Chinese Journal of Computers   Full Text
Title	Representation for a Group of Parametric Curves Based on the Orthogonal Complete U-System
Authors	QI Dong-Xu1),2),3) TAO Chen-Jun2) SONG Rui-Xia1) MA Hui2) SUN Wei3) CAI Zhan-Chuan3)
Address	1)(College of Sciences, North China University of Technology, Beijing 100041) 2)(Faculty of Information Technology, Macao University of Science and Technology, Macao)
Year	2006
Issue	No.5(778—785)
Abstract &   Background	Abstract In order to probe into the properties of frequency spectrum for a group of parametric curves, a class of orthogonal complete piecewise k-degree polynomials in L2［0,1］，called U-system， is introduced. The expansion in U-series has advantageous properties for approximations in both quadratic norm and uniform, and it can be realized to express a group of parametric curves which are piecewise k-degree polynomials in a number of finite terms of U-series. Based on U-system, the transfer process is described, by which U-spectrum is obtained for a given group of parametric curves; The algorithm is visible, simple and fast; The program for processing data can be used to analyze and synthesize geometric information, and may have applications in the field such as information security (information hiding, watermarking), and pattern recognition etc. Some graphic examples of check test for expressing a group of parametric curves in U-system are given. keywords parametric curves; orthogonal; complete; frequency spectrum background This project is supported by “Mathematics Mechanization Method and Its Application on Information Technology”(973 Program) under grant No.2004CB3180000 and the National Natural Science Foundation of China (important special project) under grant No.60133020. The work aims to provide a new method to express the geometric data of parametric curves. In this method, authors have presented a class of orthogonal complete piecewise k-degree polynomials (so-called U-system). It is well known that many classes of orthogonal functions is applied to solve and analyze signals of physics or industry etc, rarely not to express the group of geometric graphics data. It is the main reason that most orthogonal functions cannot exactly express the geometric graphics, especially which have some discontinuous sub-graphics in the group. But U-system can do it very well. This is the most important property of U-system to other classes of orthogonal functions. Generally, people always have much interesting on presenting a group of geometric graphics by the common basic elements of graphic. By using Fourier-system to deal with these graphics of parametric curves, the results will be much more different from the original ones, especially at some places where there are disconnections between the sub-graphics. But the reproduction created by U-system is exactly the same as the original group of graphics. This paper gives the realization algorithm of U-system, and does some significant researches for future works.