TECHNICAL NOTES

Parallel optical arithmetic on images by a redundant binary number representation Giuseppe A. De Biase and Annalisa Massini

La Sapienza University of Rome, Astronomy Institute, 29 Via G. M. Lancisi, I-00161 Rome, Italy. Received 12 June 1989. 0003-6935/90/111587-03$02.00/0. © 1990 Optical Society of America.

A redundant binary number representation allows the algebraic sum on signed numbers in constant time. This number representation is suitable for parallel arithmetic on images made by symbolic substitution on optical comput­ers. Key words: Optical computing, image arithmetic, re­dundant number representation.

Quick and efficient processing of images is an important request of various disciplines and several applied fields. Many authors have studied specialized computing architec­tures (with electronic technology) devoted to image process­ing (see, for example, Refs. 1-5), but other authors have pointed out that optical computing architectures are more suitable for massive computation on images because of the 2-D inherent structure both of data (images) and DOC (digi­tal optical computing) architectures.

In recent works6,7 Huang et al. showed a complete formal approach to a 2-D binary algebra and related arithmetic oriented to 2-D objects. In the second work (regarding a parallel optical arithmetic on images, with emphasis on the symbolic substitution method8-11) the use of redundant number representations is neglected in spite of several works about this subject related to optical computing systems. Redundant number systems can have important properties such as the carry-free addition (namely, the possibility to carry out addition in constant time independent of the bit strings length k).12-16

In optical image arithmetic a novel redundant binary (RB) number representation17 can be used. In the RB representa­tion an integer D is given by

where αi ε {0,1}, i is the position index, and k is the length of the bits string. Each RB number has a canonical form and several redundant representations. If, following the work of Huang et al.,7 the bits are symbol coded as

(the formal definition of the RB and p-RB redundant num­ber representations, their properties, and efficiencies will be presented in Ref. 17).

Using RB numbers, a constant time addition [instead of O(k) time of Ref. 7] can be performed by symbolic substitu­tion, by applying the rules introduced in Fig. 1 twice on the operands (see also Fig. 2).

The RB representation has several advantages if com­pared with other redundant number representations, as, for example, the modified signed digit (MSD) one (widely stud­ied for optical computing).13-15 RB representation is particu­larly suitable for machines with a two-valued logic, it allows the parallel encoding of signed numbers in 2s complement (in simply one step), and consequently, it allows use of the addition algorithm as the algebraic sum on signed numbers. This avoids the necessity of separate rules for the subtrac­tion, and one table only is needed for the whole arithmetic. Finally, the multiplication can be obtained in O(log2k) time.18

The decoding of an RB number can be performed simply

the RB numbers canonical form can be obtained—from the symbol coded natural binary representation—by the follow­ing simple rules:

Fig. 1. Symbolic substitution rules for the addition of unsigned RB numbers or for the algebraic sum of signed ones. These rules act on symbol pairs on two rows (see Fig. 2), the output lower pair is shifted

left one position.

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Fig. 2. Parallel application of addition symbolic substitution rules on two strings of symbols. At each step the input and the output are on two strings: ai and bi are the bits of the inputs operands, ci and di are the bits of the intermediate sums, and si are the bits of the result. The symbol 0 is a padded zero. At the last step the upper string is

zeroed.

by one shift and one binary addition, in fact the decoded value B—in natural binary representation—is given by

where

and

with k even. The nonconstant time complexity in the decod­ing operation is not particularly important because this oper­ation is used only when the data must be given to the external world.

A property of the RB numbers addition, performed with the rules of Fig. 1, is that the addition of two strings consid­ered as the concatenations of many RB numbers gives a resulting string which is the concatenation of RB sums (if the numbers on each string are separated by at least a pair □ □ in the same positions on the two operands).17 This property guarantees that the addition of all pixel pairs on a row coded image (see Fig. 3, and Ref. 7) or the addition of two row coded images can always be performed in parallel in two steps, independent of the number of bits per pixel used (and of the image dimension).

An RB numbers arithmetic can easily be implemented on both optical and electronic systems making easier the attain­ment of hybrid machines particularly suitable for image processing.19,20

This work was carried out within the framework of Pro-getto Finalizzato Sistemi Informatici e Calcolo Parallelo of CNR (National Research Council).

