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22
Automatic Generator of Minimal Problem Solvers
, 2008
"... Finding solutions to minimal problems for estimating epipolar geometry and camera motion leads to solving systems of algebraic equations. Often, these systems are not trivial and therefore special algorithms have to be designed to achieve numerical robustness and computational efficiency. The state ..."
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Cited by 32 (6 self)
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Finding solutions to minimal problems for estimating epipolar geometry and camera motion leads to solving systems of algebraic equations. Often, these systems are not trivial and therefore special algorithms have to be designed to achieve numerical robustness and computational efficiency. The state of the art approach for constructing such algorithms is the Gröbner basis method for solving systems of polynomial equations. Previously, the Gröbner basis solvers were designed ad hoc for concrete problems and they could not be easily applied to new problems. In this paper we propose an automatic procedure for generating Gröbner basis solvers which could be used even by non-experts to solve technical problems. The input to our solver generator is a system of polynomial equations with a finite number of solutions. The output of our solver generator is the Matlab or C code which computes solutions to this system for concrete coefficients. Generating solvers automatically opens possibilities to solve more complicated problems which could not be handled manually or solving existing problems in a better and more efficient way. We demonstrate that our automatic generator constructs efficient and numerically stable solvers which are comparable or outperform known manually constructed solvers. The automatic generator is available at
M.: Pose estimation with radial distortion and unknown focal length
- In: Computer Vision and Pattern Recognition
, 2009
"... Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download a ..."
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Cited by 19 (0 self)
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Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal? Take down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim.
Polynomial Eigenvalue Solutions to the 5-pt and 6-pt Relative Pose Problems
"... In this paper we provide new fast and simple solutions to two important minimal problems in computer vision, the five-point relative pose problem and the six-point focal length problem. We show that these two problems can easily be formulated as polynomial eigenvalue problems of degree three and two ..."
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Cited by 19 (3 self)
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In this paper we provide new fast and simple solutions to two important minimal problems in computer vision, the five-point relative pose problem and the six-point focal length problem. We show that these two problems can easily be formulated as polynomial eigenvalue problems of degree three and two and solved using standard efficient numerical algorithms. Our solutions are somewhat more stable than state-of-the-art solutions by Nister and Stewenius and are in some sense more straightforward and easier to implement since polynomial eigenvalue problems are well studied with many efficient and robust algorithms available. The quality of the solvers is demonstrated in experiments 1. 1
3D reconstruction from image collections with a single known focal length
- In ICCV 2009
"... In this paper we aim at reconstructing 3D scenes from images with unknown focal lengths downloaded from photosharing websites such as Flickr. First we provide a minimal solution to finding the relative pose between a completely calibrated camera and a camera with an unknown focal length given six po ..."
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Cited by 8 (3 self)
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In this paper we aim at reconstructing 3D scenes from images with unknown focal lengths downloaded from photosharing websites such as Flickr. First we provide a minimal solution to finding the relative pose between a completely calibrated camera and a camera with an unknown focal length given six point correspondences. We show that this problem has up to nine solutions in general and present two efficient solvers to the problem. They are based on Gröbner basis, resp. on generalized eigenvalues, computation. We demonstrate by experiments with synthetic and real data that both solvers are correct, fast, numerically stable and work well even in some situations when the classical 6-point algorithm fails, e.g. when optical axes of the cameras are parallel or intersecting. Based on this solution we present a new efficient method for large-scale structure from motion from unordered data sets downloaded from the Internet. We show that this method can be effectively used to reconstruct 3D scenes from collection of images with very few (in principle single) images with known focal lengths 1. 1.
On the Global Optimum of Planar, Range-based Robot-to-Robot Relative Pose Estimation
"... In this paper, we address the problem of determining the relative position and orientation (pose) of two robots navigating in 2D, based on known egomotion and noisy robot-to-robot distance measurements. We formulate this as a weighted Least Squares (WLS) estimation problem, and determine the exact ..."
