University West, Department of Technology, Mathematics and Computer Science, Division for Mathematics and Sciences.

Galois Module Structure of Field Extensions2007In: International Electronic Journal of Algebra, ISSN 1306-6048, E-ISSN 1306-6048, Vol. 2, no 8, p. 100-105Article in journal (Refereed)

Abstract [en]

We show, in two different ways, that every finite field extension has a basis with the property that the Galois group of the extension acts faithfully on it. We use this to prove a Galois correspondence theorem for general finite field extensions. We also show that if the characteristic of the base field is different from two and the field extension has a normal closure of odd degree, then the extension has a self-dual basis upon which the Galois group acts faithfully.

3.

Lundström, Patrik

University West, Department of Technology, Mathematics and Computer Science, Division for Mathematics and Sciences.

Separable Groupoid Rings2006In: Communications in Algebra, ISSN 0092-7872, E-ISSN 1532-4125, Vol. 34, no 8, p. 3029-3041Article in journal (Refereed)

Abstract [en]

We show that groupoid rings are separable over their ring of coefficients if and only if the groupoid is finite and the orders of the associated principal groups are invertible in the ring of coefficients. We use this to show that if we are given a finite groupoid, then the associated groupoid ring is semisimple (or hereditary) if and only if the ring of coefficients is semisimple (or hereditary) and the orders of the principal groups are invertible in the ring of coefficients. To this end, we extend parts of the theory of graded rings and modules from the group graded case to the category graded, and, hence, groupoid graded situation. In particular, we show that strongly groupoid graded rings are separable over their principal components if and only if the image of the trace map contains the identity

This paper presents novel strategies for better calibration and pose calculations of a system for determining the pose, i.e. position and orientation, of a camera. The system in question has a camera aimed to be placed on the hand of an industrial robot for welding, but is useful for any application with a need for measuring position and/or orientation. To calculate the pose of the camera circular reference points that can be recognized in the images are distributed in the working area. From their 2D image coordinates the 6D pose of the camera can be calculated. First the system is calibrated, i.e. the positions of the reference points and the camera parameters are determined. This is done by first taking images of the reference points from different locations, and then do a "total calibration" procedure to calculate the unknown parameters. For a specific system, called PosEye, it was concluded that the accuracy needs to be improved for welding applications. Also a method for making the calculations converge more easily, was needed. To meet these demands a new camera model is proposed, and three preprocessing calculation steps are presented. The new camera model increases accuracy by considering more distortion effects. The preprocessing steps give better initial values for more robust convergence of the algorithms and increased accuracy

A position and orientation (pose) measurement system is being developed. The system, called PosEye, is based on a camera and by using the information in the image, the pose of the camera taking the image can be calculated. The system is aimed to be placed on an industrial robot for welding, but it is flexible and can also be used in many other applications. The accuracy has been measured, and it is concluded that the accuracy needs to be improved for welding applications. To make the pose measurement, reference points, that can be recognized in the image, are distributed in the working area. The positions of the reference points and the parameters in a camera model are initially computed automatically from sample images from a number of directions to the reference points. After this calibration, the pose can be calculated at each sample image. For high accuracy there is a need to have a camera model that takes into account a number of distortion effects, which are further developed in this paper. The new model is used to express an optimization cost function that can be used for both the pose calculation, and the extensive calibration, that determines camera parameters in the camera model and the positions of the reference points