Abstract

In the last third of the 20th century, fuzzy logic has risen from a mathematical concept to an applicable approach in soft computing. Today, fuzzy logic is used in control systems for various applications, such as washing machines, train-brake systems, automobile automatic gear, and so forth. The approach of optical implementation of fuzzy inferencing was given by the authors in previous papers, giving an extra emphasis to applications with two dominant inputs. In this paper the authors introduce a real-time optical rule generator for the dual-input fuzzy-inference engine. The paper briefly goes over the dual-input optical implementation of fuzzy-logic inferencing. Then, the concept of constructing a set of rules from given data is discussed. Next, the authors show ways to implement this procedure optically. The discussion is accompanied by an example that illustrates the transformation from raw data into fuzzy set rules.

© 2002 Optical Society of America

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References

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  1. L. A. Zadeh, “Fuzzy sets,” Inf. Control 8, 338–353 (1965).
    [CrossRef]
  2. E. Gur, D. Mendlovic, Z. Zalevsky, “Optical implementation of fuzzy logic controllers. Part I,” Appl. Opt. 37, 6937–6945 (1998).
    [CrossRef]
  3. L. A. Zadeh, “Fuzzy algorithms,” Inf. Control 12, 94–102 (1968).
    [CrossRef]
  4. M. Schneider, A. Kandel, G. Langholz, G. Chew, Fuzzy Expert System Tools (John Wiley and Sons, New York, 1996), Chaps. 9, 10.
  5. W. Pridzik, Fuzzy Control and Fuzzy Systems, 2nd ed. (Research Studies Press, Wiley and Sons, New York, 1993), Chap. 6.
  6. E. Gur, D. Mendlovic, Z. Zalevsky, “Optical implementation of fuzzy logic controllers. Part II,” Appl. Opt. 38, 4354–4358 (1999).
    [CrossRef]
  7. Z. Zalevsky, E. Gur, D. Mendlovic, “Discussion on multi-dimensional fuzzy control,” Appl. Opt. 39, 333–336 (2000).
    [CrossRef]
  8. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

2000

1999

1998

1972

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

1968

L. A. Zadeh, “Fuzzy algorithms,” Inf. Control 12, 94–102 (1968).
[CrossRef]

1965

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

Chew, G.

M. Schneider, A. Kandel, G. Langholz, G. Chew, Fuzzy Expert System Tools (John Wiley and Sons, New York, 1996), Chaps. 9, 10.

Gerchberg, R. W.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Gur, E.

Kandel, A.

M. Schneider, A. Kandel, G. Langholz, G. Chew, Fuzzy Expert System Tools (John Wiley and Sons, New York, 1996), Chaps. 9, 10.

Langholz, G.

M. Schneider, A. Kandel, G. Langholz, G. Chew, Fuzzy Expert System Tools (John Wiley and Sons, New York, 1996), Chaps. 9, 10.

Mendlovic, D.

Pridzik, W.

W. Pridzik, Fuzzy Control and Fuzzy Systems, 2nd ed. (Research Studies Press, Wiley and Sons, New York, 1993), Chap. 6.

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Schneider, M.

M. Schneider, A. Kandel, G. Langholz, G. Chew, Fuzzy Expert System Tools (John Wiley and Sons, New York, 1996), Chaps. 9, 10.

Zadeh, L. A.

L. A. Zadeh, “Fuzzy algorithms,” Inf. Control 12, 94–102 (1968).
[CrossRef]

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

Zalevsky, Z.

Appl. Opt.

Inf. Control

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

L. A. Zadeh, “Fuzzy algorithms,” Inf. Control 12, 94–102 (1968).
[CrossRef]

Optik (Stuttgart)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Other

M. Schneider, A. Kandel, G. Langholz, G. Chew, Fuzzy Expert System Tools (John Wiley and Sons, New York, 1996), Chaps. 9, 10.

W. Pridzik, Fuzzy Control and Fuzzy Systems, 2nd ed. (Research Studies Press, Wiley and Sons, New York, 1993), Chap. 6.

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Figures (8)

Fig. 1
Fig. 1

Dual input (I1, I2) single output (O1) optical fuzzy controller. The input beam representing a membership function is shifted according to the input data before hitting the rule plate. Each section of the rule plate adds a different phase to the beam in accordance with precalculated rules. The result is an intensity distribution on the surface of the PSD and the center of gravity (COG) of this distribution is the output.