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Cited by 8 (3 self)
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In this paper, we address the problem of determining the relative position and orientation (pose) of two robots navigating in 2D, based on known egomotion and noisy robot-to-robot distance measurements. We formulate this as a weighted Least Squares (WLS) estimation problem, and determine the exact global optimum by directly solving the multivariate polynomial system resulting from the first-order optimality conditions. Given the poor scalability of the original WLS problem, we propose an alternative formulation of the WLS problem in terms of squared distance measurements (squared distances WLS or SD-WLS). Using a hybrid algebraic-numeric technique, we are able to solve the corresponding first-order optimality conditions of the SD-WLS in 125 ms in Matlab. Both methods solve the minimal (3 distance measurements) as well as the overdetermined problem (more than 3 measurements) in a
3D Relative Pose Estimation from Six Distances
"... Abstract—In this paper, we present three fast, hybrid numericalgebraic methods to solve polynomial systems in floating point representation, based on the eigendecomposition of a so-called multiplication matrix. In particular, these methods run using standard double precision, use only linear algebra ..."
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Cited by 7 (7 self)
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Abstract—In this paper, we present three fast, hybrid numericalgebraic methods to solve polynomial systems in floating point representation, based on the eigendecomposition of a so-called multiplication matrix. In particular, these methods run using standard double precision, use only linear algebra packages, and are easy to implement. We provide the proof that these methods do indeed produce valid multiplication matrices, and show their relationship. As a specific application, we use our algorithms to compute the 3D relative translation and orientation between two robots, based on known egomotion and six robotto-robot distance measurements. Equivalently, the same system of equations arises when solving the forward kinematics of the general Stewart-Gough mechanism. Our methods can find all 40 solutions, trading off speed (0.08s to 1.5s, depending on the choice of method) for accuracy.
Two Efficient Solutions for Visual Odometry Using Directional Correspondence
, 2007
"... This paper presents two new, efficient solutions to the two-view, relative pose problem from three image point correspondences and one common reference direction. This three-plus-one problem can be used either as a substitute for the classic five-point algorithm using a vanishing point for the refer ..."
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Cited by 5 (1 self)
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This paper presents two new, efficient solutions to the two-view, relative pose problem from three image point correspondences and one common reference direction. This three-plus-one problem can be used either as a substitute for the classic five-point algorithm using a vanishing point for the reference direction, or to make use of an inertial measurement unit commonly available on robots and mobile devices, where the gravity vector becomes the reference direction. We provide a simple, closed-form solution and a solution based on algebraic geometry which offers numerical advantages. In addition, we introduce a new method for computing visual odometry with RANSAC and four point correspondences per hypothesis. In a set of real experiments, we demonstrate the power of our approach by comparing it to the five-point method in a hypothesize-and-test visual odometry setting.
Optimizing Polynomial Solvers for Minimal Geometry Problems
"... In recent years polynomial solvers based on algebraic geometry techniques, and specifically the action matrix method, have become popular for solving minimal problems in computer vision. In this paper we develop a new method for reducing the computational time and improving numerical stability of al ..."
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Cited by 3 (2 self)
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In recent years polynomial solvers based on algebraic geometry techniques, and specifically the action matrix method, have become popular for solving minimal problems in computer vision. In this paper we develop a new method for reducing the computational time and improving numerical stability of algorithms using this method. To achieve this, we propose and prove a set of algebraic conditions which allow us to reduce the size of the elimination template (polynomial coefficient matrix), which leads to faster LU or QR decomposition. Our technique is generic and has potential to improve performance of many solvers that use the action matrix method. We demonstrate the approach on specific examples, including an image stitching algorithm where computation time is halved and single precision arithmetic can be used. 1.
COOPERATIVE LOCALIZATION: ON MOTION-INDUCED INITIALIZATION AND JOINT STATE ESTIMATION UNDER COMMUNICATION CONSTRAINTS
"... This thesis would not have been possible without the support of a number of people. First of all, my thanks go to my adviser, Professor Stergios Roumeliotis, for his constant encouragement and guidance, for the long hours of passing along his knowledge and experience, for pushing me beyond my own li ..."
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Cited by 3 (0 self)
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This thesis would not have been possible without the support of a number of people. First of all, my thanks go to my adviser, Professor Stergios Roumeliotis, for his constant encouragement and guidance, for the long hours of passing along his knowledge and experience, for pushing me beyond my own limitations, and for his seemingly endless supply of interesting research problems. I am also thankful for the time and invaluable advice from
