Abstract

A new optical cellular continuous-logic array for gray-scale image processing is presented. The array involves two steps of multichannel neighborhood ranked-order operations and interimage threshold decomposition continuous-logic operation. Image processing transforms available from this array include gray-scale morphological transforms, stack–nonstack filters, and gray-scale hit-or-miss transforms. An area-coding technique is used to implement neighborhood operations in a correlation thresholding system. A similar one-channel grayscale-correlation and multithresholding system is used for interimage threshold decomposition continuous logic. Primary experimental results are given.

© 1993 Optical Society of America

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  1. K. Preston, M. J. B. Duff, S. Levialdi, P. E. Norgren, J. Toriwaki, “Basis of cellular logic with some applications in medical image processing,” Proc. IEEE 67, 826–856 (1979).
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  5. M. H. Brenner, S. D. Mukherjee, M. P. Bendett, A. R. Tanguay, “Integrated optoelectronic cellular array for fine-grained parallel processing systems,” in Optical Computing, Vol. 9 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989), pp. 223–226.
  6. I. Cramb, C. Upstill, “An opto-electronic computer architecture for binary image algebra,” in 1990 International Topical Meeting on Optical Computing, Y. Ichioka, S. Ishihara, J. Tsujiuchi, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1359, 130–131 (1990).
  7. M. Wu, S. Zhou, J. Cai, G. Chin, “Optical binary image algebra processor,” Opt. Commun. 86, 454–460 (1991).
    [CrossRef]
  8. R. G. A. Craig, B. S. Wherrett, A. C. Walker, F. A. P. Tooley, S. D. Smith, “Optical cellular logic image processor: implementation and programming of a single channel digital optical circuit,” Appl. Opt. 30, 2297–2308 (1991).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  20. P. Maragos, “Tutorials on advances in morphological image processing and analysis,” Opt. Eng. 26, 623–632 (1987).
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    [CrossRef]
  22. J. Serra, Image Analysis and Mathematical Morphology (Academic, New York, 1982).
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    [CrossRef]
  24. Z. Zhu, L. Liu, “Non-stack filters and their optical implementation,” submitted to Opt. Commun.
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    [CrossRef]
  26. P. Cambon, J. L. de Bougrenet de la Tocnaye, “Mathematical morphology processor using ferroelectric liquid crystal light valves: principle,” Appl. Opt. 28, 3456–3460 (1989).
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    [CrossRef] [PubMed]

1991 (3)

1990 (1)

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

1989 (5)

1988 (1)

J. M. Hereford, W. T. Rhodes, “Nonlinear optical image filtering by time-sequential threshold decomposition,” Opt. Eng. 27, 274–279 (1988).

1987 (2)

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

E. Ochoa, J. Allebach, D. Sweeney, “Optical median filtering using threshold decomposition,” Appl. Opt. 26, 252–260 (1987).
[CrossRef] [PubMed]

1986 (2)

T. Yatgai, “Cellular logic architectures for optical computers,” Appl. Opt. 25, 1571–1577 (1986).
[CrossRef]

P. D. Wendt, E. J. Coyle, N. C. Gallagher, “Stack filters,” IEEE Trans. Acoust. Speech Signal Proc. ASSP-34, 898–911 (1986).
[CrossRef]

1985 (1)

1981 (1)

K. C. Smith, “The prospects for multi-valued logic: a technology and applications view,” IEEE Trans. Comput. C-30, 619–634 (1981).
[CrossRef]

1979 (1)

K. Preston, M. J. B. Duff, S. Levialdi, P. E. Norgren, J. Toriwaki, “Basis of cellular logic with some applications in medical image processing,” Proc. IEEE 67, 826–856 (1979).
[CrossRef]

1978 (1)

Y. Nakagawa, A. Rosenfeld, “A note on the use of min and max operations in digital picture processing,” IEEE Trans. Syst. Man Cybern. SMC-8, 632–635 (1978).

1965 (1)

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

Allebach, J.

Arce, G. R.

G. R. Arce, N. Gallagher, T. A. Nodes, “Median filters: theory for one- and two-dimensional filters,” in Advances in Computer Vision and Image Processing, T. S. Huang, ed. (JAI, London, 1986), Vol. 2.

Awwal, A.

Bendett, M. P.

M. H. Brenner, S. D. Mukherjee, M. P. Bendett, A. R. Tanguay, “Integrated optoelectronic cellular array for fine-grained parallel processing systems,” in Optical Computing, Vol. 9 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989), pp. 223–226.

Brenner, M. H.

