A Modified Hermite Interpolation with Exponential Parameterization

This work discusses the problem of fitting a regular curve γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} based on reduced data pointsQm=(q0,q1,⋯,qm)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q_m=(q_0,q_1,\dots ,q_m)$$\end{document} in arbitrary Euclidean space. The corresponding interpolation knots T=(t0,t1,⋯,tm)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal T}=(t_0,t_1,\dots ,t_m)$$\end{document} are assumed to be unknown. In this paper the missing knots are estimated by Tλ=(t^0,t^1,⋯,t^m)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal T}_{\lambda }=(\hat{t}_0,\hat{t}_1,\dots ,\hat{t}_m)$$\end{document} in accordance with the so-called exponential parameterization (see Kvasov in Methods of shape-preserving spline approximation, World Scientific Publishing Company, Singapore, 2000) controlled by a single parameter λ∈[0,1]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \in [0,1]$$\end{document}. In order to fit (T^λ,Qm)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\hat{\mathcal T}_{\lambda },Q_m)$$\end{document}, a modified Hermite interpolant γ^H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\gamma }^H$$\end{document} (a C1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^1$$\end{document} piecewise-cubic) introduced in Kozera and Noakes (Fundam Inf 61(3–4):285–301, 2004) is used. The sharp quartic convergence order for estimating γ∈C4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in C^4$$\end{document} by γ^H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\gamma }^H$$\end{document} is proved in Kozera (Stud Inf 25(4B-61):1–140, 2004) and Kozera and Noakes (2004) only for λ=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda =1$$\end{document} and within the general class of admissible samplings. The main result of this paper extends the latter to the remaining cases of exponential parameterization covering all λ∈[0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \in [0,1)$$\end{document}. A slower linear convergence order in trajectory estimation is established for any λ∈[0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \in [0,1)$$\end{document} and arbitrary more-or-less uniform sampling. The numerical tests conducted in Mathematica indicate the sharpness of the latter and confirm the necessity of more-or-less uniformity. Other interpolation schemes used to fit reduced data Qm\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q_m$$\end{document} and based on Tλ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal T}_{\lambda }$$\end{document} together with some relevant applications are also briefly recalled in this paper.


Introduction
Let γ : [0, T ] → E n be a smooth regular curve (i.e.γ (t) = 0) defined over t ∈ [0, T ], for 0 < T < ∞. The term reduced data Q m is defined by the sequence of m +1 interpolation points {q i } m i=0 in arbitrary Euclidean space E n . In addition, we assume that for q i = γ (t i ) we have q i+1 = q i (i = 0, 1, . . . , m − 1). The respective knots T = {t i } m i=0 (admitted in ascending order) are here unavailable. Any fitting schemeγ interpolating reduced data Q m is called non-parametric interpolation. In order to derive an interpolantγ , first the knot estimates {t i } m i=0 ≈ {t i } m i=0 should be somehow guessed subject to the interpolation constraintγ (t i ) = q i andt i <t i+1 . Upon selecting the specific schemeγ : [0,T ] → E n fitting Q m and the associated substitutesT = {t i } m i=0 of the unknown knots {t i } m i=0 , the question concerning the intrinsic asymptotic order α in approximating γ withγ arises naturally. In doing so, recall now some background notions (see e.g. [1,20]). The distribution of the knots {t} m i=0 is called to be more-or-less uniform if (see e.g. [9]): Definition 1. 2 The sampling T mol is more-or-less uniform if for some constants 0 < K l ≤ K u and sufficiently large m the inequalities: hold for all i = 1, 2, . . . , m. Equivalently, more-or-less uniformity stipulates the existence of some constant 0 < β ≤ 1 satisfying βδ m ≤ t i − t i−1 ≤ δ m , for all i = 1, 2, . . . , m.
In this work, the missing knots {t i } m i=0 are estimated by {t λ i } m i=0 according to the exponential parameterization (see e.g. [18]): for i = 1, 2, . . . , m, with superscript λ ∈ [0, 1] representing a free parameter. 1 Visibly the condition q i = q i+1 enforcest λ i <t λ i+1 . The case λ = 0 yields uniform knotst 0 i = i, whereas λ = 1 results in cumulative chord parameterization (see e.g. [18,19]) for which we have: Recall also the following notion (see [1]): be a sequence of sites not necessarily distinct. We say that the function p agrees with the function g at τ provided that, for every site ξ that occurs l times in the sequence τ 1 , . . . , τ k , p and g agree l-fold at ξ , that is: Newton osculatory interpolation theorem [1] for possibly repeated ordered sites τ 1 ≤ τ 2 ≤ ... ≤ τ k reads: Theorem 1. 6 If g ∈ C n class and τ = {τ i } k i=1 is a sequence of k arbitrary sites (not necessarily distinct), then there exists exactly one polynomial p n of order n that agrees with g at τ equal to: In addition, for all t we have: (1.5) Section 2 recalls the construction of the modified Hermite interpolantγ H : [0,T ] → E n . The respective asymptotics to approximate γ ∈ C 4 (sampled admissibly-see Definition 1.1) byγ H is so-far addressed only for λ = 1 appearing in (1.2)-see [7,8]. More specifically, a fast sharp quartic convergence order α(1) = 4 (see Definition 1.3) in estimating γ byγ H • φ H is proved. Note that bothγ H and φ H are precisely determined in Sect. 2 below.
Remark 1 For a given interpolation schemeγ based on reduced data Q m and fixed estimatesT of the unknown knots T (and subject to some selected mapping φ : [0, T ] → [0,T ]) we say that asymptotics γ −γ • φ = O(δ α m ) over [0, T ] is sharp within the prescribed family of curves γ ∈ J and family of samplings T ∈ K, if for some γ ∈ J and some some sampling from K, there exists t * ∈ [0, T ] and some positive constant K such that For example, for sharp quartic order from (3.1): J = C 4 , the admissible samplings (1.1) forms K,γ =γ H with φ = φ H andT is defined according to (1.2), where λ = 1. This paper's main contribution (see Sect. 3) extends the above result to the remaining cases of λ ∈ [0, 1) controlling (1.2) with K representing more-or-less uniform samplings-see Definition 1.2.
Section 5 supplements the main theoretical contribution of this work established in Theorem 3.2. The numerical tests (conducted in Mathematica) verify in affirmative the sharpness of the asymptotics from (3.2)-see also Remark 1. In addition, more-or-less uniformity of T assumed in Theorem 3.2 is also justified numerically as indispensable.
The last Sect. 6 concludes this paper and reiterates its main gist. The alternative spline schemesγ to approximate γ and based on Q m and (1.2) are briefly recalled in the context of underlying asymptotics. Some applications and open problems are also outlined.

