A class of orthonormal complete piecewise polynomial systems and applications thereof  

作　　者：XIONG GangQiang QI DongXu GUO FenHong 

机构地区：[1]School of Information Engineering, GuangDong Medical College [2]School of Information Science and Technology, Sun Yat-Sen University [3]Faculty of Information Technology, Macao University of Science and Technology [4]Mathematics Department, North China University of Technology

出　　处：《Science China(Information Sciences)》2013年第9期202-214,共13页中国科学（信息科学）（英文版）

基　　金：supported by the National Natural Science Foundation of China (Grant No. 10631080);the Science and Technology Development Fund of Macao SAR of China (Grant No. 045/2006/A);the National Basic Research Program of China (Grant No. 2011CB302400);the Scientific Research Common Program of the Beijing Municipal Commission of Education (Grant No. KM200910009001)

摘　　要：In the application of geometric graphs and image shape analysis, the Gibbs phenomenon appears if we approximate discontinuous geometric graphs using trigonometric functions, while the approximation effect of Walsh functions is not very good because of its slow convergence. This paper constructs a class of piecewise polynomials systems （referred to as quaternary U-Systems）, whose breakpoints only appear at quaternary rational numbers. Such quaternary U-Systems are a class of complete orthonormal systems in L2[0,1]. In addition, we also investigate their properties, formulae for basis values and Fourier-QU coefficients, and present a set of explicit expressions for a quaternary U-system of degree r（r=2,3,4）. Next, we apply a finite Fourier-QU series to represent image edges, and propose using the finite Fourier-QU coefficients to depict geometric graphs and image shapes. As a result, we obtain a new class of polynomial descriptors, called QU descriptors, and prove that unified QU descriptors are invariant under translation, scale, and rotation. Finally, we verify experimentally that the convergence rate of Fourier-QU series is faster than that of Fourier series, Walsh series, and Fourier-BU series in terms of the approximation of the function of a single variable. Furthermore, the experimental results prove that the QU descriptors are a class of practical shape descriptors, and that the QU distance between images can accurately measure their similarity.In the application of geometric graphs and image shape analysis, the Gibbs phenomenon appears if we approximate discontinuous geometric graphs using trigonometric functions, while the approximation effect of Walsh functions is not very good because of its slow convergence. This paper constructs a class of piecewise polynomials systems （referred to as quaternary U-Systems）, whose breakpoints only appear at quaternary rational numbers. Such quaternary U-Systems are a class of complete orthonormal systems in L2[0,1]. In addition, we also investigate their properties, formulae for basis values and Fourier-QU coefficients, and present a set of explicit expressions for a quaternary U-system of degree r（r=2,3,4）. Next, we apply a finite Fourier-QU series to represent image edges, and propose using the finite Fourier-QU coefficients to depict geometric graphs and image shapes. As a result, we obtain a new class of polynomial descriptors, called QU descriptors, and prove that unified QU descriptors are invariant under translation, scale, and rotation. Finally, we verify experimentally that the convergence rate of Fourier-QU series is faster than that of Fourier series, Walsh series, and Fourier-BU series in terms of the approximation of the function of a single variable. Furthermore, the experimental results prove that the QU descriptors are a class of practical shape descriptors, and that the QU distance between images can accurately measure their similarity.

关 键 词：piecewise polynomials orthogonal functions quaternary U-System Fourier series Walsh func- tions image edges shape descriptors 

分 类 号：TP391.41[自动化与计算机技术—计算机应用技术]