Abstract

A generic language for optical parallel processing image logic algebra (ILA), is proposed. In ILA a neighborhood configuration pattern (NCP) is introduced, and image transformations are defined by the use of NCP operations. The comprehensive relationship of ILA to symbolic substitution, optical array logic, mathematical morphology, and binary image algebra are clarified. Furthermore, an architecture that is suited for ILA and its optical implementations is proposed.

© 1992 Optical Society of America

Full Article  |  PDF Article

Errata

Masaki Fukui and Ken-ichi Kitayama, "Image logic algebra and its optical implementations: errata," Appl. Opt. 31, 4630-4630 (1992)
https://www.osapublishing.org/ao/abstract.cfm?uri=ao-31-23-4630

References

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  1. A. Huang, “Parallel algorithms for optical digital computers,” in Technical Digest of the IEEE Tenth International Optical Computing Conference (Institute of Electrical and Electronics Engineers, New York, 1983), pp. 13–17.
  2. K-H. Brenner, A. Huang, N. Streibl, “Digital optical computing with symbolic substitution,” Appl. Opt. 25, 3054–3060 (1986).
    [CrossRef] [PubMed]
  3. K-H. Brenner, “New implementation of symbolic substitution logic,” Appl. Opt. 25, 3061–3064 (1986).
    [CrossRef] [PubMed]
  4. J. Tanida, Y. Ichioka, “Optical-logic-array processor using shadowgrams. III. Parallel neighborhood operations and an architecture of an optical digital-computing system.” J. Opt. Soc. Am. A 2, 1245–1253 (1985).
    [CrossRef]
  5. J. Tanida, Y. Ichioka, “OPALS: optical parallel array logic system,” Appl. Opt. 25, 1565–1570 (1986).
    [CrossRef] [PubMed]
  6. K-S. Huang, B. K. Jenkins, A. A. Sawchuk, “Image algebra representation of parallel optical binary arithmetic,” Appl. Opt. 28, 1263–1278 (1989).
    [CrossRef] [PubMed]
  7. T. J. Drabik, S. H. Lee, “Shift-connected SIMD array architectures for digital optical computing systems, with algorithms for numerical transforms and partial differential equations,” Appl. Opt. 25, 4053–4064 (1986).
    [CrossRef] [PubMed]
  8. G. Eichmann, J. Zhu, Y. Li, “Optical parallel image skeletonization using content-addressable memory-based symbolic substitution,” Appl. Opt. 27, 2905–2911 (1988).
    [CrossRef] [PubMed]
  9. S. D. Goodman, W. T. Rhodes, “Symbolic substitution applications to image processing,” Appl. Opt. 27, 1708–1714 (1988).
    [CrossRef] [PubMed]
  10. J. Tanida, Y. Ichioka, “Programming of optical array logic. 1: Image data processing,” Appl. Opt. 27, 2926–2930 (1988).
    [CrossRef] [PubMed]
  11. J. Tanida, M. Fukui, Y. Ichioka, “Programming of optical array logic. 2: Numerical data processing based on pattern logic,” Appl. Opt. 27, 2931–2939 (1988).
    [CrossRef] [PubMed]
  12. J. Tanida, J. Nakagawa, E. Yagyu, M. Fukui, Y. Ichioka, “Experimental verification of parallel processing on a hybrid optical parallel array logic system,” Appl. Opt. 29, 2510–2521 (1990).
    [CrossRef] [PubMed]
  13. M. Fukui, J. Tanida, Y. Ichioka, “Flexible-structured computation based on optical array logic,” Appl. Opt. 29, 1604–1609 (1990).
    [CrossRef] [PubMed]
  14. R. M. Haralick, S. R. Sternberg, X. Zhuang, “Image analysis using mathematical morphology,” IEEE Tran. Pattern Anal. Machine Intell. PAMI-9, 532–550 (1987).
    [CrossRef]
  15. P. Maragos, “Tutorial on advances in morphological image processing and analysis,” Opt. Eng. 26, 623–632 (1987).
    [CrossRef]
  16. P. Maragos, R. W. Schafer, “Morphological systems for multidimensional signal processing,” Proc. IEEE 78, 690–710 (1990).
    [CrossRef]
  17. D. A. B. Miller, D. S. Chemla, T. C. Damen, T. H. Wood, C. A. Burrus, A. C. Gossard, W. Wiegmann, “The quantum well self-electrooptic effect device: optoelectronic bistability and oscillation, and self-linearized modulation,” IEEE J. Quantum Electron. QE-21, 1462–1476 (1985).
    [CrossRef]
  18. J. N. Mait, K-H. Brenner, “Optical symbolic substitution: system design using phase-only holograms,” Appl. Opt. 27, 1692–1700 (1988).
    [CrossRef] [PubMed]
  19. B. D. Clymer, J. W. Goodman, “Optical clock distribution to silicon chips,” Opt. Eng. 25, 1103–1108 (1986).
    [CrossRef]