Modified Hermite Interpolation
Interpolating Q m by a piecewise Lagrange cubicγ (see e.g. [1]) defined as a track-sum of cubicsγ i : [t i ,t i+3 ] → E n (where γ i (t i+k ) = q i+k for k = 0, 1, 2, 3) yields generically non-smooth curveγ at all junction points q 3 , q 6 , . . . , q m−3 . In order to rectify this deficiency, the data points Q m can be first supplemented with some good estimates of the unsupplied exact velocities i.e. with some v(q i ) ≈γ (t i ) (each pair in R n ). In sequel, a C 1 Hermite interpolation (see e.g. [1]) can now be applied to each pair of points (q i , q i+1 ) and the respective velocities ). The precise definition ofγ H goes along the following pattern (see [7,8]): . • Note that for the last three points (q m−2 , q m−1 , q m ) the respective velocities {v(q j )} m j=m−3 are analogously computed with the aid of reversing the last {t i } m−5 i=0 -for more details see [8].
of γ is also to be estimated by d(γ ) (see e.g. [5,16]). In contrast with the latter, some other applications (e.g. path modelling and planning in robotics or drone navigation) may require multiple or single loops (see e.g. [11]).
A similar approach as used above can be adopted to generate φ i = φ H i : I i →Î i . Indeed, the following procedure for the construction of φ H is proposed in [7]: For the last three knots the respective derivatives can be found as forγ H .

The Main Result
The following sharp result holding for λ = 1 (see Remark 1) is established in [7,8]: This paper complements (3.1) to the remaining λ ∈ [0, 1) and T more-or-less uniform. Contrasting Theorem 3.1, some choices of λ ∈ [0, 1), reduced data Q m and γ sampled more-or-less uniformly may not yield φ H i and ψ i as genuine reparameterizations (see also [9][10][11]). In particular, to cover the cases of The main result of this paper reads as: The next section proves Theorem 3.2. The sharpness of (3.2) is numerically verified in Sect. 5.

The Proof of Theorem 3.2
The proof of Theorem 3.2 is given for an arbitrary λ ∈ [0, 1). The case of λ = 1 in (1.2) addressed in Theorem 3.1 is already justified in [7,8].
Proof In order to extract the underlying asymptotics of it suffices to examine the latter for any . The first step splits the asymptotics of each f i into separate components.

