一类离散正交分段多项式及其应用

A class of discrete orthogonal piecewise polynomials and its applications

在离散的几何图形应用中,经常需要正交表示有突变或间断的几何图形,若采用连续的正交函数系(如三角函数系)必然会出现Gibbs现象,而用间断的Walsh函数系表示因其收敛速度慢而效果欠佳。从Tchebichef离散正交多项式出发,构造了一类分段点(N-1)/2p处的离散正交分段多项式基(discrete piecewise tchebichef basis,DPTB)。该类基函数既有平滑过渡的部分,又有间断突变的部分,因而可以用它较准确地表示由离散分段多项式建模的几何图形。给出了正交基的性质与构造实例。最后通过离散信号逼近仿真实验验证了该算法的可行性,实验结果表明该分段离散正交多项式基表示分段跳跃突变信号的结果明显优于离散余弦基。

For computer geometric figures representation,there is Gibbs phenomenon if continuous basis functions are used to approximate the discontinuous signals with breakpoints.The rate of convergence is very slow if Walsh basis functions are used to represent the discontinuous signals.Thus a class of discrete piecewise orthogonal polynomials basis（DPTB） was constructed from discrete orthogonal Tchebichef polynomials,whose breakpoints appear at（N-1）/2p.Since this class of basis consists of smooth and piecewise polynomials parts,finite discrete geometric figures with breakpoints at（N-1）/2p can be precisely expressed by using the constructed orthogonal basis.Then its properties and a set of explicit basis expressions with degree k（k=1,2,3） are given.Finally,the new discrete orthogonal base is used to decompose and reconstruct the signal with breakpoints.The experimental results show that this method outperforms the algorithm based on cosine orthogonal basis for expressing the signals with breakpoint.