Area-preserving geometric Hermite interpolation

Abstract

In this paper we establish a framework for planar geometric interpolation with exact area preservation using cubic Bézier polynomials. We show there exists a family of such curves which are 5 th order accurate, one order higher than standard geometric cubic Hermite interpolation. We prove this result is valid when the curvature at the endpoints does not vanish, and in the case of vanishing curvature, the interpolation is 4 th order accurate. The method is computationally efficient and prescribes the parametrization speed at endpoints through an explicit formula based on the given data. Additional accuracy (i.e. same order but lower error constant) may be obtained through an iterative process to find optimal parametrization speeds which further reduces the error while still preserving the prescribed area exactly.

Parametric Interpolation Framework for 1-D Scalar Conservation Laws with Non-Convex Flux Functions

“…The main result in [14] shows that the interpolation is fifth order accurate in the L ∞ norm provided the parameters r 1 and r 2 satisfy the Equal-area constraint (2.2) and an appropriate decay rate ( as || D − A|| → 0). It is important to note that this result assumes the portion of the parametric curve being interpolated is small enough such that it may be represented by some function x, f (x) after an appropriate rotation.…”

“…In this section we present a brief overview of the area-preserving Bézier interpolation discussed in [14]. The main objective that work may be summarized as follows: given a planar parametric curve f (s), g(s) , parametrized by s ∈ [s 0 , s 1 ], find a cubic Bézier polynomial defined by (2.1)…”

“…The conservative nature of (1.1) is fundamental to the resulting solution, for example, conserving area, as in the equal-area principle, is sufficient to select the appropriate weak solution in the convex flux function case. With this in mind, we are interested in applying the area-preserving parametric interpolation method discussed in [14]. In that work the authors construct a family of exactly areapreserving parametric Hermite polynomials which are fifth order accurate, one order higher than the standard parametric cubic Hermite.…”

“…This paper is organized as follows: First, in Section 2 we present a brief overview of cubic Bézier interpolation and discuss the area-preserving framework of [14]. Next, in Section 3, we show how the parametric interpolation framework can be applied to homogeneous scalar conservation laws in one-space dimension.…”

Exploring Optimality of Piecewise Polynomial Interpolation Functions for Lung Field Modeling in 2D Chest X-Ray Images

“…Piecewise cubic Hermite interpolation polynomial (PCHIP) [29,30] is a third order polynomial which has a shape preserving characteristic by matching only the first order derivatives at the data points with their neighbors (before and after) [31]. This characteristic makes it differ from the cubic spline function.…”