1990

1989

1988

1987

R. M. Haralick, S. R. Sternberg, X. Zhuang, “Image analysis using mathematical morphology,” IEEE Tran. Pattern Anal. Machine Intell. PAMI-9, 532–550 (1987).
[CrossRef]

P. Maragos, “Tutorial on advances in morphological image processing and analysis,” Opt. Eng. 26, 623–632 (1987).
[CrossRef]

1986

1985

J. Tanida, Y. Ichioka, “Optical-logic-array processor using shadowgrams. III. Parallel neighborhood operations and an architecture of an optical digital-computing system.” J. Opt. Soc. Am. A 2, 1245–1253 (1985).
[CrossRef]

D. A. B. Miller, D. S. Chemla, T. C. Damen, T. H. Wood, C. A. Burrus, A. C. Gossard, W. Wiegmann, “The quantum well self-electrooptic effect device: optoelectronic bistability and oscillation, and self-linearized modulation,” IEEE J. Quantum Electron. QE-21, 1462–1476 (1985).
[CrossRef]

Brenner, K-H.

Burrus, C. A.

D. A. B. Miller, D. S. Chemla, T. C. Damen, T. H. Wood, C. A. Burrus, A. C. Gossard, W. Wiegmann, “The quantum well self-electrooptic effect device: optoelectronic bistability and oscillation, and self-linearized modulation,” IEEE J. Quantum Electron. QE-21, 1462–1476 (1985).
[CrossRef]

Chemla, D. S.

D. A. B. Miller, D. S. Chemla, T. C. Damen, T. H. Wood, C. A. Burrus, A. C. Gossard, W. Wiegmann, “The quantum well self-electrooptic effect device: optoelectronic bistability and oscillation, and self-linearized modulation,” IEEE J. Quantum Electron. QE-21, 1462–1476 (1985).
[CrossRef]

Clymer, B. D.

B. D. Clymer, J. W. Goodman, “Optical clock distribution to silicon chips,” Opt. Eng. 25, 1103–1108 (1986).
[CrossRef]

Damen, T. C.

D. A. B. Miller, D. S. Chemla, T. C. Damen, T. H. Wood, C. A. Burrus, A. C. Gossard, W. Wiegmann, “The quantum well self-electrooptic effect device: optoelectronic bistability and oscillation, and self-linearized modulation,” IEEE J. Quantum Electron. QE-21, 1462–1476 (1985).
[CrossRef]

Drabik, T. J.

Eichmann, G.

Fukui, M.

Goodman, J. W.

B. D. Clymer, J. W. Goodman, “Optical clock distribution to silicon chips,” Opt. Eng. 25, 1103–1108 (1986).
[CrossRef]

Goodman, S. D.

Gossard, A. C.

