Carathéodory–Toeplitz based mathematical methods and their algorithmic applications in biometric image processing

In this paper the application of bounded series theory due to Carathéodory and Toeplitz is explored to study Brune positive real rational function (PRF). The main goal is to find the necessary and sufficient conditions for PRF coefficients. The introduced algorithms and assertions present an appropriate mathematical model derived from the developed analytical functions. The suggested solution is based on the results of Carathéodory, Toeplitz, Schur and their achievements at the beginning of the twentieth century. Toeplitz matrix lowest eigenvalues are constructed by the coefficients of the bounded power series representing Carathéodory function to establish a new simple and general algorithm for testing the nonnegativeness of real rational functions. The achieved results have shown engineering interests in two different areas of research: the electrical and mechanical circuit theory from one side and the image analysis and processing from the other side. The involvement in these methods has recently drawn the attention of researchers due to the increasing demand for simple methods of electrical and mechanical network synthesis. The author has proved the reasonability of Carathéodory–Toeplitz theory and modified it for using in other new areas of research. The most important achievements that describe relevant applications in such fields as digital filter design, speech signal and object image processing are discussed in the paper. Examples are introduced to illustrate these applications with emphasis on biometrics.