一类完备的正交分段多项式函数系及其应用

A class of orthonormal complete piecewise polynomial systems and applications thereof

在几何图形或图像边界的频谱分析应用中,用Fourier三角基表示间断图形时必然会出现Gibbs现象,而用Walsh函数表示时,因其收敛速度慢而效果欠佳.本文首先构造了一类分段点在四进制有理数点处的分段多项式函数集(简称四进制U-系统,QU-系统),它是L2[0,1]空间上的完备的正交函数系,并研究了它的性质、基函数与Fourier-QU系数的计算公式,同时,也给出了1～3次QU-系统的一组显式表达式.然后,使用Fourier-QU级数的有限项和表示图像轮廓线,提出用有限的Fourier-QU系数描述几何图形或图像轮廓线,并由此得到了一类新的多项式描述子——QU描述子,而归一化QU描述子是一类基于平移、旋转与尺度变换的特征不变量.最后,通过数值实验证实了使用Fourier-QU级数逼近一元平方可积函数时,其收敛速率要优于Fourier级数、Walsh级数和Fourier-BU级数,同样也验证了QU描述子是一类有效的形状描述子,用图像间的QU距离能准确地描述图像间的相似性.

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/j2 [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~ 31 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.