Ove Edlund

Robust registration of point sets using iteratively reweighted least squares

Registration of point sets is done by finding a rotation and translation that produces a best fit between a set of data points and a set of model points. We use robust M-estimation techniques to limit the influence of outliers, more specifically a modified version of the iterative closest point algorithm where we use iteratively re-weighed least squares to incorporate the robustness. We prove convergence with respect to the value of the objective function for this algorithm. A comparison is also done of different criterion functions to figure out their abilities to do appropriate point set fits, when the sets of data points contains outliers. The robust methods prove to be superior to least squares minimization in this setting.

Robust registration of surfaces using a refined iterative closest point algorithm with a trust region approach

The problem of finding a rigid body transformation, which aligns a set of data points with a given surface, using a robust M-estimation technique is considered. A refined iterative closest point (ICP) algorithm is described where a minimization problem of point-to-plane distances with a proposed constraint is solved in each iteration to find an updating transformation. The constraint is derived from a sum of weighted squared point-to-point distances and forms a natural trust region, which ensures convergence. Only a minor number of additional computations are required to use it. Two alternative trust regions are introduced and analyzed. Finally, numerical results for some test problems are presented. It is obvious from these results that there is a significant advantage, with respect to convergence rate of accuracy, to use the proposed trust region approach in comparison with using point-to-point distance minimization as well as using point-to-plane distance minimization and a Newton- type update without any step size control.

Computing the constrained M-estimates for regression

Constrained M-estimates for regression have been previously proposed as an alternative class of robust regression estimators with high breakdown point and high asymptotic efficiency. These are closely related to S-estimates, and it is shown that in some cases they will necessarily coincide. It has been difficult to use the CM-estimators in practice for two reasons. Adequate computational methods have been lacking and there has also been some confusion concerning the tuning parameters. Both of these problems are addressed; an updated table for choice of suitable parameter value is given, and an algorithm to compute CM-estimates for regression is presented. It can also be used to compute S-estimates. The computational problem to be solved is global optimization with an inequality constraint. The algorithm consists of two phases. The first phase is finding suitable starting values for the local optimization. The second phase, the efficient finding of a local minimum, is described in more detail. There is a MATLAB code generally available from the net. A Monte Carlo simulation is performed, using this code, to test the performance of the estimator as well as the algorithm.

Linear M-estimation with bounded variables

A subproblem in the trust region algorithm for non-linear M-estimation by Ekblom and Madsen is to find the restricted step. It is found by calculating the M-estimator of the linearized model, subject to anL2-norm bound on the variables. In this paper it is shown that this subproblem can be solved by applying Hebden-iterations to the minimizer of the Lagrangian function. The new method is compared with an Augmented Lagrange implementation.

Algorithms for Robustified Error-in-Variables Problems

We consider the problem of fitting a model of the form y = f (x, β) to a set of points (x i , y i ), i = 1,..., n. If there are measurement or observation errors in x as well as in y, we have the so called errors-in-variables-problem with model equation

Repeated surface registration for on-line use

{y_i} = f\left( {{x_i} + {\delta _i},\beta } \right) + {\varepsilon _i},\left( {i = 1, \ldots ,n} \right)

A software package for sparse orthogonal factorization and updating

(1) where δ i ∈ ℝm, i = 1,..., n are the errors in x i ∈ ℝm. Then the problem is to find a vector of parameters β ∈ ℝ p that minimizes the errors e i and δ i in some loss function subject to (1). We will present algorithms using more robust alternatives to the least squares criterion. Figure 1 gives examples where the least squares (L2), the least absolute deviation (L1) and the Huber criteria are used.