Geodesic
Automatic knot finding for piecewise-cubic approximation
Matematičeskoe modelirovanie, Tome 18 (2006) no. 3, pp. 23-40

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The paper proposes within the frame of four point transforms a method for piecewise-cubic approximation that detects the knots of the segments in auto-tracking mode. A 3-point cubic parametric spline (TPS) is used as a model of a local approximant. The free parameter $\theta $ (a coefficient at $x^{3}$) is searching using step-by-step averaging. An analytical expression for $\theta $ is received via a length of the segment and values of a function and derivatives that shows the dependence of the $C^{1}$-smoothness on the accuracy of the $\theta$-estimate. The stability of the method w.r.t. input errors is shown as well. The key parameters are: the parameters of the basis functions, the variance of the input errors, and a sampling step. The efficiency of the method is shown by numerical calculations on test examples. 
@article{MM_2006_18_3_a2,
     author = {N. D. Dikoussar and C. T\"or\"ok},
     title = {Automatic knot finding for piecewise-cubic approximation},
     journal = {Matemati\v{c}eskoe modelirovanie},
     pages = {23--40},
     publisher = {mathdoc},
     volume = {18},
     number = {3},
     year = {2006},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MM_2006_18_3_a2/}
}
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N. D. Dikoussar; C. Török. Automatic knot finding for piecewise-cubic approximation. Matematičeskoe modelirovanie, Tome 18 (2006) no. 3, pp. 23-40. http://geodesic.mathdoc.fr/item/MM_2006_18_3_a2/