Smoothing and Filling Holes with Dirichlet Boundary Conditions

Linda Stals () and Stephen Roberts
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Linda Stals: Australian National University, Department of Mathematics
Stephen Roberts: Australian National University, Department of Mathematics

A chapter in Modeling, Simulation and Optimization of Complex Processes, 2008, pp 521-530 from Springer

Abstract: Abstract A commonly used method for the fitting of smooth functions to noisy data sets is the thin-plate spline method. Traditional thin-plate splines use radial basis functions and consequently requires the solution of a dense linear system of equations that grows with the number of data points. We present a method based instead on low order polynomial basis functions with local support defined on finite element grids. An advantage of such an approach is that the resulting system of equations is sparse and its size depends on the number of nodes in the finite element grid. A potential problem with local basis functions is an inability to fill holes in the data set; by their nature local basis functions are not defined on the whole domain like radial basis functions. Our particular formulation automatically fills any holes in the data. In this paper we present the discrete thin-plate spline method and explore how Dirichlet boundary conditions affect the way holes are filled in the data set. Theory is developed for general d-dimensional data sets and model problems are presented in 2D and 3D.

Date: 2008
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DOI: 10.1007/978-3-540-79409-7_38

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