Shape-preserving approximation of multiscale univariate data by cubic L1 spline fits Original Research Article

Spline fits are calculated by minimizing a data fitting functional over a manifold of splines. Smoothing splines are calculated by minimizing a linear combination of a data fitting functional and an interpolating spline functional. For multiscale data, that is, data with abrupt changes in magnitude and/or spacing, currently available spline fits and smoothing splines typically have extraneous oscillation. In this paper, we introduce a class of cubic L1 spline fits that do not have extraneous oscillation. In contrast to conventional cubic spline fits, which are based on L2 functionals (that is, on quadratic functionals, which consist of sums and integrals of squares), cubic L1 splines are based on minimizing L1 functionals (sums and integrals of absolute values). Cubic L1 spline fits are calculated by a Lagrange-multiplier-based primal affine (interior point) algorithm. Cubic L1 spline fits are compared with conventional L2 cubic spline fits and with cubic L1 and L2 smoothing splines. Computational results indicate that L1 spline fits preserve shape well, nearly as well as L1 smoothing splines with good balance parameters and better than conventional L2 spline fits and L2 smoothing splines. L1 spline fits are computationally more expensive than L1 smoothing splines and considerably more expensive than L2 spline fits and L2 smoothing splines. However, L1 spline fits require only half the storage required by smoothing splines and do not involve a balance parameter that is required by smoothing splines.