Degree reduction of $B\acute{e}zier$ curves and filter bank

Abstract

The degree reduction of Bezier curves is considered as a filter bank process. The representation of the degree reduced curve and its error curve forms a system of analysis filters. The analysis filters and their synthesis filters are shown to correspond to the matrices of the basis conversion between the basis of Bernstein polynomials $B^n_i$ ($i=0,1,…,n$) of degree $n$ and the basis consisting of $B^{n-1}_i$ ($i=0,1,…,n-1$) and an extremal polynomial of degree n with respect to the norms $L^2$, $L^{∞}$ and $L^1$. In each case of $L^2$-, $L^{∞}$- and $L^1$-norm, we determine the synthesis filters and the analysis filters. The synthesis filters is shown to be the degree elevation matrix augmented with a column corresponding to the appropriate extremal polynomial. The analysis filters can be obtained as the inversion of the synthesis filters.