Step 1: Decomposition of the Asymptotics for f i
Observe that each function f i (over the respective I i ) satisfies: where ψ i is the Lagrange cubic (see Sect. 2). As γ ∈ C 4 and λ ∈ [0, 1) by [12] (or [9]) we have: and thus to justify f i = O(δ m ) it suffices to show that the second component in (4.1) is also of order O(δ m ) (uniformly over I i ). Newton interpolation formula (1.5) (with the remainder) applied to the up to the 9-th order Thus (4.3) reduces into: (4.6) Before proceeding further recall that for k = 0, 1 by [12] we have (see also [9]): anď both holding over I i+k and over ψ i+k (I i+k ), respectively. Hence by (4.7) and (4.8) we obtain: Similarly, coupling together (2.1) and (2.4) with (4.7) and (4.8) leads to: Thus, combining (4.9) with (4.10) reduces (4.6) into: . Hence as t ∈ I i , the latter simplifies (4.5) into: We examine now the asymptotics of the second term ρ i [t i , t i , t i+1 , t i+1 , t] appearing in (4.11). By standard property of the divided differences (see e.g. [1]) each component of ρ i (with 1 ≤ j ≤ n) reads as: Hence, by the chain rule the expression 4!ρ i [t, t i , t i , t i+1 , t i+1 ] j (here the index j in ρ i and in (4.12) stands for the j-th component, with 1 ≤ j ≤ n) att i j is equal to: (4.14) Furthermore, by (4.7) asψ . The latter combined with more-or-less uniformity, (2.3), (2.4) and (4.7) results into (and also by Theorem 1.6): Newton interpolation formula (1.3) (applied over each I i ) yields: which in turn combined with (4.13) and (4.15) leads to: We examine now the asymptotics of the respective divided differences forγ H . In doing so, note that by (2.1) and (4.8) the following holds (upon using again Theorem 1.6): (4.17) Coupling (2.1) and (4.8) again with Theorem 1.6 renders: . (4.19) Note that in the latter we exploited: Indeed, recall that each regular curve γ can be parameterized by arc-length (see [2]) giving γ (t) = 1 which upon differentiating γ (t) 2 = γ (t)|γ (t) = 1 yields γ (t)|γ (t) = 0. The latter combined with Thus by (4.17) and (4.19) we obtain: The asymptotics for (4.12) and (4.11) is next computed yielding the asymptotics for f i (see (4.1)).
In the next section we report on the numerical tests performed in Mathematica.

Experiments
In this section, the sharpness of Theorem 3.2 (see Remark 1) is numerically verified in affirmative. Additionally, the more-or-less uniformity condition assumed in Theorem 3.2 is also indicated as necessary via illustrative example. All numerical tests are carried out in Mathematica 10.0-see also [23]. Three types of more-or-less uniform samplings T mol are used here to generate ordered sequences of reduced data Q m : For the first family T 1 (see (5.1)(i))K l = 1 2 , K u = 3 2 and β = 1 3 -see Definition 1.2. The second family of samplings T 2 (see (5.1)(ii)) yields K l = 1 3 , K u = 5 3 and β = 1 5 . Lastly, the uniform sampling T u (see (5.1)(iii)) renders K l = K u = β = 1.
To estimate α(λ) withᾱ(λ) a linear regression is applied with 96 ≤ m ≤ 144. The generated numerical results are presented in Table 1. Visibly they are all consistent with the asymptotics established in Th 3.2. Note also that the accelerated quartic convergence orders established for either uniform sampling T u or for λ = 1 (see (3.1)) are also numerically confirmed as listed in Table 1. However, a closer look for λ ε = 1 − ε ≈ 1 (with ε > 0 very small) suggests for this particular case (i.e. for γ sp sampled along (5.1)(ii)) faster convergence ratesᾱ(λ ε ) ranging continuously from 1 to 4-see Table 2. Evidently, the results from Table 2 do not contradict (3.2). Indeed, the latter establishes merely the lower bounds on the respective α(λ) ≥ 1, for arbitrary λ ∈ [0, 1). Taking into account Remark 1, this example verifies numerically the sharpness of Theorem 3.2 at least for λ ∈ [0, 0.9]. It also suggests that some continuous variations in accelerated α(λ) ranging from 1 to 4 are possible. Noticeably, any λ ≥ 0.9999995 is rounded up by Mathematica to the value λ = 1.0 and the resulting computed asymptotics stands forᾱ(1) ≈ α(1) = 4-see also Table 1.
(ii) In an effort to verify the sharpness of (3.2) for λ ε ≈ 1 consider now a straight line γ l introduced in (5.3)(ii) and sampled according to (5.1)(ii). The corresponding numerical results are presented in Table 3. In fact for λ ∈ {0, 0.1, 0.3, 0.5, 0.7, 0.9} numerical estimatesᾱ(λ) = 1.00, for all λ ∈ [0, 1). Thus the sharpness of the asymptotics in Theorem 3.2 is confirmed numerically. Note also that for λ = 1 both curves γ l andγ H • φ H coincide yielding the machine error while estimating (5.5).     Table 4 are consistent with the asymptotics proved in Theorem 3.2. Moreover, faster convergence orders established for either uniform sampling T u or for λ = 1 (see (3.1)) are also numerically confirmed in Table 4.

