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

We investigate C1-smooth univariate curvature-based cubic L1 interpolating splines in Cartesian and polar coordinates. The coefficients of these splines are calculated by minimizing the L1 norm of curvature. We compare these curvature-based cubic L1 splines with second-derivative-based cubic L1 splines and with cubic L2 splines based on the L2 norm of curvature and of the second derivative. In computational experiments in Cartesian coordinates, cubic L1 splines based on curvature preserve the shape of multiscale data well, as do cubic L1 splines based on the second derivative. Cartesian-coordinate cubic L1 splines preserve shape much better than analogous Cartesian-coordinate cubic L2 splines. In computational experiments in polar coordinates, cubic L1 splines based on curvature preserve the shape of multiscale data better than cubic L1 splines based on the second derivative and much better than analogous cubic L2 splines. Extensions to splines in general curvilinear coordinate systems, to bivariate splines in spherical coordinate systems and to nonpolynomial splines are outlined.