D. A. B. Miller, D. S. Chemla, T. C. Damen, T. H. Wood, C. A. Burrus, A. C. Gossard, W. Wiegmann, “The quantum well self-electrooptic effect device: optoelectronic bistability and oscillation, and self-linearized modulation,” IEEE J. Quantum Electron. QE-21, 1462–1476 (1985).
[CrossRef]

Haralick, R. M.

R. M. Haralick, S. R. Sternberg, X. Zhuang, “Image analysis using mathematical morphology,” IEEE Tran. Pattern Anal. Machine Intell. PAMI-9, 532–550 (1987).
[CrossRef]

Huang, A.

K-H. Brenner, A. Huang, N. Streibl, “Digital optical computing with symbolic substitution,” Appl. Opt. 25, 3054–3060 (1986).
[CrossRef] [PubMed]

A. Huang, “Parallel algorithms for optical digital computers,” in Technical Digest of the IEEE Tenth International Optical Computing Conference (Institute of Electrical and Electronics Engineers, New York, 1983), pp. 13–17.

Huang, K-S.

Ichioka, Y.

Jenkins, B. K.

Lee, S. H.

Li, Y.

Mait, J. N.

Maragos, P.

P. Maragos, R. W. Schafer, “Morphological systems for multidimensional signal processing,” Proc. IEEE 78, 690–710 (1990).
[CrossRef]

P. Maragos, “Tutorial on advances in morphological image processing and analysis,” Opt. Eng. 26, 623–632 (1987).
[CrossRef]

Miller, D. A. B.

D. A. B. Miller, D. S. Chemla, T. C. Damen, T. H. Wood, C. A. Burrus, A. C. Gossard, W. Wiegmann, “The quantum well self-electrooptic effect device: optoelectronic bistability and oscillation, and self-linearized modulation,” IEEE J. Quantum Electron. QE-21, 1462–1476 (1985).
[CrossRef]

Nakagawa, J.

Rhodes, W. T.

Sawchuk, A. A.

Schafer, R. W.

P. Maragos, R. W. Schafer, “Morphological systems for multidimensional signal processing,” Proc. IEEE 78, 690–710 (1990).
[CrossRef]

Sternberg, S. R.

R. M. Haralick, S. R. Sternberg, X. Zhuang, “Image analysis using mathematical morphology,” IEEE Tran. Pattern Anal. Machine Intell. PAMI-9, 532–550 (1987).
[CrossRef]

Streibl, N.

Tanida, J.

Wiegmann, W.

D. A. B. Miller, D. S. Chemla, T. C. Damen, T. H. Wood, C. A. Burrus, A. C. Gossard, W. Wiegmann, “The quantum well self-electrooptic effect device: optoelectronic bistability and oscillation, and self-linearized modulation,” IEEE J. Quantum Electron. QE-21, 1462–1476 (1985).
[CrossRef]

Wood, T. H.

D. A. B. Miller, D. S. Chemla, T. C. Damen, T. H. Wood, C. A. Burrus, A. C. Gossard, W. Wiegmann, “The quantum well self-electrooptic effect device: optoelectronic bistability and oscillation, and self-linearized modulation,” IEEE J. Quantum Electron. QE-21, 1462–1476 (1985).
[CrossRef]

Yagyu, E.

Zhu, J.

Zhuang, X.

R. M. Haralick, S. R. Sternberg, X. Zhuang, “Image analysis using mathematical morphology,” IEEE Tran. Pattern Anal. Machine Intell. PAMI-9, 532–550 (1987).
[CrossRef]

Appl. Opt.

