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A. Asano, Y. Kasai, and S. Yokozeki,
Linear Separability of Positive Self-Dual Logical Filters
Optical Review, 2, 5, 327-330 (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.


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