Example 3
The next example reports on numerical experiments testing the sharpness of Theorem 3.2 for the quadratic elliptical helix γ qh (see (5.4)) sampled according to (5.1). To examine numerically α(λ) withᾱ(λ) a linear regression is used for m varying within the set 110 ≤ m ≤ 162. Since (3.2) has asymptotic character (i.e. it holds for sufficiently large m) for some λ ∈ [0, 1] other intervals of integers are used. The results presented in Table  5 confirm the asymptotics from (3.2). In addition, fast quartic convergence orders prevailing for either uniform sampling T u or for λ = 1 (see (3.1)) are numerically confirmed as listed in Table 5.
The last example shows that more-or-less uniformity assumption cannot be dropped in Theorem 3.2.

Conclusions
This work's main contribution extends the claim of Theorem 3.1 (established for λ = 1-see [7] or [8]) to all remaining λ ∈ [0, 1) controlling exponential parameterization ( 1) and (3.2). The necessity of more-or-less uniformity stipulated by Theorem 3.2 is also substantiated numerically. Future work may e.g. focus on proving analytically the sharpness of the above asymptotics or on enforcing its acceleration within certain subfamilies of more-or-less uniform samplings. The other possible extension of this work is to examine the impact of both δ m and Q P m representing a small perturbation of data points Q m on the difference between γ andγ fitting Q P m . Lastly establishing sufficient conditions ensuring that φ H defines a reparameterization of [0, T ] → [0,T ] is also in priority demands. Related problem was addressed in the context of C 0 piecewisequadratic Lagrange interpolation in [9][10][11]. The reparameterization problem is vital for curve modelling (e.g. for a trajectory planning) and length estimation d(γ ) based on either sparse or dense reduced data. For example, in trajectory planning some applications like inspecting electrical poles by drones may require extra looping. For such a task, no reparameterization condition imposed on φ is needed. In contrast, the length estimation d(γ ) by d(γ ) relies on the existence of genuine re-reparameterization φ : [0, T ] → [0,T ], at best preserving fast convergence rates (see e.g. [5,16]).
In particular, the complete cubic splineγ C S ∈ C 2 (see e.g. [1]) incorporating Q m , both velocitiesγ (t 0 ) = v 0 ,γ (t m ) = v m and cumulative chord parameterization (i.e.for λ = 1 in (1.2)) is discussed in [5]. The latter proves the quartic convergence rate α(1) = 4 in estimating γ byγ C S . However, the remaining cases of λ ∈ [0, 1) in (1.2) are not addressed in [5]. On the other hand, if both initial and terminal velocities are not supplied, the modified complete splineγ C S M ∈ C 2 defined exclusively by Q m and (1.2) is numerically tested in [13]. Here, the ending velocities v 0 and v n are similarly approximated as outlined in Sect. 2. The latter results in numerically estimated quartic (linear) convergence order in approximating γ byγ C S M , for λ = 1 (or for λ ∈ [0, 1). Alternatively, for v 0 and v n unavailable, the natural splineγ N S ∈ C 2 (see e.g. [1]) based on Q m , (1.2) and ad hock adopted constraintŝ γ N S (t 0 ) =γ N S (t m ) = 0 can be applied. As numerically verified in [14], the decelerated quadratic (linear) convergence order in approximating γ byγ N S prevails, for λ = 1 (for λ ∈ [0, 1). In contrast to Theorem 3.2, no analytic proof is so-far given to justify the underlying asymptotics to approximate γ by eitherγ C S M or byγ N S .
Fitting reduced data forms an important problem in computer vision and graphics, engineering, microbiology, physics and other applications like medical image processing (e.g. for area and boundary computation or trajectory and length estimation)-see e.g. [4,6,15,16,18].
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