Narrow Results

In this work, we study the realization and bifurcation of Boolean functions of four variables via a Cellular Neural Network (CNN). We characterize the basic relations between the genes and the offsets of an uncoupled CNN as well as the basis of the binary input vectors set. Based on the analysis, we have rigorously proved that there are exactly 1882 linearly separable Boolean functions of four variables, and found an effective method for realizing all linearly separable Boolean functions via an uncoupled CNN. Consequently, any kind of linearly separable Boolean function can be implemented by an uncoupled CNN, and all CNN genes that are associated with these Boolean functions, called the CNN gene bank of four variables, can be easily determined. Through this work, we will show that the standard CNN invented by Chua and Yang in 1988 indeed is very essential not only in terms of engineering applications but also in the sense of fundamental mathematics.

Some image processing research are restudied via CNN genes with five variables, and this include edge detection, corner detection, center point extraction and horizontal-vertical line detection. Although they were implemented with nine variables, the results of computer simulation show that the effect with five variables is identical to or better than that with nine variables.

Recently, an effective method for realizing linearly separable Boolean functions via Cellular Neural Networks (CNN), called the threshold bifurcation method, was introduced, with a CNN gene bank of four variables established [Chen & Chen, 2005]. Based on this success, the present paper is to further explore the realization of all linearly separable Boolean functions of five variables via CNN with von Neumann neighborhoods. This paper provides: (i) important and essential relations among the genes (or templates) and the offsets of an uncoupled CNN as well as the basis of the binary input vectors set, (ii) a neat truth table of uncoupled CNN with five input variables, (iii) 94572 linearly separable Boolean functions (LSBF) in the family of 2²⁵ = 4.294967296 × 10⁹ Boolean functions of five variables, realizable by a single CNN, and (iv) all 94572 CNN linearly separable Boolean genes (LSBG), which can be determined to form the CNN gene bank of five variables.

In this paper, we study fingerprint feature extraction via CNN with Von Neumann neighborhood. The extraction was implemented by using CNN with nine input variables, and we find that the process could also be implemented with only five variables, and an easier algorithm without compromising the effectiveness. According to the CNN model with five input variables and the corresponding CNN gene bank done by Chen et al. [2006, Http1], we can determine the CNN gene easily. Simultaneously, we also find some results in one of the references are incorrect.