Analytical investigations for the design of fast approximation methods for fitting curves and surfaces to scattered data

We present an analytical framework for linear and nonlinear least squares methods and adopt it to the construction of fast iterative methods for fitting curves and surfaces to scattered data. The results are directly applicable to curves and surfaces that have a representation as a linear combination of smooth basis functions associated with the control points. Standard Bézier and B-spline curves / surfaces as well as subdivision schemes have this property. In the global approximation step for the control points our approach couples the standard linear approximation part with the reparameterization to heavily reduce the number of overall steps in the iteration process. This can be formulated in such a way that we have a standard least squares problem in each step. For the local nonlinear parameter corrections our results allow for an optimal choice of the methods used in different stages of the process. Furthermore, regularization terms that express the fairness of the intermediate and / or final result can be added. Adaptivity is easily integrated in our concept. Moreover our approach is well suited for reparameterization occurring in grid generation.