Robust fitting of implicitly defined surfaces using Gauss-Newton-type techniques

We describe Gauss-Newton-type methods for fitting implicitly defined curves and surfaces to given unorganized data points. The methods are suitable not only for least-squares approximation, but they can also deal with general error functions, such as approximations to the a"" (1) or a"" (a) norm of the vector of residuals. Two different definitions of the residuals will be discussed, which lead to two different classes of methods: direct methods and data-based ones. In addition we discuss the continuous versions of the methods, which furnish geometric interpretations as evolution processes. It is shown that the data-based methods-which are less costly, as they work without the computation of the closest points-can efficiently deal with error functions that are adapted to noisy and uncertain data. In addition, we observe that the interpretation as evolution process allows to deal with the issues of regularization and with additional constraints.