J. Tanida, Y. Ichioka, “OPALS: optical parallel array logic system,” Appl. Opt. 25, 1565–1570 (1986).
[CrossRef] [PubMed]

K-S. Huang, B. K. Jenkins, A. A. Sawchuk, “Image algebra representation of parallel optical binary arithmetic,” Appl. Opt. 28, 1263–1278 (1989).
[CrossRef] [PubMed]

T. J. Drabik, S. H. Lee, “Shift-connected SIMD array architectures for digital optical computing systems, with algorithms for numerical transforms and partial differential equations,” Appl. Opt. 25, 4053–4064 (1986).
[CrossRef] [PubMed]

G. Eichmann, J. Zhu, Y. Li, “Optical parallel image skeletonization using content-addressable memory-based symbolic substitution,” Appl. Opt. 27, 2905–2911 (1988).
[CrossRef] [PubMed]

S. D. Goodman, W. T. Rhodes, “Symbolic substitution applications to image processing,” Appl. Opt. 27, 1708–1714 (1988).
[CrossRef] [PubMed]

J. Tanida, Y. Ichioka, “Programming of optical array logic. 1: Image data processing,” Appl. Opt. 27, 2926–2930 (1988).
[CrossRef] [PubMed]

J. Tanida, M. Fukui, Y. Ichioka, “Programming of optical array logic. 2: Numerical data processing based on pattern logic,” Appl. Opt. 27, 2931–2939 (1988).
[CrossRef] [PubMed]

J. Tanida, J. Nakagawa, E. Yagyu, M. Fukui, Y. Ichioka, “Experimental verification of parallel processing on a hybrid optical parallel array logic system,” Appl. Opt. 29, 2510–2521 (1990).
[CrossRef] [PubMed]

M. Fukui, J. Tanida, Y. Ichioka, “Flexible-structured computation based on optical array logic,” Appl. Opt. 29, 1604–1609 (1990).
[CrossRef] [PubMed]

K-H. Brenner, A. Huang, N. Streibl, “Digital optical computing with symbolic substitution,” Appl. Opt. 25, 3054–3060 (1986).
[CrossRef] [PubMed]

K-H. Brenner, “New implementation of symbolic substitution logic,” Appl. Opt. 25, 3061–3064 (1986).
[CrossRef] [PubMed]

J. N. Mait, K-H. Brenner, “Optical symbolic substitution: system design using phase-only holograms,” Appl. Opt. 27, 1692–1700 (1988).
[CrossRef] [PubMed]

IEEE J. Quantum Electron.

D. A. B. Miller, D. S. Chemla, T. C. Damen, T. H. Wood, C. A. Burrus, A. C. Gossard, W. Wiegmann, “The quantum well self-electrooptic effect device: optoelectronic bistability and oscillation, and self-linearized modulation,” IEEE J. Quantum Electron. QE-21, 1462–1476 (1985).
[CrossRef]

IEEE Tran. Pattern Anal. Machine Intell.

R. M. Haralick, S. R. Sternberg, X. Zhuang, “Image analysis using mathematical morphology,” IEEE Tran. Pattern Anal. Machine Intell. PAMI-9, 532–550 (1987).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Eng.

B. D. Clymer, J. W. Goodman, “Optical clock distribution to silicon chips,” Opt. Eng. 25, 1103–1108 (1986).
[CrossRef]

P. Maragos, “Tutorial on advances in morphological image processing and analysis,” Opt. Eng. 26, 623–632 (1987).
[CrossRef]

Proc. IEEE

P. Maragos, R. W. Schafer, “Morphological systems for multidimensional signal processing,” Proc. IEEE 78, 690–710 (1990).
[CrossRef]

Other

A. Huang, “Parallel algorithms for optical digital computers,” in Technical Digest of the IEEE Tenth International Optical Computing Conference (Institute of Electrical and Electronics Engineers, New York, 1983), pp. 13–17.

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Figures (16)

Fig. 1
Fig. 1

Example of binary images.

Fig. 2
Fig. 2

Examples of image casting transformation.

Fig. 3
Fig. 3

Example of multiple imaging transformation.

Fig. 4
Fig. 4

Example of extended erosion. B[12;12] is the result of the extended erosion of image A[12;12].

Fig. 5
Fig. 5

Example of dilation. B[12;12] is the result of the dilation of image A[12;12].

Fig. 6
Fig. 6

Four SS rules expressing exclusive or operation.