M. H. Brenner, S. D. Mukherjee, M. P. Bendett, A. R. Tanguay, “Integrated optoelectronic cellular array for fine-grained parallel processing systems,” in Optical Computing, Vol. 9 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989), pp. 223–226.

Cai, J.

M. Wu, S. Zhou, J. Cai, G. Chin, “Optical binary image algebra processor,” Opt. Commun. 86, 454–460 (1991).
[CrossRef]

Cambon, P.

Chavel, P.

Chin, G.

M. Wu, S. Zhou, J. Cai, G. Chin, “Optical binary image algebra processor,” Opt. Commun. 86, 454–460 (1991).
[CrossRef]

Coyle, E. J.

P. D. Wendt, E. J. Coyle, N. C. Gallagher, “Stack filters,” IEEE Trans. Acoust. Speech Signal Proc. ASSP-34, 898–911 (1986).
[CrossRef]

Craig, R. G. A.

Cramb, I.

I. Cramb, C. Upstill, “An opto-electronic computer architecture for binary image algebra,” in 1990 International Topical Meeting on Optical Computing, Y. Ichioka, S. Ishihara, J. Tsujiuchi, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1359, 130–131 (1990).

Cui, B.

de Bougrenet de la Tocnaye, J. L.

Devos, F.

Duff, M. J. B.

K. Preston, M. J. B. Duff, S. Levialdi, P. E. Norgren, J. Toriwaki, “Basis of cellular logic with some applications in medical image processing,” Proc. IEEE 67, 826–856 (1979).
[CrossRef]

M. J. B. Duff, T. J. Fountain, Cellular Logic Image Processing (Academic, London, 1986).

M. J. B. Duff, D. M. Watson, “CLIP 3: a cellular logic processor,” in New Concepts and Technologies in Parallel Information Processing, E. R. Caianeillo, ed. (Noordhoff, Leyden, The Netherlands, 1975).
[CrossRef]

Farhat, N. H.

Fountain, T. J.

M. J. B. Duff, T. J. Fountain, Cellular Logic Image Processing (Academic, London, 1986).

Gallagher, N.

G. R. Arce, N. Gallagher, T. A. Nodes, “Median filters: theory for one- and two-dimensional filters,” in Advances in Computer Vision and Image Processing, T. S. Huang, ed. (JAI, London, 1986), Vol. 2.

Gallagher, N. C.

P. D. Wendt, E. J. Coyle, N. C. Gallagher, “Stack filters,” IEEE Trans. Acoust. Speech Signal Proc. ASSP-34, 898–911 (1986).
[CrossRef]

Hereford, J. M.

J. M. Hereford, W. T. Rhodes, “Nonlinear optical image filtering by time-sequential threshold decomposition,” Opt. Eng. 27, 274–279 (1988).

Huang, K. S.

K. S. Huang, B. K. Jenkins, A. A. Sawchuk, “Binary image algebra and optical cellular logic processor design,” Comput. Vis. Graphics Image Proc. 45, 295–345 (1989).
[CrossRef]

Jenkins, B. K.

K. S. Huang, B. K. Jenkins, A. A. Sawchuk, “Binary image algebra and optical cellular logic processor design,” Comput. Vis. Graphics Image Proc. 45, 295–345 (1989).
[CrossRef]

Karim, M.

Levialdi, S.

K. Preston, M. J. B. Duff, S. Levialdi, P. E. Norgren, J. Toriwaki, “Basis of cellular logic with some applications in medical image processing,” Proc. IEEE 67, 826–856 (1979).
[CrossRef]

Liu, L.

L. Liu, X. Liu, B. Cui, “Optical programmable cellular logic array for image processing,” Appl. Opt. 30, 943–949 (1991).
[CrossRef] [PubMed]

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

Z. Zhu, L. Liu, “Non-stack filters and their optical implementation,” submitted to Opt. Commun.

Liu, X.

Maragos, P.

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

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

Mukherjee, S. D.

M. H. Brenner, S. D. Mukherjee, M. P. Bendett, A. R. Tanguay, “Integrated optoelectronic cellular array for fine-grained parallel processing systems,” in Optical Computing, Vol. 9 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989), pp. 223–226.

Nakagawa, Y.

Y. Nakagawa, A. Rosenfeld, “A note on the use of min and max operations in digital picture processing,” IEEE Trans. Syst. Man Cybern. SMC-8, 632–635 (1978).

Nodes, T. A.

G. R. Arce, N. Gallagher, T. A. Nodes, “Median filters: theory for one- and two-dimensional filters,” in Advances in Computer Vision and Image Processing, T. S. Huang, ed. (JAI, London, 1986), Vol. 2.

Norgren, P. E.