Fig. 3. Parallel addition on a row coded image (the pixels are 10 bits wide and in RB representation): (a) example of the addition, (b) operands on a row coded image, (c) first step (image with intermedi­

ate sums), (d) second step (image with results).

References 1. M. J. B. Duff and S. Levialdi, Eds., Languages and Architec­

tures for Image Processing, (Academic, London, 1981). 2. H. Matzushima, T. Uno, and M. Ejiri, "An Array Processor for

image processing," in Real-Time Parallel Computing, M. Onoe, K. Preston, and A. Rosenfeld Eds. (Plenum, New York, 1981), p. 325.

3. P. E. Danielsson and S. Levialdi, "Computer Architectures for Pictorial Information Processing," Computer, Nov., 53-67, (1981).

4. A. P. Reeves, "Parallel Computer Architectures for Image Processing," Comput. Vision Graphics, Image Process. 25, 68-88 (1984).

5. E. W. Kent, M. 0. Shneier, and R. Lumia, "PIPE (Pipelined Image-Processing Engine)," J. Parallel Distributed Comput. 2, 50-78 (1985).

6. K.-S. Huang, B. K. Jenkins, and A. A. Sawchuk, "Binary Image Algebra and Optical Cellular Logic Processor Design," Comput­er Vision, Graphics, Image Process. 45, 295-345 (1989).

7. K.-S. Huang, B. K. Jenkins, and A. A. Sawchuk, "Image Algebra Representation of Parallel Optical Binary Arithmetic," Appl. Opt. 28, 1263-1278 (1989).

8. C. C. Guest and T. K. Gaylord, "Truth-Table Look-Up Optical Processing Utilizing Binary and Residue Arithmetic," Appl. Opt. 19, 1201-1207 (1980).

9. M. M. Mirsalehi and T. K. Gaylord, "Truth-Table Look-Up Parallel Data Processing Using an Optical Content-Addressable Memory," Appl. Opt. 25, 2277-2283 (1986).

10. K.-H. Brenner, A. Huang, and N. Streibl, "Digital Optical Com­puting with Symbolic Substitution," Appl. Opt. 25, 3054-3060 (1986).

11. K.-H. Brenner, "New Implementation of Symbolic Substitution Logic," Appl. Opt. 25, 3061-3064 (1986).

12. A. Avizienis, "Signed-Digit Number Representations for Fast Parallel Arithmetic," IRE Trans. Electron. Comput. EC-10, 389-400 (1961).

13. R. P. Bocker, B. L. Drake, M. E. Lasher, and T. B. Henderson, "Modified Signed-Digit Addition and Subtraction Using Opti­cal Symbolic Substitution," Appl. Opt. 25, 2456-2457 (1986).

14. Y. Li and G. Eichmann, "Conditional Symbolic Modified Signed-Digit Arithmetic Using Optical Content-Addressable Memory Logic Elements," Appl. Opt. 26, 2328-2333 (1987).

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15. A. K. Cherri and M. A. Karim, "Modified-Signed Digit Arithme­tic Using an Efficient Symbolic Substitution," Appl. Opt. 27, 3824-3827 (1988).

16. B. Parhami, "Carry-Free Addition of Recoded Binary Signed-Digit Numbers," IEEE Trans. Comput. C-37, 1470-1476 (1988).

17. G. A. De Biase and A. Massini, "Redundant Binary Number Representation for Parallel Arithmetic on Optical and Electron­ic Computers," IEEE Trans. Comput. (1989), submitted.

18. N. Takagi, H. Yasuura, and S. Yajima, "High-Speed VLSI Mul­tiplication Algorithm with a Redundant Binary Addition Tree," IEEE Trans. Comput. C-34, 789-796 (1985).

19. G. A. De Biase, "Trends in Astronomical Image Processing," Comput. Vision Graphics Image Process. 43, 347-360 (1988).

20. G. A. De Biase, "Interconnection Structures and Parallel Com­puting," in Advances in Parallel Computing, D. J. Evans, Ed. (JAI Press, London, 1990), in press.

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