Multi-Valued and Universal Binary Neurons deals with two new types of neurons: multi-valued neurons and universal binary neurons. These neurons are based on complex number arithmetic and are hence much more powerful than the typical neurons used in artificial neural networks. Therefore, networks with such neurons exhibit a broad functionality. They can not only realise threshold input/output maps but can also implement any arbitrary Boolean function. Two learning methods are presented whereby these networks can be trained easily. The broad applicability of these networks is proven by several case studies in different fields of application: image processing, edge detection, image enhancement, super resolution, pattern recognition, face recognition, and prediction. The book is hence partitioned into three almost equally sized parts: a mathematical study of the unique features of these new neurons, learning of networks of such neurons, and application of such neural networks. Most of this work was developed by the first two authors over a period of more than 10 years and was only available in the Russian literature. With this book we present the first comprehensive treatment of this important class of neural networks in the open Western literature. Multi-Valued and Universal Binary Neurons is intended for anyone with a scholarly interest in neural network theory, applications and learning. It will also be of interest to researchers and practitioners in the fields of image processing, pattern recognition, control and robotics.

f(z)=1 if 0<=arg(z) f(z)=0 if Pi/2<=arg(z)Then it is easy to demonstrate that XOR logical function is realizable by this extension of perceptron. The whole book consists of different extensions of the above simple idea that are able to realize more complicated Boolean functions (in particular the so-called k-valued Boolean threshold functions). The notion of linear separability is extended to the so-called P-realizable functions, then multi-valued Boolean threshold functions may be correctly realized. Moreover, it is demonstrated that an incremental perceptron learning may be modified to adjust complex weight coefficients, so that multi-valued Boolean threshold functions are realized. At the end of the book illustrative applications are presented that demonstrate an effectiveness of the proposed method (e.g. an associative memory for gray-scale images processing). The book is written in a highly sophisticated style employing mathematical concepts (e.g. group theory) unusual in neural networks. What we may accept from the book, that is substantial enough to be included in neural-network lectures? The two extensions of percetron to overcome a linear-separability block suggested by Minsky and Papert may be completed by the third possible extension based on complex weight coefficients. This is an interesting fact, but I would recommend the book only to a reader, which already knows neural networks really well, likes mathematics, and is specifically interested in perceptron learning.