Shape-preserving, first-derivative-based parametric and nonparametric cubic L1 spline curves

We investigate parametric and nonparametric cubic L1 interpolating spline curves (“L1 splines”) in two and three dimensions with the goal of achieving shape-preserving interpolation of irregular data. We introduce five types of parametric L1 and L2 splines calculated by minimizing expressions involving L1 norms, L2 norms and squares of L2 norms of second derivatives and five types of parametric L1 and L2 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” L1 splines preserve the shape of irregular data best. Nonparametric first-derivative-based L1 splines are introduced and shown to preserve shape better than the previously known nonparametric second-derivative-based L1 splines, than nonparametric first- and second-derivative-based L2 splines and than the simple monotonicity-based and Brodlie–Fritsch–Butland interpolants.