Image algebra representation of parallel optical binary arithmetic

Kung-Shiuh Huang; B. Keith Jenkins; Alexander A. Sawchuk

doi:10.1364/AO.28.001263

Applied Optics
Vol. 28,
Issue 6,
pp. 1263-1278
(1989)
•https://doi.org/10.1364/AO.28.001263

Image algebra representation of parallel optical binary arithmetic

Kung-Shiuh Huang, B. Keith Jenkins, and Alexander A. Sawchuk

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Kung-Shiuh Huang, B. Keith Jenkins, and Alexander A. Sawchuk, "Image algebra representation of parallel optical binary arithmetic," Appl. Opt. 28, 1263-1278 (1989)

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Abstract

A binary image algebra (BIA) that gives a mathematical description of parallel processing operations is described. Rigorous and concise BIA representations of parallel arithmetic and symbolic substitution operations are given. A sequence of programming steps for implementation of these operations on a parallel architecture is specified by the BIA representation. Examples of arithmetic operations implemented on a digital optical cellular image processor architecture are given.

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Number Representation	Binary Row-coding	Binary Stack-coding	Symbolic Substitution (single-pixel coding)	Symbolic Substitution (two-pixel coding)
No. of Dilations (or Erosions)	k	0	9(k+1)	8(k+1)
No. of Unions (or Intersections)	4k+3	16k-12	15(k+1)	16(k+1)
No. of Complements	7k+4 ^*2k+2	20k-13 ^*7k-5	12(k+1) ^*3(k+1)	16(k+1) ^*4(k+1)
Total No. of Parallel Fundamental O Operations	12k+7 ^*7k+5	36k-25 ^*23k-17	36(k+1) ^*27(k+1)	40(k+1) ^*28(k+1)
No. of Processing P Elements	kN²	N²	2 k N²	4k N²
Total No. of Computations O×P	(12k+7)kN² ^* (7k+5)kN²	(36k-25)N² ^* (22k-16)N²	76k(k+1)N² ^* 54k(k+1)N²	160k(k+1)N² ^* 112k(k+1)N²
DOCIP T Execution Time	5k+3	17k-13	17(k+1)	18(k+1) or 28(k+1)
P×T	(5k+3)kN²	(17k-13)N²	34k(k+1)N²	72k(k+1 )N² or 112k(k+1)N²

Number Representation	Binary Row-coding	Binary Stack-coding	Symbolic Substitution (single-pixel coding)	Symbolic Substitution (two-pixel coding)
No. of Dilations (or Erosions)	k	0	6(k+1)	8(k+1)
No. of Unions (or Intersections)	4k+3	16k-12	10(k+1)	16(k+1)
No. of Complements	6k+4 ^*3k+2	22k-18 ^*11 k-8	8(k+1) ^*2(K+1)	16(k+1) ^*4(k+1)
Total No. of Parallel Fundamental O Operations	11k+7 ^*8k+5	43k-33 ^*33k-26	24(k+1) ^*18(k+1)	40(k+1) ^*28(k+1)
No. of Processing P Elements	k N²	N²	2 k N²	4k N²
Total No. of Computations O×P	(11k+7)kN² ^*(8k+5)kN²	(43k-33)N² ^*(33k-26)N²	48k(k+1)N² ^*36k(k+1)N²	160k(k+1)N² ^*112k(k+1)N²
DOCIP T Execution Time	4k+3	20k-17	11(k+1)	18(k+1) or 28(k+1)
P×T	(4k+3)k N²	(20k-17)N²	22k(k+1)N²	72k(k+1)N² or 112k(k+1)N²

Number Representation	Binary Row-coding	Binary Stack-coding	Symbolic Substitution (single-pixel coding)	Symbolic Substitution (two-pixel coding)
No. of Dilations (or Erosions)	k	0	9(k+1)	8(k+1)
No. of Unions (or Intersections)	4k+3	16k-12	15(k+1)	16(k+1)
No. of Complements	7k+4 ^*2k+2	20k-13 ^*7k-5	12(k+1) ^*3(k+1)	16(k+1) ^*4(k+1)
Total No. of Parallel Fundamental O Operations	12k+7 ^*7k+5	36k-25 ^*23k-17	36(k+1) ^*27(k+1)	40(k+1) ^*28(k+1)
No. of Processing P Elements	kN²	N²	2 k N²	4k N²
Total No. of Computations O×P	(12k+7)kN² ^* (7k+5)kN²	(36k-25)N² ^* (22k-16)N²	76k(k+1)N² ^* 54k(k+1)N²	160k(k+1)N² ^* 112k(k+1)N²
DOCIP T Execution Time	5k+3	17k-13	17(k+1)	18(k+1) or 28(k+1)
P×T	(5k+3)kN²	(17k-13)N²	34k(k+1)N²	72k(k+1 )N² or 112k(k+1)N²

Number Representation	Binary Row-coding	Binary Stack-coding	Symbolic Substitution (single-pixel coding)	Symbolic Substitution (two-pixel coding)
No. of Dilations (or Erosions)	k	0	6(k+1)	8(k+1)
No. of Unions (or Intersections)	4k+3	16k-12	10(k+1)	16(k+1)
No. of Complements	6k+4 ^*3k+2	22k-18 ^*11 k-8	8(k+1) ^*2(K+1)	16(k+1) ^*4(k+1)
Total No. of Parallel Fundamental O Operations	11k+7 ^*8k+5	43k-33 ^*33k-26	24(k+1) ^*18(k+1)	40(k+1) ^*28(k+1)
No. of Processing P Elements	k N²	N²	2 k N²	4k N²
Total No. of Computations O×P	(11k+7)kN² ^*(8k+5)kN²	(43k-33)N² ^*(33k-26)N²	48k(k+1)N² ^*36k(k+1)N²	160k(k+1)N² ^*112k(k+1)N²
DOCIP T Execution Time	4k+3	20k-17	11(k+1)	18(k+1) or 28(k+1)
P×T	(4k+3)k N²	(20k-17)N²	22k(k+1)N²	72k(k+1)N² or 112k(k+1)N²

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Equations (73)

Applied Optics