - a ( B + A ) / 2 a ( B - A ) , y - 0.5 b b ] } ,
q = f 1 - b Λ λ ,
Δ f = f 1 λ Λ .
F = , A B = min ( A , B ) , A ¯ B ¯ = min ( A ¯ , B ¯ ) , A ¯ = 1 - A , B ¯ = 1 - B , A ¯ B = { A ¯ + B A ¯ + B 1 1 A ¯ + B > 1 , A B ¯ = { A + B ¯ A + B ¯ 1 1 A + B ¯ > 1 , A B + A ¯ B ¯ = min ( A , B ) + min ( A ¯ , B ¯ ) , A - B = min ( A , B ¯ )             for A B , B - A = min ( B ¯ , A )             for B A , A - B = ( A - B ) + ( B - A ) , A B = max ( A , B ) , A B ¯ = 1 - min ( A , B ) , T = 1.
R = [ μ R ( 11 ) μ R ( 21 ) μ R ( 12 ) μ R ( 22 ) μ R ( 13 ) μ R ( 23 ) μ R ( 31 ) μ R ( 41 ) μ R ( 32 ) μ R ( 42 ) μ R ( 33 ) μ R ( 43 ) μ R ( 14 ) μ R ( 24 ) μ R ( 15 ) μ R ( 25 ) μ R ( 16 ) μ R ( 26 ) μ R ( 34 ) μ R ( 44 ) μ R ( 35 ) μ R ( 45 ) μ R ( 36 ) μ R ( 46 ) μ R ( 17 ) μ R ( 27 ) μ R ( 18 ) μ R ( 28 ) μ R ( 19 ) μ R ( 29 ) μ R ( 37 ) μ R ( 47 ) μ R ( 38 ) μ R ( 48 ) μ R ( 39 ) μ R ( 49 ) ] ,
A = [ μ A ( 1 ) μ A ( 2 ) μ A ( 3 ) μ A ( 4 ) ] ,
Δ i 1 = f 2 ( 1 - q 1 f 1 ) λ Λ 3 × 3
Δ i 2 = f 5 q 2 q 2 + q 3 - f 5 λ Λ 2 × 2
q 2 = f 2 f 4 b Λ f 1 f 3 λ ,
M 5 = f 5 q 2 + q 3 - f 5 .
b min Λ .
Δ i 1 2 f 2 f 1 D in ,
b min λ D OFE ,
IT = ( D in b min ) 2 = [ f 1 D OFE 2 Λ 3 × 3 ( 1 - q 1 f 1 ) ] 2 .

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