K. Preston, M. J. B. Duff, S. Levialdi, P. E. Norgren, J. Toriwaki, “Basis of cellular logic with some applications in medical image processing,” Proc. IEEE 67, 826–856 (1979).
[CrossRef]

Ochoa, E.

Paek, E.

Prata, A.

Preston, K.

K. Preston, M. J. B. Duff, S. Levialdi, P. E. Norgren, J. Toriwaki, “Basis of cellular logic with some applications in medical image processing,” Proc. IEEE 67, 826–856 (1979).
[CrossRef]

Psaltis, D.

Rhodes, W. T.

J. M. Hereford, W. T. Rhodes, “Nonlinear optical image filtering by time-sequential threshold decomposition,” Opt. Eng. 27, 274–279 (1988).

W. T. Rhodes, A. A. Sawchuk, “Incoherent optical processing,” in Optical Information Processing, S. H. Lee, ed. (Springer-Verlag, Berlin, 1981).
[CrossRef]

Rosenfeld, A.

Y. Nakagawa, A. Rosenfeld, “A note on the use of min and max operations in digital picture processing,” IEEE Trans. Syst. Man Cybern. SMC-8, 632–635 (1978).

Sawchuk, A. A.

K. S. Huang, B. K. Jenkins, A. A. Sawchuk, “Binary image algebra and optical cellular logic processor design,” Comput. Vis. Graphics Image Proc. 45, 295–345 (1989).
[CrossRef]

W. T. Rhodes, A. A. Sawchuk, “Incoherent optical processing,” in Optical Information Processing, S. H. Lee, ed. (Springer-Verlag, Berlin, 1981).
[CrossRef]

Schafer, R.

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

Serra, J.

J. Serra, Image Analysis and Mathematical Morphology (Academic, New York, 1982).

Smith, K. C.

K. C. Smith, “The prospects for multi-valued logic: a technology and applications view,” IEEE Trans. Comput. C-30, 619–634 (1981).
[CrossRef]

Smith, S. D.

Sweeney, D.

Taboury, J.

Tanguay, A. R.

M. H. Brenner, S. D. Mukherjee, M. P. Bendett, A. R. Tanguay, “Integrated optoelectronic cellular array for fine-grained parallel processing systems,” in Optical Computing, Vol. 9 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989), pp. 223–226.

Tooley, F. A. P.

Toriwaki, J.

K. Preston, M. J. B. Duff, S. Levialdi, P. E. Norgren, J. Toriwaki, “Basis of cellular logic with some applications in medical image processing,” Proc. IEEE 67, 826–856 (1979).
[CrossRef]

Upstill, C.

I. Cramb, C. Upstill, “An opto-electronic computer architecture for binary image algebra,” in 1990 International Topical Meeting on Optical Computing, Y. Ichioka, S. Ishihara, J. Tsujiuchi, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1359, 130–131 (1990).

Walker, A. C.

Wang, J. M.

Watson, D. M.

M. J. B. Duff, D. M. Watson, “CLIP 3: a cellular logic processor,” in New Concepts and Technologies in Parallel Information Processing, E. R. Caianeillo, ed. (Noordhoff, Leyden, The Netherlands, 1975).
[CrossRef]

Wendt, P. D.

P. D. Wendt, E. J. Coyle, N. C. Gallagher, “Stack filters,” IEEE Trans. Acoust. Speech Signal Proc. ASSP-34, 898–911 (1986).
[CrossRef]

Wherrett, B. S.

Wu, M.

M. Wu, S. Zhou, J. Cai, G. Chin, “Optical binary image algebra processor,” Opt. Commun. 86, 454–460 (1991).
[CrossRef]

Yatgai, T.

Zadeh, L. A.

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

Zhou, S.

M. Wu, S. Zhou, J. Cai, G. Chin, “Optical binary image algebra processor,” Opt. Commun. 86, 454–460 (1991).
[CrossRef]

Zhu, Z.

Z. Zhu, L. Liu, “Non-stack filters and their optical implementation,” submitted to Opt. Commun.

Appl. Opt. (8)

Comput. Vis. Graphics Image Proc. (1)

K. S. Huang, B. K. Jenkins, A. A. Sawchuk, “Binary image algebra and optical cellular logic processor design,” Comput. Vis. Graphics Image Proc. 45, 295–345 (1989).
[CrossRef]

IEEE Trans. Acoust. Speech Signal Proc. (1)

P. D. Wendt, E. J. Coyle, N. C. Gallagher, “Stack filters,” IEEE Trans. Acoust. Speech Signal Proc. ASSP-34, 898–911 (1986).
[CrossRef]

IEEE Trans. Comput. (1)

K. C. Smith, “The prospects for multi-valued logic: a technology and applications view,” IEEE Trans. Comput. C-30, 619–634 (1981).
[CrossRef]

IEEE Trans. Syst. Man Cybern. (1)

Y. Nakagawa, A. Rosenfeld, “A note on the use of min and max operations in digital picture processing,” IEEE Trans. Syst. Man Cybern. SMC-8, 632–635 (1978).

