Shape-preserving, first-derivative-based parametric and nonparametric cubic [formula omitted] spline curves

Abstract

We investigate parametric and nonparametric cubic L 1 interpolating spline curves (“ L 1 splines”) in two and three dimensions with the goal of achieving shape-preserving interpolation of irregular data. We introduce five types of parametric L 1 and L 2 splines calculated by minimizing expressions involving L 1 norms, L 2 norms and squares of L 2 norms of second derivatives and five types of parametric L 1 and L 2 splines calculated by minimizing analogous expressions involving first derivatives minus first differences. We compare these splines among themselves, with a simple monotonicity-based interpolant and with the interpolant of Brodlie, Fritsch and Butland. Of all of the parametric splines, first-derivative-based “interactive-component” L 1 splines preserve the shape of irregular data best. Nonparametric first-derivative-based L 1 splines are introduced and shown to preserve shape better than the previously known nonparametric second-derivative-based L 1 splines, than nonparametric first- and second-derivative-based L 2 splines and than the simple monotonicity-based and Brodlie–Fritsch–Butland interpolants.