Results 1  10
of
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 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
Optimal algorithms in multiview geometry
 IN: ASIAN CONF. COMPUTER VISION
, 2007
"... This is a survey paper summarizing recent research aimed at finding guaranteed optimal algorithms for solving problems in Multiview Geometry. Many of the traditional problems in Multiview Geometry now have optimal solutions in terms of minimizing residual imageplane error. Success has been achieved ..."
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Cited by 30 (7 self)
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This is a survey paper summarizing recent research aimed at finding guaranteed optimal algorithms for solving problems in Multiview Geometry. Many of the traditional problems in Multiview Geometry now have optimal solutions in terms of minimizing residual imageplane error. Success has been achieved in minimizing L2 (leastsquares) or L∞ (smallest maximum error) norm. The main methods involve Second Order Cone Programming, or quasiconvex optimization, and Branchandbound. The paper gives an overview of the subject while avoiding as far as possible the mathematical details, which can be found in the original papers.
Fast and Robust Numerical Solutions to Minimal Problems for Cameras with Radial Distortion
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Polynomial Eigenvalue Solutions to the 5pt and 6pt Relative Pose Problems
"... In this paper we provide new fast and simple solutions to two important minimal problems in computer vision, the fivepoint relative pose problem and the sixpoint 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 fivepoint relative pose problem and the sixpoint 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 stateoftheart 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
Motion estimation for nonoverlapping multicamera rigs: Linear algebraic and l∞ geometric solutions
 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE
, 2010
"... We investigate the problem of estimating the egomotion of a multicamera rig from two positions of the rig. We describe and compare two new algorithms for finding the 6 degrees of freedom (3 for rotation and 3 for translation) of the motion. One algorithm gives a linear solution and the other is a ..."
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Cited by 14 (4 self)
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We investigate the problem of estimating the egomotion of a multicamera rig from two positions of the rig. We describe and compare two new algorithms for finding the 6 degrees of freedom (3 for rotation and 3 for translation) of the motion. One algorithm gives a linear solution and the other is a geometric algorithm that minimizes the maximum measurement error—the optimal L1 solution. They are described in the context of the General Camera Model (GCM), and we pay particular attention to multicamera systems in which the cameras have nonoverlapping or minimally overlapping field of view. Many nonlinear algorithms have been developed to solve the multicamera motion estimation problem. However, no linear solution or guaranteed optimal geometric solution has previously been proposed. We made two contributions: 1) a fast linear algebraic method using the GCM and 2) a guaranteed globally optimal algorithm based on the L1 geometric error using the branchandbound technique. In deriving the linear method using the GCM, we give a detailed analysis of degeneracy of camera configurations. In finding the globally optimal solution, we apply a rotation space search technique recently proposed by Hartley and Kahl. Our experiments conducted on both synthetic and real data have shown excellent results.
A ColumnPivoting Based Strategy for Monomial Ordering in Numerical Gröbner Basis Calculations
"... Abstract. This paper presents a new fast approach to improving stability in polynomial equation solving. Gröbner basis techniques for equation solving have been applied successfully to several geometric computer vision problems. However, in many cases these methods are plagued by numerical problems. ..."
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Cited by 10 (3 self)
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Abstract. This paper presents a new fast approach to improving stability in polynomial equation solving. Gröbner basis techniques for equation solving have been applied successfully to several geometric computer vision problems. However, in many cases these methods are plagued by numerical problems. An interesting approach to stabilising the computations is to study basis selection for the quotient space C[x]/I. In this paper, the exact matrix computations involved in the solution procedure are clarified and using this knowledge we propose a new fast basis selection scheme based on QRfactorization with column pivoting. We also propose an adaptive scheme for truncation of the Gröbner basis to further improve stability. The new basis selection strategy is studied on some of the latest reported uses of Gröbner basis methods in computer vision and we demonstrate a fourfold increase in speed and nearly as good overall precision as the previous SVDbased method. Moreover, we get typically get similar or better reduction of the largest errors. 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 6point 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 largescale 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.
A threepoint minimal solution for panoramic stitching with lens distortion
 In Proceeding of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR
, 2008
"... We present a minimal solution for aligning two images taken by a rotating camera from point correspondences. The solution particularly addresses the case where there is lens distortion in the images. We assume to know the two camera centers but not the focal lengths and allow the latter to vary. Our ..."
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Cited by 8 (0 self)
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We present a minimal solution for aligning two images taken by a rotating camera from point correspondences. The solution particularly addresses the case where there is lens distortion in the images. We assume to know the two camera centers but not the focal lengths and allow the latter to vary. Our solution uses a minimal number (three) of point correspondences and is well suited to be used in a hypothesis testing framework. It does not suffer from numerical instabilities observed in other algebraic minimal solvers and is also efficient. We validate our solution in multiimage panoramic stitching on real images with lens distortion. 1.
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 socalled 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 socalled 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 robottorobot distance measurements. Equivalently, the same system of equations arises when solving the forward kinematics of the general StewartGough mechanism. Our methods can find all 40 solutions, trading off speed (0.08s to 1.5s, depending on the choice of method) for accuracy.