53IV. CONCLUDING REMARKS The objective of this brief was to extend the CNN paradigm to multilevel halftoning of digital images. The primary contributions of the brief include using the CNN transient mode, one of the general working modes of the CNN [6], for multilevel halftoning and developing a stopping criterion for selection of the network output. Potential applications of CNN based multilevel halftoning include image preprocessing, e.g., enhancement or segmentation, and image compression.The proposed system lends itself to real-time processing and simple VLSI realization. Moreover, an analog programmable computing machine [7] could accept CNN based multilevel halftoning as the algorithmic step while perform more complex image processing tasks. Real-time realization of VMSE measure would allow utilization of the dynamic stopping criterion in applications that require fast processing. We are presently developing methods of realization of the CNN for multilevel halftoning using the multilayer [1], modularized, or programmable analogic [7] CNN.Abstract-Residue number systems have computational advantages for addition and multiplication since operations on residue digits are performed independently and so these processes can be performed in parallel. However other operations such as input/output conversions are significantly more difficult. A method for conversion from a specific residue number system with moduli of the form (2 k 0 1; 2 k ; 2 k + 1) to a weighted number system is presented here. The digit parallel method is significant, in that the largest number which must be handled is of the same order as the moduli, the digits of the result are calculated in parallel, and the required moduli operations are accomplished with addition or subtraction of a constant.
IEEE?lethods of implementing division are considered in terms of reducing the time required per iteration and the number of iterations required in a system which includes parallel operations. Several division schemes are described and an evaluation measure is proposed and is used to evaluate them.
This paper presents a method for residue number system ( R N S ) to weighted binary number system (BNS) conversion for moduli of the form 2k-1, 2k, 2k+l. The conversion method, which will be called the Srinivasan/Gallaher (SG) method, is an application of work by Srinivasan [l] on division by 2"-1 and 2'6 1 .Specifically the method for computing remainders is reversed to provide a convenient digitparallel RNS to binary conversion procedure. In comparison to other methods recently presented in the literature [2][3][4] [5], this technique is faster and requires less area for implementation. After the SG method is described and proved, this and other conversion approaches are classified and compared.Residue number system arithmetic (RNS) has proved to be very attractive for the implementation of fast signal processing hardware such as discrete fast Fourier transform processors and digital filters, because arithmetic units based on RNS are fast and simple as far addition, subtraction and multiplication are concemed. However, RNS is found to be inferior to the binary number system (BNS) in generalpurpose computing since general division, sign determination, and magnitude comparison are too slow. Contemporary RNS algorithms for these operations are defined in terms of conversion out of RNS. Even when the application requires only a fast special-purpose computing system, RNS is used in a special purpose processor hooked up to a generalpurpose host, which usually uses binary representations, and input-output conversion between binary and RNS representations is routine. RNS-BNS conversion is therefore an operation that requires carefd optimization.We present an original procedure for RNS decoding that has been developed based upon previous research on remainder generation in integer division.In its current form, the algorithm is designed for RNS with the set of three moduli (2n-1,2n.2n+l). The deconversion algorithm incoprates the remainder generation procedure in reverse, applied in a combined form to the residues (remainders) input, to produce the weighted binary representation (dividend). Aside from being a new and competitive (0 (logN)) process for RNS decoding, this procedure is especially interesting the since the weighted binary representation is produced in n-bit units, with each of these "digits" computed entirely independently of each other TH0301-2/90~/M36$01.00 0 1990 IEEE 636 P. Srinivasan University of Southwestern Louisianaand in parallel. This fact may be used to great advantage in developing algorithms for other "hard RNS operations such as sign &temninatim and magnitude comparison, because these operations require a fast conversion but may not require the entire binary representation.Implementation issues are examined and a hardware design is presented, as well as a proof and comparison with existing techniques. The implementation is presented for the restricted but very popular modulus set (2"-1,2". 2"+ 1 1. Extensions of the algorithm and implementation to less restrictive sets of moduli are also exami...
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