Inform. Contr. (1)

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

Opt. Commun. (2)

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

M. Wu, S. Zhou, J. Cai, G. Chin, “Optical binary image algebra processor,” Opt. Commun. 86, 454–460 (1991).
[CrossRef]

Opt. Eng. (2)

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

J. M. Hereford, W. T. Rhodes, “Nonlinear optical image filtering by time-sequential threshold decomposition,” Opt. Eng. 27, 274–279 (1988).

Proc. IEEE (2)

K. Preston, M. J. B. Duff, S. Levialdi, P. E. Norgren, J. Toriwaki, “Basis of cellular logic with some applications in medical image processing,” Proc. IEEE 67, 826–856 (1979).
[CrossRef]

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

Other (8)

Z. Zhu, L. Liu, “Non-stack filters and their optical implementation,” submitted to Opt. Commun.

W. T. Rhodes, A. A. Sawchuk, “Incoherent optical processing,” in Optical Information Processing, S. H. Lee, ed. (Springer-Verlag, Berlin, 1981).
[CrossRef]

J. Serra, Image Analysis and Mathematical Morphology (Academic, New York, 1982).

G. R. Arce, N. Gallagher, T. A. Nodes, “Median filters: theory for one- and two-dimensional filters,” in Advances in Computer Vision and Image Processing, T. S. Huang, ed. (JAI, London, 1986), Vol. 2.

M. J. B. Duff, T. J. Fountain, Cellular Logic Image Processing (Academic, London, 1986).

M. J. B. Duff, D. M. Watson, “CLIP 3: a cellular logic processor,” in New Concepts and Technologies in Parallel Information Processing, E. R. Caianeillo, ed. (Noordhoff, Leyden, The Netherlands, 1975).
[CrossRef]

M. H. Brenner, S. D. Mukherjee, M. P. Bendett, A. R. Tanguay, “Integrated optoelectronic cellular array for fine-grained parallel processing systems,” in Optical Computing, Vol. 9 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989), pp. 223–226.

I. Cramb, C. Upstill, “An opto-electronic computer architecture for binary image algebra,” in 1990 International Topical Meeting on Optical Computing, Y. Ichioka, S. Ishihara, J. Tsujiuchi, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1359, 130–131 (1990).

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Figures (10)

Fig. 1
Fig. 1

Schematic of TC-logic operation of {a(1), a(2),…, a(P)}.

Fig. 2
Fig. 2

Edge enhancement by gray-scale hit-or-miss transforms: (a) original one-dimensional gray-scale signal, (b) resulting one-dimensional gray-scale signal by gray-scale hit-or-mis transforms.

Fig. 3
Fig. 3

Two types of edge pixel on a one-dimensional gray-scale signal.

Fig. 4
Fig. 4

Implementation of RO operations: (a) area coding of variables {a(1),…, a(P)} into P bright cells, (b) intensity distribution along one dimension of the overlapping cell of P coded cells in (a), (c) thresholding curve for Rth RO operation, (d) intensity distribution along one dimension of the thresholded output bright cell.

Fig. 5
Fig. 5

Image configurations on the two correlation planes of a correlation system oriented to the optical implementation of neighborhood RO and TC-logic operations: (a) the pixel of the input image is represented by an area-coded cell centered at (iDx, jDy), (b) the correlation kernel is represented by an array of transparent points centered at (mDx, nDy). For RO operation, all c(m, n) values are 1; for TC-logic operation, c(m, n) values are 1, 1/2, 1/4,….

Fig. 6
Fig. 6

Area and intensity coding of variables {a(1),…, a(P)}.

Fig. 7
Fig. 7

Thresholding curve for implementing absolute-difference operation |a(1) − a(2)|.

Fig. 8
Fig. 8

Schematic of an optical CCLA: 1, spatially extended light source; 2, liquid-crystal panel displaying input image A and Ā; 3, correlation kernel mask array; 4, lens array; 5, multichannel thresholding devices; 6, correlation kernel mask; 7, lens; 8, CCD camera connected to a computer.

Fig. 9
Fig. 9

(a) Input area-coded image A; (b)–(f) RO operation results of image A; (b), first; (c), second; (d), third; (e), fourth; and (f), fifth.

