128 research outputs found
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
Duality, Barycentric Coordinates and Intersection Computation in Projective Space with GPU support
This paper presents solution of selected problems using principle of duality and projective space representation. It will be shown that alternative formulation in the projective space offers quite surprisingly simple solutions that lead to more robust and faster algorithms which are convenient for use within parallel architectures as GPU (Graphical Processor Units-NVIDIA-TESLA/Fermi) or SCC (Intel-Single-chip Cloud Computing), which can speed up solutions of numerical problems in magnitude of 10-100. There are many geometric algorithms based on computation of intersection of lines, planes etc. Sometimes, very complex
mathematical notations are used to express simple mathematical solutions, even if their formulation in the projective space offers much more simple solution. It is shown that a solution of a system of linear equations is equivalent to generalized cross product, which leads with the duality principle to new algorithms. This is presented on a new formulation of a line in 3D given as intersection of two planes which is robust and fast, based on duality of PlĂĽcker coordinates. The presented approach can be used also for reformulation of barycentric coordinates computations on parallel architectures. The presented approach for intersection computation is well suited especially for applications where robustness is required, e.g. large GIS/CAD/CAM systems etc
New Results on Triangulation, Polynomial Equation Solving and Their Application in Global Localization
This thesis addresses the problem of global localization from images. The overall goal is to find the location and the direction of a camera given an image taken with the camera relative a 3D world model. In order to solve the problem several subproblems have to be handled. The two main steps for constructing a system for global localization consist of model building and localization. For the model construction phase we give a new method for triangulation that guarantees that the globally optimal position is attained under the assumption of Gaussian noise in the image measurements. A common framework for the triangulation of points, lines and conics is presented. The second contribution of the thesis is in the field of solving systems of polynomial equations. Many problems in geometrical computer vision lead to computing the real roots of a system of polynomial equations, and several such geometry problems appear in the localization problem. The method presented in the thesis gives a significant improvement in the numerics when Gröbner basis methods are applied. Such methods are often plagued by numerical problems, but by using the fact that the complete Gröbner basis is not needed, the numerics can be improved. In the final part of the thesis we present several new minimal, geometric problems that have not been solved previously. These minimal cases make use of both two and three dimensional correspondences at the same time. The solutions to these minimal problems form the basis of a localization system which aims at improving robustness compared to the state of the art
SINGULAB - A Graphical user Interface for the Singularity Analysis of Parallel Robots based on Grassmann-Cayley Algebra
This paper presents SinguLab, a graphical user interface for the singularity
analysis of parallel robots. The algorithm is based on Grassmann-Cayley
algebra. The proposed tool is interactive and introduces the designer to the
singularity analysis performed by this method, showing all the stages along the
procedure and eventually showing the solution algebraically and graphically,
allowing as well the singularity verification of different robot poses.Comment: Advances in Robot Kinematics, Batz sur Mer : France (2008
Solving Equations Using Khovanskii Bases
We develop a new eigenvalue method for solving structured polynomial
equations over any field. The equations are defined on a projective algebraic
variety which admits a rational parameterization by a Khovanskii basis, e.g., a
Grassmannian in its Pl\"ucker embedding. This generalizes established
algorithms for toric varieties, and introduces the effective use of Khovanskii
bases in computer algebra. We investigate regularity questions and discuss
several applications.Comment: 25 pages, 1 figure, 2 table
{A Classification of Spherical Schubert Varieties in the Grassmannian}
Let be a Levi subgroup of which acts by left multiplication on a Schubert variety in the Grassmannian . We say that is a spherical Schubert variety if is a spherical variety for the action of . In earlier work we provide a combinatorial description of the decomposition of the homogeneous coordinate ring of into irreducible -modules for the induced action of . In this work we classify those decompositions into irreducible -modules that are multiplicity-free. This is then applied towards giving a complete classification of the spherical Schubert varieties in the Grassmannian
On a category of cluster algebras
We introduce a category of cluster algebras with fixed initial seeds. This
category has countable coproducts, which can be constructed combinatorially,
but no products. We characterise isomorphisms and monomorphisms in this
category and provide combinatorial methods for constructing special classes of
monomorphisms and epimorphisms. In the case of cluster algebras from surfaces,
we describe interactions between this category and the geometry of the
surfaces.Comment: 37 page
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