Fitting Dense and Sparse Reduced Data

Abstract

This paper addresses the topic of fitting reduced data represented by the sequence of interpolation points $\mathcal{M}=\{q_i\}_{i=0}^n$ in arbitrary Euclidean space $\mathbb {E}^m$. The parametric curve $\gamma$ together with its knots $\mathcal{T}=\{t_i\}_{i=0}^n$ (for which $\gamma (t_i)=q_i$) are both assumed to be unknown. We look at some recipes to estimate $\mathcal{T}$ in the context of dense versus sparse $\mathcal{M}$ for various choices of interpolation schemes $\hat{\gamma }$. For $\mathcal{M}$ dense, the convergence rate to approximate $\gamma$ with $\hat{\gamma }$ is considered as a possible criterion to force a proper choice of new knots $\hat{\mathcal{T}}=\{\hat{t}_i\}_{i=0}^n \approx \mathcal{T}$. The latter incorporates the so-called exponential parameterization “retrieving” the missing knots $\mathcal{T}$ from the geometrical spread of $\mathcal{M}$. We examine the convergence rate in approximating $\gamma$ by commonly used interpolants $\hat{\gamma }$ based here on $\mathcal{M}$ and exponential parameterization. In contrast, for $\mathcal{M}$ sparse, a possible optional strategy is to select $\hat{\mathcal{T}}$ which optimizes a certain cost function depending on the family of admissible knots $\hat{\mathcal{T}}$. This paper focuses on minimizing “an average acceleration” within the family of natural splines $\hat{\gamma }=\hat{\gamma }^{NS}$ fitting $\mathcal{M}$ with $\hat{\mathcal{T}}$ admitted freely in the ascending order. Illustrative examples and some applications listed supplement theoretical component of this work.