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25
NLEVP: A Collection of Nonlinear Eigenvalue Problems
, 2010
"... We present a collection of 46 nonlinear eigenvalue problems in the form of a MATLAB toolbox. The collection contains problems from models of reallife applications as well as ones constructed specifically to have particular properties. A classification is given of polynomial eigenvalue problems acco ..."
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Cited by 49 (12 self)
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We present a collection of 46 nonlinear eigenvalue problems in the form of a MATLAB toolbox. The collection contains problems from models of reallife applications as well as ones constructed specifically to have particular properties. A classification is given of polynomial eigenvalue problems according to their structural properties. Identifiers based on these and other properties can be used to extract particular types of problems from the collection. A brief description of each problem is given. NLEVP serves both to illustrate the tremendous variety of applications of nonlinear Eigenvalue problems and to provide representative problems for testing, tuning, and benchmarking of algorithms and codes.
A minimal solution to the autocalibration of radial distortion
, 2007
"... Epipolar geometry and relative camera pose computation are examples of tasks which can be formulated as minimal problems and solved from a minimal number of image points. Finding the solution leads to solving systems of algebraic equations. Often, these systems are not trivial and therefore special ..."
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Cited by 39 (12 self)
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Epipolar geometry and relative camera pose computation are examples of tasks which can be formulated as minimal problems and solved from a minimal number of image points. Finding the solution 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. In this paper we provide a solution to the problem of estimating radial distortion and epipolar geometry from eight correspondences in two images. Unlike previous algorithms, which were able to solve the problem from nine correspondences only, we enforce the determinant of the fundamental matrix be zero. This leads to a system of eight quadratic and one cubic equation in nine variables. We simplify the system by eliminating six of these variables. Then, we solve the system by finding eigenvectors of an action matrix of a suitably chosen polynomial. We show how to construct the action matrix without computing complete Gröbner basis, which provides an efficient and robust solver. The quality of the solver is demonstrated on synthetic and real data. 1.
A general solution to the P4P problem for camera with unknown focal length
, 2008
"... This paper presents a general solution to the determination of the pose of a perspective camera with unknown focal length from images of four 3D reference points. Our problem is a generalization of the P3P and P4P problems previously developed for fully calibrated cameras. Given four 2Dto3D corres ..."
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Cited by 33 (8 self)
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This paper presents a general solution to the determination of the pose of a perspective camera with unknown focal length from images of four 3D reference points. Our problem is a generalization of the P3P and P4P problems previously developed for fully calibrated cameras. Given four 2Dto3D correspondences, we estimate camera position, orientation and recover the camera focal length. We formulate the problem and provide a minimal solution from four points by solving a system of algebraic equations. We compare the Hidden variable resultant and Gröbner basis techniques for solving the algebraic equations of our problem. By evaluating them on synthetic and on realdata, we show that the Gröbner basis technique provides stable results.
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 31 (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 nonexperts 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
HandEye Calibration without Hand Orientation Measurement Using Minimal Solution
"... Abstract. In this paper we solve the problem of estimating the relative pose between a robot’s gripper and a camera mounted rigidly on the gripper in situations where the rotation of the gripper w.r.t. the robot global coordinate system is not known. It is a variation of the so called handeye calib ..."
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Abstract. In this paper we solve the problem of estimating the relative pose between a robot’s gripper and a camera mounted rigidly on the gripper in situations where the rotation of the gripper w.r.t. the robot global coordinate system is not known. It is a variation of the so called handeye calibration problem. We formulate it as a problem of seven equations in seven unknowns and solve it using the Gröbner basis method for solving systems of polynomial equations. This enables us to calibrate from the minimal number of two relative movements and to provide the first exact algebraic solution to the problem. Further, we describe a method for selecting the geometrically correct solution among the algebraically correct ones computed by the solver. In contrast to the previous iterative methods, our solution works without any initial estimate and has no problems with error accumulation. Finally, by evaluating our algorithm on both synthetic and real scene data we demonstrate that it is fast, noise resistant, and numerically stable. 1
Singlybordered blockdiagonal form for minimal problem solvers
 In ACCV’14, 2014. 8
"... Abstract. The Grobner basis method for solving systems of polynomial equations became very popular in the computer vision community as it helps to nd fast and numerically stable solutions to difficult problems. In this paper, we present a method that potentially signicantly speeds up Grobner basis s ..."
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Cited by 2 (2 self)
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Abstract. The Grobner basis method for solving systems of polynomial equations became very popular in the computer vision community as it helps to nd fast and numerically stable solutions to difficult problems. In this paper, we present a method that potentially signicantly speeds up Grobner basis solvers. We show that the elimination template matrices used in these solvers are usually quite sparse and that by permuting the rows and columns they can be transformed into matrices with nice blockdiagonal structure known as the singlybordered blockdiagonal (SBBD) form. The diagonal blocks of the SBBD matrices constitute independent subproblems and can therefore be solved, i.e. eliminated or factored, independently. The computational time can be further reduced on a parallel computer by distributing these blocks to different processors for parallel computation. The speedup is visible also for serial processing since we perform O(n3) GaussJordan eliminations on smaller (usually two, approximately n 2 n
Fast and Robust Numerical Solutions to Minimal Problems for Cameras with Radial Distortion
"... ..."
Two minimal problems for cameras with radial distortion
 In OMNIVIS 2007, Rio de Janeiro
"... Epipolar geometry and relative camera pose computation for uncalibrated cameras with radial distortion has recently been formulated as a minimal problem and successfully solved in floating point arithmetics. The singularity of the fundamental matrix has been used to reduce the minimal number of poin ..."
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Cited by 18 (4 self)
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Epipolar geometry and relative camera pose computation for uncalibrated cameras with radial distortion has recently been formulated as a minimal problem and successfully solved in floating point arithmetics. The singularity of the fundamental matrix has been used to reduce the minimal number of points to eight. It was assumed that the cameras were not calibrated but had same distortions. In this paper we formulate two new minimal problems for estimating epipolar geometry of cameras with radial distortion. First we present a minimal algorithm for partially calibrated cameras with same radial distortion. Using the trace constraint which holds for the epipolar geometry of calibrated cameras to reduce the number of necessary points from eight to six. We demonstrate that the problem is solvable in exact rational arithmetics. Secondly, we present a minimal algorithm for uncalibrated cameras with different radial distortions. We show that the problem can be solved using nine points in two views by manipulating polynomials by a sequence of GaussJordan eliminations in exact rational arithmetics. We demonstrate the algorithms on synthetic and real data. 1.
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