A learning procedure of a three layer neural network with limited structure, called a multi-valued neural network, is proposed. The three layer net has a single linear neuron in its output layer. All input weights of a number of hidden neurons are identical. The network takes k+1 distinct stable values, where k is the number of hidden neurons. The proposed learning procedure consists of two parts, Phase and Phase . The former is one for the learning of weights between the hidden and output layers, and the latter is one for those between the input and the hidden layers. The network is applied to classification of numerals, which shows the effectiveness of the proposed learning procedure.

A function approximation scheme for image restoration is presented to resolve conflicting demands for smoothing within each object and differentiation between objects. Images are defined by probability distributions in the augmented functional space composed of image values and image planes. According to the fuzzy Hough transform, the probability distribution is assumed to take a robust form and its local maxima are extracted to yield restored images. This statistical scheme is implemented by a feedforward neural network composed of radial basis function neurons and a local winner-takes-all subnetwork.