Focal metric space in the identification problem
Mechanical Engeneering Research Institute A.A. Blagoravova RAS
The problem of decision making in the focus paradigm is considered, which makes it possible to use the focus model in the features space both at the learning stage for the construction of the classification metric space and for the decision making at the identification stage. The difficulties of traditional approaches to automating the identification process are mainly related to the piecewise nature of the description of the boundaries of class areas.Edge processing is associated with a procedure for searching through logical constructions, which is rapidly becoming more complex with increasing dimensionality of space, the complexity of which is compensated by the high level of IT technologies. The convex-concave character of the boundaries or their discontinuity significantly increase the volume and complexity of the solution of the problem.
This paper is devoted to the use of the focal representation of curves and surfaces developed by the author in the class of multifocal lemniscates  for describing class boundaries and making a decision in the recognition problem. The application of the focus model gives a single analytical description of the boundaries of each class in the form of a lemniscate through the focus system and the radius determined at the learning stage. The focal representation is not critical to the convexity and coherence of class boundaries.
The theoretical substantiation and methods of algorithmic realization of the description of class areas and their boundaries using a focal model for representing curves and surfaces are given in different statements of problems depending on the format of the initial data. The selection of training samples determines the construction of the classification set by the formation for each class of a focus system with foci at given learning points and the radius determined by the statistical characteristics of the samples. In this case, a complete metric classification space for K classes is formed in the form of K focal mk-systems with corresponding families of confocal lemniscates and membership functions.The solution of the binary problem, in contrast to the basic functional , allows us to use another, fractional-rational functional. The solution of a problem specified by boundaries in a piecewise or discrete-point format is possible in different analytic-algorithmic real or complex variants that admit a multidimensional generalization .
The traditional task of recognition is to compress the initial information. The peculiarity of recognition tasks with respect to boundaries is that, as a rule, these tasks do not have high requirements for detailing their form. The description of class boundaries without significant loss of information can be compressed by eliminating the redundancy of the focus system representing the training sample. An algorithm and procedure for optimizing the focus system are developed while maintaining a significant shape of the boundary.
An essential advantage is that the classification metric space forms a continuous membership function that allows to reduce the multidimensional identification problem to optimization in a one-dimensional space of radii.
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