Fig. 10
Fig. 10

Cross-shaped correlation kernel.

Tables (1)

Tables Icon

Table 1 Four Possible Resulting Intensities of the Overlapped Cell of Two Coded Cells: a(1) with an Initial Intensity IQ and a(2) with an Initial Intensity 2IQ

Equations (21)

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

R O [ a ( p ) ; R ] = R largest of [ a ( p ) : p = 1 , , P ] p = 1 , , P .
dilation a ( i , j ) = MAX [ a ( i + m , j + n ) ] ( m , n ) W ,
median filter a ( i , j ) = MED [ a ( i + m , j + n ) ] ( m , n ) W ,
erosion a ( i , j ) = MIN [ a ( i + m , j + n ) ] ( m , n ) W .
a ( i , j ) = MAX [ a ( i + m , j + n ) , a ¯ ( i + m , j + n ) ] ( m , n ) W F , ( m , n ) W B ,
a t ( p , q ) = { 1 when a ( p ) t ( q ) q = 1 , , P + 1 0 otherwise p = 1 , , P ,
a ( q ) = B [ a t ( 1 , q ) , a t ( 2 , q ) , , a t ( P , q ) ] q = 1 , , P + 1 .
a = T C { a ( p ) ; B ( a 1 , , a p ) } = q = 1 P + 1 a ( q ) [ t ( q ) t ( q 1 ) p = 1 , , P ,
a ( i , j ) = TC { RO [ a ( i + m , j + n ) , a ¯ ( i + m , j + n ) ; R p ] ; B ( a 1 , , a p ) } p = 1 , , P ; ( m , n ) , ( m , n ) W p ,
φ ( F ) : a ( i , j ) = MAX { MIN [ a ( i + m , j + n ) ] } G β ( φ ) ( m , n ) G ,
opening a ( i , j ) = MAX { MIN [ a ( i + m , j + n ] ) } p = 1 , , 4 ( m , n ) W p ,
closing a ( i , j ) = MIN { MAX [ a ( i + m , j + n ) ] } p = 1 , , 4 ( m , n ) W p ,
a ( i , j ) = min { a ( i , j ) , med [ a ( i + m , j + n ) ] } ( m , n ) W ,
a ( i ) = MAX { MIN [ a ¯ ( i 2 ) , a ¯ ( i 1 ) , a ( i ) , a ( i + 1 ) ] , MIN [ a ( i 1 ) , a ( i ) , a ¯ ( i + 1 ) , a ¯ ( i + 2 ) ] } .
a ( i , j ) = MAX [ a ( i + m , j + n ) ] MIN [ a ( i + m , j + n ) ] = TC { MAX [ a ( i + m , j + n ) ] ; MIN [ a ( i + m , j + n ) ] ; B ( a 1 , a 2 ) = a 1 a ¯ 2 + a ¯ 1 a 2 } ( m , n ) W , ( m , n ) W ,
a ( i , j ) = MIN { a ( i , j ) ; a ( i + m , j + n ) } ( m , n ) W B ,
a ( i , j ) = a ( i , j ) MAX { MIN [ a ( i + m , j + n ) ] } p = 1 , , 9 ( m , n ) W p = TC { MIN [ a ( i + m , j + n ) ] ; B ( a 0 , , a 9 ) = a 0 ( a 1 + a 2 + + a 9 ) ¯ } p = 0 , , 9 , ( m , n ) W p ,
I 1 ( x , y ) = ( i , j ) rect [ x i D x a ( i , j ) d x 1 2 ] rect ( y i D y d y 1 2 ) .
I 2 ( x , y ) = Σ δ ( x m D x ) δ ( y n D y ) ( m , n ) W ,
I ( x , y ) = I 1 ( α x , β y ) I 2 ( α , β ) d α d β = ( i , j ) { ( m , n ) W rect [ x + i D x a ( i + m , j + n ) d x + 1 2 ] × rect ( y + i D y d y + 1 2 ) } .
a ( i , j ) = MAX { MIN [ a ( i + 1 , j ) , a ( i , j ) , a ¯ ( i 1 , j ) , a ¯ ( i 2 , j ) ] , MIN [ a ( i 1 , j ) , a ( i , j ) , a ¯ ( i + 1 , j ) , a ¯ ( i + 2 , j ) ] , MIN [ a ( i , j + 1 ) , a ( i , j ) , a ¯ ( i , j 1 ) , a ¯ ( i , j 2 ) ] , MIN [ a ( i , j 1 ) , a ( i , j ) , a ¯ ( i , j + 1 ) , a ¯ ( i , j + 2 ) ] } .

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