Per-Åke Wedin

Determining the movements of the skeleton using well-configured markers

The problem of determining skeletal movements in three dimensions by using a number of landmarks is treated. We present a method that determines the motion of a rigid body by using the positions of the landmarks in least-squares sense. The method uses the singular value decomposition of a matrix derived from the positions of the landmarks. We show how one can use this method to express movement of skeleton segments relative to each other. As many others have pointed out, the movement can be very ill determined if the landmarks are badly configured. We present a condition number for the problem with good geometrical properties. The condition number depends on the configuration of the landmarks and indicates how to distribute the landmarks in a suitable way.

Numerical tools for parameter estimation in ode-systems

The numerical problem of estimating unknown parameters in systems of ordinary differential equations from complete or incomplete data is treated. A new numerical method for the optimization part, based on the Gauss-Newton method with a trust region approach to subspace minimization for the weighted nonlinear least squares problem, is presented. The method is implemented in the framework of a toolbox (called diffpar) in Matlab and several test problems from applications, giving non-stiff and stiff ODE-systems, are treated

A note on some identification problems arising in roentgen stereo photogrammetric analysis

When studying skeleton movements by using implanted landmarks and Roentgen stereo photogrammetric analysis (RSA), some identification problems arise. One is to put the same label on a landmark at different examinations, another is to use the correct pairs of landmark images in order to compute a correct three-dimensional position of each landmark. We will present some methods that can be used as a support for solving these problems. We will also show how it is possible to detect and exclude landmarks that are loose and landmarks that have large measurement errors in their positions.

ON CONDITION NUMBERS AND ALGORITHMS FOR DETERMINING A RIGID-BODY MOVEMENT

Using a set of landmarks to represent a rigid body, a rotation of the body can be determined in least-squares sense as the solution of an orthogonal Procrustes problem. We discuss some geometrical properties of the condition number for the problem of determining the orthogonal matrix representing the rotation. It is shown that the condition number critically depends on the configuration of the landmarks. The problem is also reformulated as an unconstrained nonlinear least-squares problem and the condition number is related to the geometry of such problems. In the common 3-D case, the movement can be represented by using a screw axis. Also the condition numbers for the problem of determining the screw axis representation are shown to closely depend on the configuration of the landmarks. The condition numbers are finally used to show that the used algorithms are stable.

Algorithms for Constrained and Weighted Nonlinear Least Squares

A hybrid algorithm consisting of a Gauss--Newton method and a second-order method for solving constrained and weighted nonlinear least squares problems is developed, analyzed, and tested. One of the advantages of the algorithm is that arbitrarily large weights can be handled and that the weights in the merit function do not get unnecessarily large when the iterates diverge from a saddle point. The local convergence properties for the Gauss--Newton method are thoroughly analyzed and simple ways of estimating and calculating the local convergence rate for the Gauss--Newton method are given. Under the assumption that the constrained and weighted linear least squares subproblems attained in the Gauss--Newton method are not too ill conditioned, global convergence towards a first-order KKT point is proved.

A new linesearch algorithm for nonlinear least squares problems

A linesearch (steplength) algorithm for unconstrained nonlinear least squares problems is described. To estimate the steplength inside the linesearch algorithm a new method that interpolates the residual vector is used together with a standards method that interpolates the sums of squares.Numerical results are reported that point out the advantage with the new steplength estimation method.

Perturbation theory for generalized and constrained linear least squares

The perturbation analysis of weighted and constrained rank-deficient linear least squares is difficult without the use of the augmented system of equations. In this paper a general form of the augm ...

The Use and Properties of Tikhonov Filter Matrices

We consider the concept of Tikhonov filter matrices in connection with discrete ill-posed and rank-deficient linear problems. Important properties of the Tikhonov filter matrices are given together with their filtering and regularization effects. We also present new perturbation identities for the Tikhonov regularized linear least squares problem using filter matrices generalizing well-known perturbation identities for the linear least squares problem and pseudoinverses.

Truncated Gauss–Newton algorithms for Ill-conditioned nonlinear least squares problems

We address numerical optimization algorithms for solving nonlinear least squares problems that lack well-defined solutions, in particular discrete parameter estimation problems. We present algorithms based on the Gauss–Newton method for both exactly and almost rank-deficient problems. Merit functions proposed have good global convergence properties. Numerical results that confirm local convergence results are presented.