F-Transform As a First Step Towards a General Approach to Data Processing and Data Fusion

In data fusion, we have several approximations to the desired objects, and we need to fuse them into a single – more accurate – approximation. In the traditional approach to data fusion, we usually assume that all the given approximations were obtained by minimizing the same distance function – most frequently, the Euclidean (L) distance. In practice, however, we sometimes need to use approximations corresponding to different distance functions. To handle such situations, a new more general approach to data processing and data fusion is needed. In this paper, we show that the simplest cases of such new situations lead to F-transform. Thus, F-transform can be viewed as a first step to such a general approach. From this viewpoint, we explain the formulas for the inverse F-transform, formulas which are empirically successful but which look somewhat strange from the viewpoint of the traditional approximation theory. 1 Traditional Approach to Data Processing and Data Fusion: Description and Limitations What is data processing: a brief reminder. In many practical situations, we need to extract the information about the objects of interest from the measurement results. The process of such extraction is known as data processing. In some situations, we are interested in the numerical characteristics: e.g., we are interested in the distance to a faraway star or in the amount of oil in a given oil field. In other situations, we are interested in a function; for example: • we want to reconstruct the signal x(t), i.e., to find out how the amplitude x changed with time;