[OPTICAL REVIEW Vol. 2, No. 5 (1995) 327-330]

## Linear Separability of Positive Self-Dual Logical Filters

Akira ASANO,^{1} Yasuhiro KASAI^{2} and Shunsuke YOKOZEKI^{1}

^{1}Department of Mechanical System Engineering, Kyushu Institute of Technology, Kawazu 680-4, Iizuka, Fukuoka, 820 Japan, ^{2}Graduate Course of Applied Physics, Osaka University, Yamadaoka 2-1, Suita, Osaka, 565 Japan

(Received November 25, 1994; Accepted July 5, 1995)

The threshold logic filter is an important nonlinear filter class, which is defined by a threshold logic function of binary input values. It is proved in this paper that not all positive self-dual logical functions are threshold functions if the number of input variables is 5 or more. The positive self-dual logical filter is a limited class but includes almost all filters for noise removal. Our result means that not all positive self-dual filters can be expressed by one operation of the threshold logic filters. In the sense of the filter expression by neural networks, the two-layer network cannot always optimize even this limited class.

Key words : image processing filters, logical filters, threshold logic, linear separability, positive self-dual logic, neural network