A class of orthonormal complete piecewise polynomial systems and applications thereof

Abstract

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 ratio- nal 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.