A mathematical model based on an optimization formulation is presented for perceptual clustering of dot patterns. The features in the present model are its nonlinearity enabling the model to reveal hysteresis phenomena and its scale invariance. The clustering of dots is given by the mutual linking of dots by virtual lines. Every dot is assumed to be perceived at locations displaced from their original places. It is exemplified with simulations that the model can produce a hierarchical clustering of dots by variation in thresholds for the wiring of virtual lines and also the model can additionally reproduce some geometrical illusions semiquantitatively. This model is further extended for perceptual grouping in line segment patterns and geometrical illusions obsrved in those patterns are reproduced by the extended model.

A learning algorithm is presented for nearest neighbor pattern classifiers for the cases where mixed supervised and unsupervised training data are given. The classification rule includes rejection of outlier patterns and fuzzy classification. This partially supervised learning problem is formulated as a multiobjective program which reduces to purely super-vised case when all training data are supervised or to the other extreme of fully unsupervised one when all data are unsupervised. The learning, i. e. the solution process of this program is performed with a gradient method for searching a saddle point of the Lagrange function of the program.

A method based on clustering is presented for restoring and segmenting gray scale images. An optimum clustering obtained by a gradient method gives an image with gray scale values which vary smoothly in each segmented region. The method is also applied to restoration from sparsely sampled data.