Adaptive scattered data fitting by extension of local approximations to hierarchical splines
It provides a practical solution for fitting scattered data in geospatial applications, but the approach is incremental, combining existing techniques.
The paper introduces an adaptive scattered data fitting method that extends local least squares approximations to hierarchical spline spaces, using variable degree polynomial approximations based on data availability and singular values. Numerical experiments demonstrate its effectiveness for approximating real terrain data.
We introduce an adaptive scattered data fitting scheme as extension of local least squares approximations to hierarchical spline spaces. To efficiently deal with non-trivial data configurations, the local solutions are described in terms of (variable degree) polynomial approximations according not only to the number of data points locally available, but also to the smallest singular value of the local collocation matrices. These local approximations are subsequently combined without the need of additional computations with the construction of hierarchical quasi-interpolants described in terms of truncated hierarchical B-splines. A selection of numerical experiments shows the effectivity of our approach for the approximation of real scattered data sets describing different terrain configurations.