Approximation de fonctions et de données discrètes au sens de la norme L1 par splines polynomiales

Abstract : Data and function approximation is fundamental in application domains like path planning or signal processing (sensor data). In such domains, it is important to obtain curves that preserve the shape of the data. Considering the results obtained for the problem of data interpolation, L1 splines appear to be a good solution. Contrary to classical L2 splines, these splines enable to preserve linearities in the data and to not introduce extraneous oscillations when applied on data sets with abrupt changes. We propose in this dissertation a study of the problem of best L1 approximation. This study includes developments on best L1 approximation of functions with a jump discontinuity in general spaces called Chebyshev and weak-Chebyshev spaces. Polynomial splines fit in this framework. Approximation algorithms by smoothing splines and spline fits based on a sliding window process are introduced. The methods previously proposed in the littérature can be relatively time consuming when applied on large datasets. Sliding window algorithm enables to obtain algorithms with linear complexity. Moreover, these algorithms can be parallelized. Finally, a new approximation approach with prescribed error is introduced. A pure algebraic algorithm with linear complexity is introduced. This algorithm is then applicable to real-time application.
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Laurent Gajny. Approximation de fonctions et de données discrètes au sens de la norme L1 par splines polynomiales. Automatique. Ecole nationale supérieure d'arts et métiers - ENSAM, 2015. Français. ⟨NNT : 2015ENAM0006⟩. ⟨tel-01177009⟩

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