Convex Shape Preservation of Cubic $$L^1$$ Spline Fits

Abstract

Cubic $$L^1$$ spline fits have been developed for geometric data approximation and shown excellent performances in shape preservation. To quantify the convex-shape-preserving capability of spline fits, we consider a basic shape of convex corner with two line segments in a given window. Given the horizontal length difference and slope change of convex corner, we conduct an analytical approach and a numerical procedure to calculate the second-derivative-based spline fit in 3-node and 5-node windows, respectively. Results in both cases show that the convex shape can be well preserved when the horizontal length difference is within the middle third of window’s length. In addition, numerical results in the 5-node window indicate that the second-derivative based and first-derivative based spline fits outperform function-value based spline fits for preserving this convex shape. Our study extends current quantitative research on shape preservation of cubic $$L^1$$ spline fits and provides more insights on improving advance spline node positions for shape preserving purpose.