Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  tackles the problem of interpolating reduced data $Q_m=\{q_i\}_{i=0}^m$ obtained by sampling an unknown curve γ in arbitrary euclidean space. The interpolation knots ${\cal T}_m= \{t_i\}_{i=0}^m$ satisfying γti=qi are assumed to be unknown non-parametric interpolation. Upon selecting a specific numerical scheme $\hat \gamma$ here a piecewise-quadratic$\hat \gamma=\hat \gamma_2$, one needs to supplement Qm with knots' estimates $\{\hat t_i\}_{i=0}^m\approx\{t_i\}_{i=0}^m$. A common choice of $\{\hat t_i^{\lambda}\}_{i=0}^m$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 $\hat \gamma_2$ combined with exponential parameterization yields in trajectory estimation either linear αλ=1 λ∈[0,1 or cubic α1=3 convergence orders, once Qm gets progressively denser. The asymtototics proved in [8] relies on the extra assumptions requiring $\hat \gamma_2$ to be reparameterizable to the domain of γ. Indeed, as shown in [14], a natural candidate i¾ź for 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 $\hat \gamma$. Our paper fills out this gap and establishes sufficient conditions imposed on ${\cal T}_m$ to render i¾ź a 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 i¾ź indeed is a reparameterization for λ∈[0,1 and for more-or-less uniform samplings ${\cal T}_m$ are also performed. The sharpness of the asymptotics in question is additionally confirmed with the aid of numerical tests.
['Ryszard Kozera', 'Lyle Noakes']
I am writing a research paper on a idea Piecewise-Quadratics and Reparameterizations for Interpolating Reduced Data
[{"from": "human", "value": "I am writing a research paper on a idea Piecewise-Quadratics and Reparameterizations for Interpolating Reduced Data"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  tackles the problem of interpolating reduced data $Q_m=\\{q_i\\}_{i=0}^m$ obtained by sampling an unknown curve \u03b3 in arbitrary euclidean space. The interpolation knots ${\\cal T}_m= \\{t_i\\}_{i=0}^m$ satisfying \u03b3ti=qi are assumed to be unknown non-parametric interpolation. Upon selecting a specific numerical scheme $\\hat \\gamma$ here a piecewise-quadratic$\\hat \\gamma=\\hat \\gamma_2$, one needs to supplement Qm with knots' estimates $\\{\\hat t_i\\}_{i=0}^m\\approx\\{t_i\\}_{i=0}^m$. A common choice of $\\{\\hat t_i^{\\lambda}\\}_{i=0}^m$\u03bb\u2208[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 $\\hat \\gamma_2$ combined with exponential parameterization yields in trajectory estimation either linear \u03b1\u03bb=1 \u03bb\u2208[0,1 or cubic \u03b11=3 convergence orders, once Qm gets progressively denser. The asymtototics proved in [8] relies on the extra assumptions requiring $\\hat \\gamma_2$ to be reparameterizable to the domain of \u03b3. Indeed, as shown in [14], a natural candidate i\u00be\u017a for such a reparameterization meets this criterion only for \u03bb=1, whereas the latter see [8] may not hold for the remaining \u03bb\u2208[0,1 which e.g. brings difficulty in length estimation of \u03b3 by using $\\hat \\gamma$. Our paper fills out this gap and establishes sufficient conditions imposed on ${\\cal T}_m$ to render i\u00be\u017a a genuine reparameterization with \u03bb\u2208[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 i\u00be\u017a indeed is a reparameterization for \u03bb\u2208[0,1 and for more-or-less uniform samplings ${\\cal T}_m$ are also performed. The sharpness of the asymptotics in question is additionally confirmed with the aid of numerical tests."}]