Fitting Dense and Sparse Reduced Data

  • Ryszard KozeraEmail author
  • Artur Wiliński
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 889)


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.


Interpolation Reduced data Computer vision and graphics 


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Authors and Affiliations

  1. 1.Faculty of Applied Informatics and MathematicsWarsaw University of Life Sciences - SGGWWarsawPoland
  2. 2.Department of Computer Science and Software EngineeringThe University of Western AustraliaPerthAustralia

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