Shape-preserving, multiscale interpolation by univariate curvature-based cubic L1 splines in Cartesian and polar coordinates

Abstract

We investigate C 1-smooth univariate curvature-based cubic L 1 interpolating splines in Cartesian and polar coordinates. The coefficients of these splines are calculated by minimizing the L 1 norm of curvature. We compare these curvature-based cubic L 1 splines with second-derivative-based cubic L 1 splines and with cubic L 2 splines based on the L 2 norm of curvature and of the second derivative. In computational experiments in Cartesian coordinates, cubic L 1 splines based on curvature preserve the shape of multiscale data well, as do cubic L 1 splines based on the second derivative. Cartesian-coordinate cubic L 1 splines preserve shape much better than analogous Cartesian-coordinate cubic L 2 splines. In computational experiments in polar coordinates, cubic L 1 splines based on curvature preserve the shape of multiscale data better than cubic L 1 splines based on the second derivative and much better than analogous cubic L 2 splines. Extensions to splines in general curvilinear coordinate systems, to bivariate splines in spherical coordinate systems and to nonpolynomial splines are outlined.