On four-point penalized Lagrange subdivision schemes

This paper is devoted to the definition and analysis of new subdivision schemes called penalized Lagrange. Their construction is based on an original reformulation for the construction of the coefficients of the mask associated to the classical 4-points Lagrange interpolatory subdivision scheme: these coefficients can be formally interpreted as the solution of a linear system similar to the one resulting from the constrained minimization problem in Kriging theory which is commonly used for reconstruction in geostatistical studies. In such a framework, the introduction in the formulation of a so-called error variance can be viewed as a penalization of the oscillations of the coefficients. Following this idea, we propose to penalize the 4-points Lagrange system. This penalization transforms the interpolatory schemes into approximating ones with specific properties suitable for the subdivision of locally noisy or strongly oscillating data. According to a so-called penalization vector, a family of schemes can be generated. A full theoretical study is first performed to analyze this new type of non stationary subdivision schemes. Then, in the framework of position dependant penalization vector, several numerical tests are provided to point out the efficiency of these schemes compared to standard approaches.