Fig. 2
Fig. 2

Distribution of final grades in an introductory course with respect to the individual grades of two questions in the test. The location of each data point indicated the grade of the first question (y axis) and second question (x axis), while the intensity indicates the final grade.

Fig. 3
Fig. 3

Original data in a 5 × 5-sectioned view. Each section represents a different rule to be determined from the data in that section.

Fig. 4
Fig. 4

(a) Rule table as obtained for the data shown before, by computer calculations finding the mean value of all data points in each subsection. (b) A rule table as obtained by a low-resolution CCD by taking the average of all intensities in each subsection.

Fig. 5
Fig. 5

Two sets of prisms containing N prisms each. (a) has many variations in phase while (b) shows a unified phase front.

Fig. 6
Fig. 6

Simple test case showing two empty regions, and two regions filled halfway with data. (a) The raw data shown in the top left-hand side for an equal number of data point with grades 50 and 100, and in the bottom-right-hand side for an equal number of data points with grades 20 and 60. (b) The shape of the prisms representing the top-left data. (c) the shape of the light beams hitting the PSD where the bright lines indicate the center of the region, left of which we place the background and block it. (d) the resulting center of gravity, results where multiplied by constant to obtain expected value, this constant may be used for the more complex data yet to come.

Fig. 7
Fig. 7

Obtaining rule conclusions for a 2 × 2 section of the test case. (a) The raw data of four cells, containing grades of students, (b) the resulting intensity distribution in the PSD plane, (c) the calculated COG values, (d) the obtained COG values on a gray-scale sketch.

Fig. 8
Fig. 8

Final results: (a) the expected rule table and three variations of the obtained result due to noise reduction (b) for a threshold of 25%, (c) for a threshold of 12.5%, (d) for a threshold of 50% below the maximum signal level.

Tables (1)

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Table 1 Segment of the Raw Dataa

Equations (15)

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yx=n=0M-1expjΦnxδ x-xn.
Yυ= -n=0M-1expjΦnxδ x-xnexp-j2πxυdx=n=0M-1expjΦnxn-2πυxn,
yu, υ ynewMu-1+1, Nυ-1+1ynewMu-1+1, NυynewMu, Nυ-1+1ynewMu, Nυ.
yx= n=0M-1expjΦnxux-xn+1-ux-xn,
Yυ=-n=0M-1expjΦnxux-xn+1-ux-xnexp-j2πxυdx=n=0M-1xnxn+1expjΦnxexp-j2πxυdx,
Φnx=φnx-nΔxif n Δx<x<n+1Δx0outside region,
Yυ=Δx n=0M-1expj2π φn2π-υ · xn+Δx2×sincφn2π-υΔx2exp-jφnnΔx,
sincα=sinπαπα.
Yυ|φn=0=2Δx exp-j2πυ xn+Δx2sincυΔx,
Yυ24 Δx2n=0M-1sinc2φn2π-υΔx,
φ0= 1Nn=1N φn= φ¯n,
Ex=E0xexp2πiφ0x+E0x-Δxexp2πiφ0x++E0x- M-1Δxexp2πiφ0x =E0x+E0x-Δx++E0x- M-1Δxexp2πiφ0x,
Eˆυ=Eˆ0υ-φ0+E0υ-φ0exp2πiΔxυ-φ0++E0υ-φ0exp2πiM-1×Δx υ-φ0 =Eˆ0υ-φ01+exp2πiΔx υ-φ0++exp2πi M-1Δx υ-φ0=Eˆ0υ-φ01-exp2πiMΔxυ-φ01-exp2πiΔxυ-φ0,
Eˆυ2= Eˆ0υ-φ021-cos2πMΔxυ-φ01-cos2πΔxυ-φ0.
EˆMυ2EˆNυ2= 1-cos2πMΔxυ-φ01-cos2πNΔxυ-φ0 υφ0M2N2,

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