Gauss-Newton-type techniques for robustly fitting implicitly defined curves and surfaces to unorganized data points

Abstract

We describe Gauss-Newton type methods for fitting implicitly defined curves and surfaces to given unorganized data points. The methods can deal with general error functions, such as approximations to the l1 or linfin norm of the vector of residuals. Depending on the definition of the residuals, we distinguish between direct and data-based methods. In addition, we show that these methods can either be seen as (discrete) iterative methods, where an update of the unknown shape parameters is computed in each step, or as continuous evolution processes, that generate a time-dependent family of curves or surfaces, which converges towards the final result. It is shown that the data-based methods - which are less costly, as they work without the need of computing 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.