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

Abstract

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 leastsquares approximation, but they can also deal with general error functions, such as approximations to the ℓ 1 or ℓ ∞ 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.

Approximate Implicitization of Space Curves

“…Following the idea in [2] we use an approximation of the exact geometric distance from a data point to a space curve. More precisely, we use the Sampson distance which was originally introduced for the case of hypersurfaces [25].…”

“…Consequently, (24) minimizes the Sampson distances from a point p j to each of the surfaces F and G independently. We adapt the evolution based-framework [2] in order to deal with the objective function (24). We consider the combination of the two evolutions for F and G which is defined by the minimization problem E → miṅ s , where…”

Identifying Unstructured Systems in the Errors-in-Variables Context

“…k-means (MacKay, 2003) LGA (Aigner and Jüttler, 2009) proposed algorithm Goal find best centers find best-fit lines find best-fit shapes Error measure distance from center distance from line distance from shape Update step compute mean estimate line parameters estimate shape parameters Assignment step assign to closest center project to line assign to closest or most feasible shape corresponding control coefficient. The curve reconstruction problem is to find a B-spline function f such that the geometric distance between the implicit curve f (x, y) = 0 and the point clouds be as small as possible.…”

“…In this section, we propose a self-organizing data-driven approach that alloys a polynomial estimation method to compute parameter estimates with a clustering scheme to discover a feasible partition of the data set. The scheme uses alternating optimization similar to the one employed by the linear grouping algorithm (Aelsta et al, 2006) but instead of fitting a linear relationship, fits quadratic or higher-order curves with a possibly iterative attractive scheme that resembles (Aigner and Jüttler, 2009). Unlike continuous curve or surface approximation methods, spline-based or moving least squares methods in particular, the proposed method not only provides a minimumerror fit in a certain sense but also a decomposition of data points into groups.…”

“…With a series of alternating optimization steps, the algorithm can discover the two separate domains and estimate the ellipse parameters over each domain. In contrast, gradient-based approaches (Aigner and Jüttler, 2009;Yang et al, 2005) cannot handle such a case: either they identify a single continuous curve outlining the composite shape, or the composite shape falls apart at intersections, depending on the starting parameter values and the number of initial continuous curves. Similarly, the moving least squares approach (Guennebaud and Gross, 2007) gives false estimates near the intersections, especially if the noise magnitude is more substantial.…”