Fig. 7
Fig. 7

System architecture for ILA.

Fig. 8
Fig. 8

Schematic overview of the optical implementation and data flow of an ILA. Meshed line, dashed line, and solid line represent data flow of a two-dimensional image, a control signal flow from system controller, and a signal flow to system controller, respectively.

Fig. 9
Fig. 9

System architecture with an optical image bus.

Fig. 10
Fig. 10

Schematic of the OICS.

Fig. 11
Fig. 11

(a) Schematic diagram of an optical implementation of image casting with M ≤ R, N ≤ S. (b) Example of image casting from A[3;3] to B[6;6].

Fig. 12
Fig. 12

(a) Schematic diagram of an optical implementation of image casting with M ≥ R, N ≥ S. (b) Example of image casting from A[6;6]to B[3;3].

Fig. 13
Fig. 13

(a) Schematic diagram of an optical implementation of multiple imaging. (b) Example of multiple imaging from A[2;2] to B[6;6].

Fig. 14
Fig. 14

Schematic diagram of an optical implementation of TEST.

Fig. 15
Fig. 15

Configuration of optically addressable memory.

Fig. 16
Fig. 16

Hybrid-type optically addressable memory.

Tables (7)

Tables Icon

Table I Operations and Transformations of ILA

Tables Icon

Table II Definition of r

Tables Icon

Table III Definition of the Disjunction of r1 and r2

Tables Icon

Table IV Definition of the Conjunction of r1 and r2

Tables Icon

Table V Definition of the Negation of r

Tables Icon

Table VI Relation between the Operation Kernel of OAL and NCP

Tables Icon

Table VII Symbols of Operation Kernels of OALa

Equations (77)

