We consider a subproblem in parameter estimation using the Gauss-Newton algorithm with regularization for NURBS curve fitting. The NURBS curve is fitted to a set of data points in least-squares sense, where the sum of squared orthogonal distances is minimized. Control-points and weights are estimated. The knot-vector and the degree of the NURBS curve are kept constant. In the Gauss-Newton algorithm, a search direction is obtained from a linear overdetermined system with a Jacobian and a residual vector. Because of the properties of our problem, the Jacobian has a particular sparse structure which is suitable for performing a splitting of variables. We are handling the computational problems and report the obtained accuracy using different methods, and the elapsed real computational time. The splitting of variables is a two times faster method than using plain normal equations.

We consider the problem of fitting a non-uniform rational B-spline (NURBS) curve to a set of data points by determining the control points and the weights using techniques aimed for solving separable least squares problems. The main technique under consideration is the variable projection method which utilises that the NURBS model depends linearly on its control points but non-linearly on the weights. The variable projection method can be used with the Gauss-Newton algorithm but also with Newton's algorithm. We investigate the efficiency of the different algorithms when fitting NURBS and observe that the variable projection methods do not perform as well as reported for its use on, e.g., exponential fitting problems.

We consider the problem of matching sets of 3D points from a measured surface to the surface of a corresponding computer-aided design (CAD) object. The problem arises in the production line where the shape of the produced items is to be compared on-line with its pre-described shape. The involved registration problem is solved using the iterative closest point (ICP) method. In order to make it suitable for on-line use, i.e., make it fast, we pre-process the surface representation of the CAD object. A data structure for this purpose is proposed and named Distance Varying Grid tree. It is based on a regular grid that encloses points sampled from the CAD surfaces. Additional finer grids are added to the vertices in the grid that are close to the sampled points. The structure is efficient since it utilizes that the sampled points are distributed on surfaces, and it provides fast identification of the sampled point that is closest to a measured point. A local linear approximation of the surface is used for improving the accuracy. Experiments are done on items produced for the body of a car. The experiments show that it is possible to reach good accuracy in the registration and decreasing the computational time by a factor 700 compared with using the common kd-tree structure.

We study the problem of automatic generation of smooth and obstacle-avoiding planar paths for efficient guidance of autonomous mining vehicles. Fast traversal of a path is of special interest. We consider four-wheel four-gear articulated vehicles and assume that we have an a priori knowledge of the mine wall environment in the form of polygonal chains. Computing quartic uniform B-spline curves, minimizing curvature variation, staying at least at a proposed safety margin distance from the mine walls, we plan high speed paths.

In solving the nonlinear least-squares problem of minimizing ||f(x)||22, difficulties arise with standard approaches, such as the Levenberg-Marquardt approach, when the Jacobian of f is rank-deficient or very ill-conditioned at the solution. To handle this difficulty, we study a special class of least-squares problems that are uniformly rank-deficient, i.e., the Jacobian of f has the same deficient rank in the neighborhood of a solution. For such problems, the solution is not locally unique. We present two solution tecniques: (i) finding a minimum-norm solution to the basic problem, (ii) using a Tikhonov regularization. Optimality conditions and algorithms are given for both of these strategies. Asymptotical convergence properties of the algorithms are derived and confirmed by numerical experiments. Extensions of the presented ideas make it possible to solve more general nonlinear least-squares problems in which the Jacobian of f at the solution is rank-deficient or ill-conditioned.

V-invariant methods for the generalised least squares problem extend the techniques based on orthogonal factorization for ordinary least squares to problems with multiscaled, even singular covariances. These methods are summarised briefly here, and the ability to handle multiple scales indicated. An application to a class of Kalman filter problems derived from generalised smoothing splines is considered. Evidence of severe illconditioning of the covariance matrices is demonstrated in several examples. This suggests that this is an appropriate application for the V-invariant techniques.

We study the problem of computing a planar curve, restricted to lie between two given polygonal chains, such that the integral of the square of arc-length derivative of curvature along the curve is minimized. We introduce the minimum variation B-spline problem, which is a linearly constrained optimization problem over curves, defined by B-spline functions only. An empirical investigation indicates that this problem has one unique solution among all uniform quartic B-spline functions. Furthermore, we prove that, for any B-spline function, the convexity properties of the problem are preserved subject to a scaling and translation of the knot sequence defining the B-spline.

We study the curvature variation functional, i.e., the integral over the square of arc-length derivative of curvature, along a planar curve. With no other constraints than prescribed position, slope angle, and curvature at the endpoints of the curve, the minimizer of this functional is known as a cubic spiral. It remains a challenge to effectively compute minimizers or approximations to minimizers of this functional subject to additional constraints such as, for example, for the curve to avoid obstacles such as other curves. In this paper, we consider the set of smooth curves that can be written as graphs of three times continuously differentiable functions on an interval, and, in particular, we consider approximations using quartic uniform B- spline functions. We show that if quartic uniform B-spline minimizers of the curvature variation functional converge to a curve, as the number of B-spline basis functions tends to infinity, then this curve is in fact a minimizer of the curvature variation functional. In order to illustrate this result, we present an example of sequences of B-spline minimizers that converge to a cubic spiral.

We study the problem of computing a planar curve restricted to lie between two given polygonal chains such that the integral of the square of arc- length derivative of curvature along the curve is minimized. We introduce the Minimum Variation B-spline problem which is a linearly constrained optimization problem over curves defined by B-spline functions only. An empirical investigation indicates that this problem has one unique solution among all uniform quartic B-spline functions. Furthermore, we prove that, for any B-spline function, the convexity properties of the problem are preserved subject to a scaling and translation of the knot sequence defining the B-spline.