Numerical Algorithms

, Volume 67, Issue 4, pp 807–826 | Cite as

L 1 C 1 polynomial spline approximation algorithms for large data sets

  • Laurent Gajny
  • Olivier GibaruEmail author
  • Eric Nyiri
Original Paper


In this article, we address the problem of approximating data points by C 1-smooth polynomial spline curves or surfaces using L 1-norm. The use of this norm helps to preserve the data shape and it reduces extraneous oscillations. In our approach, we introduce a new functional which enables to control directly the distance between the data points and the resulting spline solution. The computational complexity of the minimization algorithm is nonlinear. A local minimization method using sliding windows allows to compute approximation splines within a linear complexity. This strategy seems to be more robust than a global method when applied on large data sets. When the data are noisy, we iteratively apply this method to globally smooth the solution while preserving the data shape. This method is applied to image denoising.


L1 spline Approximation Smooth spline Noisy data 


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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Arts et Métiers ParisTechLSIS - UMR CNRS 7296Lille CedexFrance
  2. 2.Arts et Métiers ParisTechLSIS - UMR CNRS 7296Villeneuve d’AscqFrance

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