Piecewise-quadratics and reparameterizations for interpolating reduced data

R. Kozera, Lyle Noakes

    Research output: Chapter in Book/Conference paperConference paper

    7 Citations (Scopus)
    12 Downloads (Pure)


    © 2015 Springer International Publishing Switzerland. This paper tackles the problem of interpolating reduced data Qm= {qi}m i=0obtained by sampling an unknown curve γ in arbitrary euclidean space. The interpolation knots Tm= {ti}mi=o satisfying γ(ti) = qi are assumed to be unknown (non-parametric interpolation).Upon selecting a specific numerical scheme γ (here a piecewise-quadraticγ = γ2), one needs to supplement Qmwith knots’ estimates {ti}m i=0 ≈{ti}m i=0. A common choice of {tλ i}m i=0 (λ∈ [0,1]) frequently used in curve modeling and data fitting (e.g. in computer graphics and vision or in computer aided design) is called exponential parameterizations(see, e.g., [11] or [16]). Recent results in [8] and [14] show that j2combined with exponential parameterization yields (in trajectory estimation) either linear α(λ) = 1 (λ ∈[0,1)) or cubic α(1) = 3 convergence orders, once Qmgets progressively denser. The asymtototics proved in [8] relies on the extra assumptions requiring γ2 to be reparameterizable to the domain of γ. Indeed, as shown in [14], a natural candidate ψfor such a reparameterization meets this criterion only for λ= 1, whereas the latter (see [8]) may not hold for the remaining λ ∈[0,1) (which e.g. brings difficulty in length estimation of γ by using γ). Our paper fills out this gap and establishes sufficient conditions imposed on Tmto render ψa genuine reparameterization with λ ∈[0,1) (see Th. 4). The derivation of a such a condition involves theoretical analysis and symbolic computation, and this constitutes a novel contribution of the present work. The numerical tests verifying whether Yindeed is a reparameterization (for λ ∈[0,1) and for more-or-less uniform samplings Tm)are also performed. The sharpness of the asymptotics in question is additionally confirmed with the aid of numerical tests.
    Original languageEnglish
    Title of host publicationComputer Algebra in Scientific Computing
    Place of PublicationUSA
    ISBN (Print)9783319240206
    Publication statusPublished - 2015
    Event17th International Workshop, CASC 2015: Computer Algebra in Scientific Computing - Aachen, Germany
    Duration: 14 Sep 201518 Sep 2015


    Conference17th International Workshop, CASC 2015: Computer Algebra in Scientific Computing

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