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

A [ 3 ; 3 ] = | 1 0 0 0 1 1 1 0 0 | , B [ 2 ; 3 ] = | 1 0 0 0 1 1 | .
C [ M ; N ] i j = A [ M ; N ] i j · B [ M ; N ] i j ( 1 i M , 1 j N ) .
C [ M ; N ] i j = A [ M ; N ] i j + B [ M ; N ] i j ( 1 i M , 1 j N ) .
C [ M ; N ] i j = A [ M ; N ] i j ¯ ( 1 i M , 1 j N ) .
B [ M ; N ] i j = A [ M ; N ] i + k , j + l .
B [ R ; S ] m n = { A [ M ; N ] i j m = k i , n = l j . 0 otherwise
B [ R ; S ] i j = A [ M ; N ] k i , l j .
A [ 6 ; 6 ] = IC ( A [ 3 ; 3 ] ) , B [ 2 ; 2 ] = IC ( B [ 4 ; 4 ] ) .
B [ R ; S ] m i , n j = A [ M ; N ] i j , m = 1 , 2 , k , n = 1 , 2 , 1 .
A [ 8 ; 8 ] = MI ( | 0 1 0 0 0 1 0 0 0 1 1 1 0 0 0 0 | ) .
B [ 1 ; 1 ] 1 , 1 = i = 1 M j = 1 N A [ M ; N ] i j ¯ .
B [ M ; N ] i j = x ( 0 ) A [ M ; N ] i + k , j + l + x ( 1 ) A [ M , N ] i + k , j + l ¯ .
B [ M ; N ] = A [ M ; N ] r ( k , l ) .
A [ M ; N ] [ r ( k , l ) s ( p , q ) ] A [ M ; N ] r ( k , l ) + A [ M ; N ] s ( p , q ) ,
A [ M ; N ] [ r ( k , l ) s ( p , q ) ] [ A [ M ; N ] r ( k , l ) ] · [ A [ M ; N ] s ( p , q ) ] ,
A [ M ; N ] r ( k , l ) A [ M ; N ] r ( k , l ) ¯ .
r ( k , l ) s ( p , q ) = r + s ( k , l ) ,
r ( k , l ) s ( p , q ) = r · s ( k , l ) ,
r ( k , l ) ¯ = r ¯ ( k , l ) .
B [ M ; N ] i j = m k , l ( x k l m ( 0 ) A [ M ; N ] i + k , j + l + x k l m ( 1 ) A [ M ; N ] i + k , j + l ¯ ) ,
B [ M ; N ] = m k , l ( A [ M ; N ] r k l m ( k , l ) ) .
B [ M ; N ] = A [ M ; N ] [ m k , l r k l m ( k , l ) ] .
S = m k , l r k l m ( k , l ) .
B [ M ; N ] = A [ M ; N ] S .
P = k , l r k l ( k , l ) .
P = r m 1 , n + 1 r m , n + 1 r m + 1 , n + 1 r m 1 , n r m , n ¯ r m + 1 , n r m 1 , n 1 r m , n 1 r m + 1 , n 1 ,
AND of NCP ' s * 1 1 0 ¯ * 1 * 1 0 ¯ 1 * 1 = * 1 1 1 0 ¯ 1 * 1 1 ,
OR of NCP ' s * 1 1 0 ¯ * 1 * 0 1 0 ¯ * 1 = 1 0 ¯ * 1 ,
NOT of NCP ' s * 1 1 0 ¯ * 1 ¯ = 0 * ¯ 0 * ¯ 1 * ¯ 0 .
A [ M ; N ] ( R S ) = ( A [ M ; N ] R ) · ( A [ M ; N ] S ) ,
A [ M ; N ] ( R S ) = ( A [ M ; N ] R ) + ( A [ M ; N ] S ) ,
A [ M ; N ] R ¯ = A [ M ; N ] R ¯ .
B [ 12 ; 12 ] = A [ 12 ; 12 ] ( 1 0 ¯ * 1 1 0 ¯ 1 1 1 * 0 ¯ ) .
B [ M ; N ] = A [ M ; N ] [ k , l r k l ( k , l ) ] ,
B [ M ; N ] = A [ M ; N ] [ k , l r k l ¯ ( k , l ) ] ¯ .
A [ M ; N ] [ k , l r k l ( k , l ) ] A [ M ; N ] [ k , l r k l ¯ ( k , l ) ] ¯ .
B [ M ; N ] = A [ M ; N ] Q ,
