On shape-preserving capability of cubic L1 spline fits

• We analytically quantify shape-preserving capability of cubic L1 spline fits.
• We propose a shape-preserving metric for the linear shape of Heaviside step function.
• We analytically calculate and numerically compute the metric.
• We find that function-value-based spline fits preserve linear shape best.

Cubic L1L1 spline fits have shown some favorable shape-preserving property for geometric data. To quantify the shape-preserving capability, we consider the basic shape of two parallel line segments in a given window. When one line segment is sufficiently longer than the other, the spline fit can preserve its linear shape in at least half of the window. We propose to use the minimum of such length difference as a shape-preserving metric because it represents the extra information that the spline fits need to preserve the shape. We analytically calculate this metric in a 3-node window for second-derivative-based, first-derivative-based and function-value-based spline fits. In a 5-node window, we compute this metric numerically. In both cases, the shape-preserving metric is rather small, which explains the observed strong shape-preserving capability of spline fits. Moreover, the function-value-based spline fits are indicated to preserve shape better than the other two types of spline fits. This study initiates a quantitative research on shape preservation of L1L1 spline fits.