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19
Polynomial shape from shading
 CVPR
"... We examine the shape from shading problem without boundary conditions as a polynomial system. This view allows, in generic cases, a complete solution for ideal polyhedral objects. For the general case we propose a semidefinite programming relaxation procedure, and an exact line search iterative proc ..."
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Cited by 8 (1 self)
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We examine the shape from shading problem without boundary conditions as a polynomial system. This view allows, in generic cases, a complete solution for ideal polyhedral objects. For the general case we propose a semidefinite programming relaxation procedure, and an exact line search iterative procedure with a new smoothness term that favors folds at edges. We use this numerical technique to inspect shading ambiguities. 1.
Two Efficient Solutions for Visual Odometry Using Directional Correspondence
, 2007
"... This paper presents two new, efficient solutions to the twoview, relative pose problem from three image point correspondences and one common reference direction. This threeplusone problem can be used either as a substitute for the classic fivepoint 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 twoview, relative pose problem from three image point correspondences and one common reference direction. This threeplusone problem can be used either as a substitute for the classic fivepoint 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, closedform 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 fivepoint method in a hypothesizeandtest 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.
Robust fitting for multiple view geometry
 In European Conference on Computer Vision
, 2012
"... Abstract. How hard are geometric vision problems with outliers? We show that for most fitting problems, a solution that minimizes the number of outliers can be found with an algorithm that has polynomial timecomplexity in the number of points (independent of the rate of outliers). Further, and perha ..."
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Abstract. How hard are geometric vision problems with outliers? We show that for most fitting problems, a solution that minimizes the number of outliers can be found with an algorithm that has polynomial timecomplexity in the number of points (independent of the rate of outliers). Further, and perhaps more interestingly, other cost functions such as the truncated L2norm can also be handled within the same framework with the same time complexity. We apply our framework to triangulation, relative pose problems and stitching, and give several other examples that fulfill the required conditions. Based on efficient polynomial equation solvers, it is experimentally demonstrated that these problems can be solved reliably, in particular for lowdimensional models. Comparisons to standard random sampling solvers are also given. 1
Exploiting pFold Symmetries for Faster Polynomial Equation Solving
"... Numerous geometric problems in computer vision involve the solution of systems of polynomial equations. This is true for problems with minimal information, but also for finding stationary points for overdetermined problems. The stateoftheart is based on the use of numerical linear algebra on the ..."
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Numerous geometric problems in computer vision involve the solution of systems of polynomial equations. This is true for problems with minimal information, but also for finding stationary points for overdetermined problems. The stateoftheart is based on the use of numerical linear algebra on the large but sparse coefficient matrix that represents the expanded original equation set. In this paper we present two simplifications that can be used (i) if the zero vector is one of the solutions or (ii) if the equations display certain pfold symmetries. We evaluate the simplifications on a few example problems and demonstrate that significant speed increases are possible without loosing accuracy. 1.
Structure from Motion with Directional Correspondence for Visual Odometry
"... This report presents two efficient solutions to the twoview, relative pose problem from three image point correspondences and one common reference direction. This threeplusone problem can be used either as a substitute for the classic fivepoint algorithm using a vanishing point for the reference ..."
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Cited by 1 (1 self)
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This report presents two efficient solutions to the twoview, relative pose problem from three image point correspondences and one common reference direction. This threeplusone problem can be used either as a substitute for the classic fivepoint 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 closedform solution and a solution based on techniques from algebraic geometry and investigate numerical and computational advantages of each approach. In a set of real experiments, we demonstrate the power of our approach by comparing it to the fivepoint method in a hypothesizeandtest visual odometry setting.
Optimal geometric fitting under the truncated L2norm
 In CVPR
, 2013
"... This paper is concerned with model fitting in the presence of noise and outliers. Previously it has been shown that the number of outliers can be minimized with polynomial complexity in the number of measurements. This paper improves on these results in two ways. First, it is shown that for a lar ..."
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This paper is concerned with model fitting in the presence of noise and outliers. Previously it has been shown that the number of outliers can be minimized with polynomial complexity in the number of measurements. This paper improves on these results in two ways. First, it is shown that for a large class of problems, the statistically more desirable truncated L2norm can be optimized with the same complexity. Then, with the same methodology, it is shown how to transform multimodel fitting into a purely combinatorial problem—with worstcase complexity that is polynomial in the number of measurements, though exponential in the number of models. We apply our framework to a series of hard registration and stitching problems demonstrating that the approach is not only of theoretical interest. It gives a practical method for simultaneously dealing with measurement noise and large amounts of outliers for fitting problems with lowdimensional models. 1.
Towards a minimal solution for the relative pose between axial cameras
"... An axial camera is a particular case of a noncentral camera where every backprojection ray intersects a line in 3D (the axis). The axial camera can be used to model vision systems and imaging situations of practical interest. Examples include any catadioptric system that combines a revolution mir ..."
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An axial camera is a particular case of a noncentral camera where every backprojection ray intersects a line in 3D (the axis). The axial camera can be used to model vision systems and imaging situations of practical interest. Examples include any catadioptric system that combines a revolution mirror with a central camera for which the viewpoint is aligned with the mirror axis (e.g. a pinhole looking at a spherical mirror) The relative pose problem has 6 unknowns meaning that in theory 6 point correspondences provide enough information for determining the relative rotation and translation of the axial camera. Stewenius et al. proposed in [6] a minimal solution for the relative pose between generalized cameras. However, their algorithm is complex, provides a large number of possible solutions (up to 64), and, as reported in Given that all backprojection rays of an axial camera intersect its axis, they belong to a linear line congruent of dimension 4 [5]. This means that all rays can be represented by 5 dimensional coordinate vectors λ i that are a linear combination of 5 base lines aligned with the axes x, y, z,ŷ,ẑ in Given a set of intersecting ray correspondences (λ i , λ i ), we can establish linear relations with the form with Φ being a 5×5 matrix that encodes the 4 essential matrices displayed in The matrix Φ has 17 free parameters, and therefore can be linearly estimated from 16 correspondences. Additionally, the following family of matrices must have the properties of an essential matrix, and therefore verify the following nonlinear constraints This makes us able to solve the problem using just 10 correspondences (λ i , λ i ) by first generating a 7 dimensional linear subspace for Φ and then
Centre for Mathematical Sciences
"... This paper presents a solution to panoramic image stitching of two images with coinciding optical centers, but unknown focal length and radial distortion. The algorithm operates with a minimal set of corresponding points (three) which means that it is well suited for use in any RANSAC style algorith ..."
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This paper presents a solution to panoramic image stitching of two images with coinciding optical centers, but unknown focal length and radial distortion. The algorithm operates with a minimal set of corresponding points (three) which means that it is well suited for use in any RANSAC style algorithm for simultaneous estimation of geometry and outlier rejection. Compared to a previous method for this problem, we are able to guarantee that the right solution is found in all cases. The solution is obtained by solving a small system of polynomial equations. The proposed algorithm has been integrated in a complete multi image stitching system and we evaluate its performance on real images with lens distortion. We demonstrate both quantitative and qualitative improvements compared to state of the art methods. 1