Q = k , l r k l ( k , l ) .
B [ 12 ; 12 ] = A [ 12 ; 12 ] 1 1 1 * ¯ 1 1 .
k = 1 N ( A [ M ; N ] P r k ) P s k ,
P r k = m , n r r k m n ( m , n ) , P s k = m , n r s k m n ( m , n ) ,
( A [ M ; N ] 0 1 0 ¯ 1 ) 0 1 ¯ + ( A [ M ; N ] 1 0 1 ¯ 0 ) 0 1 ¯ + ( A [ M ; N ] 0 1 1 ¯ 0 ) 1 0 ¯ + ( A [ M ; N ] 1 0 0 ¯ 1 ) 1 0 ¯ .
[ A [ M ; N ] ( 0 1 0 ¯ 1 1 0 1 ¯ 0 ) ] 0 1 ¯ + [ A [ M ; N ] + ( 0 1 1 ¯ 0 1 0 0 ¯ 1 ) ] 1 0 ¯ .
k = 1 N = 1 { ( ( A [ M ; N ] · B [ M ; N ] ) P A B k ) · ( ( A [ M ; N ] ¯ · B [ M ; N ] ) P A ¯ B k ) ¯ × { ( ( A [ M ; N ] · B [ M ; N ] ¯ ) P A B ¯ k ) · ( ( A [ M ; N ] ¯ · B [ M ; N ] ¯ ) P A ¯ B ¯ k ) } ¯
( ( A [ M ; N ] ¯ · B [ M ; N ] ) [ 0 ] ) · ( ( A [ M ; N ] · B [ M ; N ] ¯ ) [ 0 ] ) ¯ .
P = m , n r m n ( m , n ) ,
A [ M ; N ] ( | ( i , j ) | B [ M ; N ] 1 ( i . j ) ) .
{ 0 [ M ; N ] i j = 0 1 i M , 1 j N 1 [ M ; N ] i j = 1 1 i M , 1 j N .
0 a . 0 [ M ; N ] = 1 [ M ; N ] ¯ 0 b . 1 [ M ; N ] = 0 [ M ; N ] ¯
1 a . A [ M ; N ] + 0 [ M ; N ] = A [ M ; N ] 1 b . A [ M ; N ] · 1 [ M ; N ] = A [ M ; N ] 2 a . 1 [ M ; N ] + A [ M ; N ] = 1 [ M ; N ] 2 b . 0 [ M ; N ] · A [ M ; N ] = 0 [ M ; N ]
3 a . A [ M ; N ] + A [ M ; N ] = A [ M ; N ] 3 b . A [ M ; N ] · A [ M ; N ] = A [ M ; N ]
4 a . A [ M ; N ] ¯ ¯ = A [ M ; N ]
5 a . A [ M ; N ] + A [ M ; N ] ¯ = 1 [ M ; N ] 5 b . A [ M ; N ] · A [ M ; N ] ¯ = 0 [ M ; N ]
6 a . A [ M ; N ] + B [ M ; N ] = B [ M ; N ] + A [ M ; N ] 6 b . A [ M ; N ] · B [ M ; N ] = B [ M ; N ] · A [ M ; N ]
7 a . A [ M ; N ] + A [ M ; N ] · B [ M ; N ] = A [ M ; N ] 7 b . A [ M ; N ] · ( A [ M ; N ] + B [ M ; N ] ) = A [ M ; N ] 8 a . ( A [ M ; N ] + B [ M ; N ] ¯ ) · B [ M ; N ] = A [ M ; N ] · B [ M ; N ] 8 b . A [ M ; N ] · B [ M ; N ] ¯ + B [ M ; N ] = A [ M ; N ] + B [ M ; N ]
9 a . A [ M ; N ] + B [ M ; N ] + C [ M ; N ] = ( A [ M ; N ] + B [ M ; N ] ) + C [ M ; N ] = A [ M ; N ] + ( B [ M ; N ] + C [ M ; N ] ) 9 b . A [ M ; N ] · B [ M ; N ] · C [ M ; N ] = ( A [ M ; N ] · B [ M ; N ] ) · C [ M ; N ] = A [ M ; N ] · ( B [ M ; N ] · C [ M ; N ] )
10 a . A [ M ; N ] · B [ M ; N ] + A [ M ; N ] · C [ M ; N ] = A [ M ; N ] · ( B [ M ; N ] + C [ M ; N ] ) 10 b . ( A [ M ; N ] + B [ M ; N ] ) · ( A [ M ; N ] + C [ M ; N ] ) = A [ M ; N ] + B [ M ; N ] · C [ M ; N ]
