Quadratic forms and linear algebraic groups

Topics discussed at the workshop Quadratic Forms and Linear Algebraic Groups included besides the algebraic theory of quadratic and Hermitian forms and their Witt groups several aspects of the theory of linear algebraic groups and homogeneous varieties, as well as some arithmetic aspects pertaining to the theory of quadratic forms over function fields over local fields

Universal tools for analysing structures and interactions in geometry

This study examined symmetry and perspective in modern geometric transformations, treating them as functions that preserve specific properties while mapping one geometric figure to another. The purpose of this study was to investigate geometric transformations as a tool for analysis, to consider invariants as universal tools for studying geometry. Materials and Methods: The Erlangen ideas of F. I. Klein were used, which consider geometry as a theory of group invariants with respect to the transformation of the plane and space. Results and Discussion: Projective transformations and their extension to two-dimensional primitives were investigated. Two types of geometric correspondences, collinearity and correlation, and their properties were studied. The group of homotheties, including translations and parallel translations, and their role in the affine group were investigated. Homology with ideal line axes, such as stretching and centre stretching, was considered. Involutional homology and harmonic homology with the centre, axis, and homologous pairs of points were investigated. In this study unified geometry concepts, exploring how different geometric transformations relate and maintain properties across diverse geometric systems. Conclusions: It specifically examined Möbius transforms, including their matrix representation, trace, fixed points, and categorized them into identical transforms, nonlinear transforms, shifts, dilations, and inversions

Autocalibration with the Minimum Number of Cameras with Known Pixel Shape

In 3D reconstruction, the recovery of the calibration parameters of the cameras is paramount since it provides metric information about the observed scene, e.g., measures of angles and ratios of distances. Autocalibration enables the estimation of the camera parameters without using a calibration device, but by enforcing simple constraints on the camera parameters. In the absence of information about the internal camera parameters such as the focal length and the principal point, the knowledge of the camera pixel shape is usually the only available constraint. Given a projective reconstruction of a rigid scene, we address the problem of the autocalibration of a minimal set of cameras with known pixel shape and otherwise arbitrarily varying intrinsic and extrinsic parameters. We propose an algorithm that only requires 5 cameras (the theoretical minimum), thus halving the number of cameras required by previous algorithms based on the same constraint. To this purpose, we introduce as our basic geometric tool the six-line conic variety (SLCV), consisting in the set of planes intersecting six given lines of 3D space in points of a conic. We show that the set of solutions of the Euclidean upgrading problem for three cameras with known pixel shape can be parameterized in a computationally efficient way. This parameterization is then used to solve autocalibration from five or more cameras, reducing the three-dimensional search space to a two-dimensional one. We provide experiments with real images showing the good performance of the technique.Comment: 19 pages, 14 figures, 7 tables, J. Math. Imaging Vi

Differential invariant signatures and flows in computer vision : a symmetry group approach

Includes bibliographical references (p. 40-44).Supported by the National Science Foundation. DMS-9204192 DMS-8811084 ECS-9122106 Supported by the Air Force Office of Scientific Research. AFOSR-90-0024 Supported by the Rothschild Foundation-Yad Hanadiv and by Image Evolutions, Ltd.Peter J. Olver, Guillermo Sapiro, Allen Tannenbaum