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.
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Kozera, R., Wiliński, A. (2019). Fitting Dense and Sparse Reduced Data. In: Pejaś, J., El Fray, I., Hyla, T., Kacprzyk, J. (eds) Advances in Soft and Hard Computing. ACS 2018. Advances in Intelligent Systems and Computing, vol 889. Springer, Cham. https://doi.org/10.1007/978-3-030-03314-9_1
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