Abstract

A method for high-accuracy analog optical computing based on interval arithmetic and the fixed-point theorem is considered. Two-variable simultaneous equations are studied to investigate the proposed method. An optical implementation is considered by the use of spatial coding of intervals, affine transformation, and image magnification. Computational simulation verifies the principle of the method.

© 1996 Optical Society of America

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References

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  1. D. G. Feitelson, Optical Computing (MIT, Cambridge, Mass., 1988), Chaps. 3 and 4, pp. 52–137.
  2. Ref. 1, Chap. 7, pp. 164–225.
  3. A. D. McAulay, Optical Computer Architectures (Wiley, New York, 1991), Chaps. 8-10, pp. 193–288.
  4. R. E. Moore, Interval Analysis (Prentice-Hall, Englewood Cliffs, N.J., 1966), Chap. 2, pp. 9–17.
  5. L. E. J. Brouwer, “U¨ber Abbildungen von Mannigfaltigkeiten,” Math. Ann. 71, 97–115 (1912).
  6. E. Kaucher, S. M. Rump, “E-Methods for fixed point equations f (x) = x,” Computing 28, 31–42 (1982).
  7. J. Tanida, A. Uemoto, Y. Ichioka, “Optical fractal synthesizer: concept and experimental verification,” Appl. Opt. 32, 653–658 (1993).

1993 (1)

1982 (1)

E. Kaucher, S. M. Rump, “E-Methods for fixed point equations f (x) = x,” Computing 28, 31–42 (1982).

1912 (1)

L. E. J. Brouwer, “U¨ber Abbildungen von Mannigfaltigkeiten,” Math. Ann. 71, 97–115 (1912).

Brouwer, L. E. J.

L. E. J. Brouwer, “U¨ber Abbildungen von Mannigfaltigkeiten,” Math. Ann. 71, 97–115 (1912).

Feitelson, D. G.

D. G. Feitelson, Optical Computing (MIT, Cambridge, Mass., 1988), Chaps. 3 and 4, pp. 52–137.

Ichioka, Y.

Kaucher, E.

E. Kaucher, S. M. Rump, “E-Methods for fixed point equations f (x) = x,” Computing 28, 31–42 (1982).

McAulay, A. D.

A. D. McAulay, Optical Computer Architectures (Wiley, New York, 1991), Chaps. 8-10, pp. 193–288.

Moore, R. E.

R. E. Moore, Interval Analysis (Prentice-Hall, Englewood Cliffs, N.J., 1966), Chap. 2, pp. 9–17.

Rump, S. M.

E. Kaucher, S. M. Rump, “E-Methods for fixed point equations f (x) = x,” Computing 28, 31–42 (1982).

Tanida, J.

Uemoto, A.

Appl. Opt. (1)

Computing (1)

E. Kaucher, S. M. Rump, “E-Methods for fixed point equations f (x) = x,” Computing 28, 31–42 (1982).

Math. Ann. (1)

L. E. J. Brouwer, “U¨ber Abbildungen von Mannigfaltigkeiten,” Math. Ann. 71, 97–115 (1912).

Other (4)

D. G. Feitelson, Optical Computing (MIT, Cambridge, Mass., 1988), Chaps. 3 and 4, pp. 52–137.

Ref. 1, Chap. 7, pp. 164–225.

A. D. McAulay, Optical Computer Architectures (Wiley, New York, 1991), Chaps. 8-10, pp. 193–288.

R. E. Moore, Interval Analysis (Prentice-Hall, Englewood Cliffs, N.J., 1966), Chap. 2, pp. 9–17.

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Figures (6)

Fig. 1
Fig. 1

Conceptual diagram of interval shrinking with reduction mapping around the fixed point. The filled circles and the lines indicate the fixed point and the intervals, respectively.

Fig. 2
Fig. 2

Spatial coding for two-dimensional data.

Fig. 3
Fig. 3

Schematic diagram of the optical fractal synthesizer. DP's, dove prisms for rotation; M's, mirrors for translation; BS's, beam splitters.

Fig. 4
Fig. 4

Relationship between global and local coordinate systems. Block (italic) letters indicate the number in a global (local) coordinate system.

Fig. 5
Fig. 5

Processing procedure of the optical implementation of high-accuracy computing algorithm. Block (italic) letters indicate boundaries of the POA in a global (local) coordinate system.

Fig. 6
Fig. 6

Simulation result of computing of Eq. (5). POA's before and after transformation at first–sixth iterations are shown.

Tables (2)

Tables Icon

Table 1 Time Development of Transformed Intervals

Tables Icon

Table 2 Time Development of Parameters of Affine Transformation and Transformed Intervals

Equations (16)

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

[ a , b ] + [ c , d ] = [ a + c , b + d ] ,
[ a , b ] [ c , d ] = [ a d , b d ] ,
[ a , b ] [ c , d ] = [ min ( a c , a d , b c , b d ) ,  max ( a c , a d , b c , b d ) ] ,
[ a , b ] / [ c , d ] = [ a , b ] [ 1 / d , 1 / c ] .
𝒜 x = b ,
( b 𝒜 x ˜ ) + ( 𝒜 ) x * = x *,
g ( x * ) = ( b 𝒜 x ˜ ) + ( 𝒜 ) x * = x *
g ( X ) = ( b 𝒜 x ˜ ) + ( 𝒜 ) X ,
X ( k ) = g [ X ( k 1 ) ] X ( k 1 ) , X ( 0 ) = X ,
[ 2 4 2 1 ] ( x 0 x 1 ) = ( 3 2 ) .
= [ 0 . 2 0 . 6 0 . 3 0 . 3 ] , x ˜ = ( 0 . 8 0 . 3 ) .
g ( x 0 x 1 ) = [ 0 . 2 0 . 2 0 0 . 1 ] ( x 0 x 1 ) + ( 0 . 02 0 . 03 ) .
g ( x y ) = [ a b c d ] ( x y ) + ( e f ) .
g ( i j ) = [ a α β b β α c d ] ( i j ) + ( α e β f ) + [ α ( a 1 ) α b β c β ( d 1 ) ] ( x bottom y bottom ) ,
α = s x top x bottom ,
β = t y top y bottom .

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