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Research Seminar
Curve Modeling via Interpolation Based on Multidimensional Reduced Data Part 2A/Prof Ryszard Kozera, UWA 10am, 29th July AbstractWe consider the problem of modeling curves together with estimation of their length and trajectory via various interpolation schemes (i.e. piecewise polynomials) based on the so-called reduced data (an ordered sequence of interpolation points stripped from the corresponding knot parameters). Such interpolation is termed as non-parametric, as opposed to the classical parametric one, where the interpolation points together with the respective knots are given (non-reduced data). The analysis in question applies to the general class of smooth curves in arbitrary Euclidean space sampled with sparse or dense data.PART B: this talk discusses the case of sparse date, where the asymptotic analysis from PART A does not apply (as dense data are needed). We establish the alternative criterion of selecting the knots to minimize the "average acceleration" of the curve. The latter forms an optimization task for which we prove the existence of the optimal knots. Any numerical scheme based on derivative computation meets difficulties due to the complicated nature of the above optimization. We propose the computationally feasible scheme (called Leap-Frog) which converges to the critical point and is based on the overlapping 1-dimensional optimizations easily handled by e.g. Mathematica package. The examples supplement the analysis. |