11 a . A [ M ; N ] + B [ M ; N ] ¯ = A [ M ; N ] ¯ · B [ M ; N ] ¯ 11 b . A [ M ; N ] · B [ M ; N ] ¯ = A [ M ; N ] ¯ + B [ M ; N ] ¯
1 a . r + ϕ = r 1 b . r · * = r 2 a . * + r = * 2 b . ϕ · r = ϕ
3 a . r + r = r 3 b . r · r = r
4 a . r ¯ ¯ = r
5 a . r + r ¯ = * 5 b . r · r ¯ = ϕ
6 a . r + s = s + r 6 b . r · s = s · r
7 a . r + r · s = r 7 b . r · ( r + s ) = r 8 a . ( r + s ¯ ) · s = r · s 8 b . r · s ¯ + s = r + s
9 a . r + s + t = ( r + s ) + t = r + ( s + t ) 9 b . r · s · t = ( r · s ) · t = r · ( s · t )
10 a . r · s + r · t = r · ( s + t ) 10 b . ( r + s ) · ( r + t ) = r + s · t
11 a . r + s ¯ = r ¯ · s ¯ 11 b . r · s ¯ = r ¯ + s ¯
* = * ( k m l ) , ϕ = ϕ ( k , l ) ( k , l = 0 , ± 1 , ± 2 , ) ,
1 a . r ( x 1 , y 1 ) ϕ = r ( x 1 , y 1 ) 1 b . r ( x 1 , y 1 ) * = r ( x 1 , y 1 ) 2 a . * r ( x 1 , y 1 ) = * 2 b . ϕ r ( x 1 , y 1 ) = θ
3 a . r ( x 1 , y 1 ) r ( x 1 , y 1 ) = r ( x 1 , y 1 ) 3 b . r ( x 1 , y 1 ) r ( x 1 , y 1 ) = r ( x 1 , y 1 )
4 a . r ( x 1 , y 1 ) ¯ ¯ = r ( x 1 , y 1 )
5 a . r ( x 1 , y 1 ) r ( x 1 , y 1 ) ¯ = * 5 b . r ( x 1 , y 1 ) r ( x 1 , y 1 ) ¯ = ϕ
6 a . r ( x 1 , y 1 ) s ( x 2 , y 2 ) = s ( x 2 , y 2 ) r ( x 1 , y 1 ) 6 b . r ( x 1 , y 1 ) s ( x 2 , y 2 ) = s ( x 2 , y 2 ) r ( x 1 , y 1 )
7 a . r ( x 1 , y 1 ) [ r ( x 1 , y 1 ) s ( x 2 , y 2 ) ] = r ( x 1 , y 1 ) 7 b . r ( x 1 , y 1 ) [ r ( x 1 , y 1 ) s ( x 2 , y 2 ) ] = r ( x 1 , y 1 ) 8 a . [ r ( x 1 , y 1 ) s ( x 2 , y 2 ) ¯ ] s x ( 2 , y 2 ) = r ( x 1 , y 1 ) s ( x 2 , y 2 ) 8 b . r ( x 1 , y 1 ) s ( x 2 , y 2 ) ¯ s ( x 2 , y 2 ) = r ( x 1 , y 1 ) s ( x 2 , y 2 )
9 a . r ( x 1 , y 1 ) s ( x 2 , y 2 ) t ( x 3 , y 3 ) = [ r ( x 1 , y 1 ) s ( x 2 , y 2 ) ] t ( x 3 , y 3 ) = r ( x 1 , y 1 ) [ s ( x 2 , y 2 ) t ( x 3 , y 3 ) ] 9 b . r ( x 1 , y 1 ) s ( x 2 , y 2 ) t ( x 3 , y 3 ) = [ r ( x 1 , y 1 ) s ( x 2 , y 2 ) ] t ( x 3 , y 3 ) = r ( x 1 , y 1 ) [ s ( x 2 , y 2 ) t ( x 3 , y 3 ) ]
10 a . r ( x 1 , y 1 ) s ( x 2 , y 2 ) r ( x 1 , y 1 ) t ( x 3 , y 3 ) = r ( x 1 , y 1 ) [ s ( x 2 , y 2 ) t ( x 3 , y 3 ) ] 10 b . [ r ( x 1 , y 1 ) s ( x 2 , y 2 ) ] [ r ( x 1 , y 1 ) t ( x 3 , y 3 ) ] = r ( x 1 , y 1 ) s ( x 2 , y 2 ) t ( x 3 , y 3 )
11 a . r ( x 1 , y 1 ) s ( x 2 , y 2 ) ¯ = r ( x 1 , y 1 ) ¯ s ( x 2 , y 2 ) ¯ 11 b . r ( x 1 , y 1 ) s ( x 2 , y 2 ) ¯ = r ( x 1 , y 1 ) ¯ s ( x 2 , y 2 ) ¯

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