| 1. |
- Aanæs, Henrik, et al.
(författare)
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Camera Resectioning from a Box
- 2009
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Ingår i: Lecture Notes in Computer Science. - 0302-9743 .- 1611-3349.
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Konferensbidrag (refereegranskat)abstract
- In this paper we describe how we can do camera resectioning from a box with unknown dimensions, i.e. determine the camera model, assuming that image pixels are square. This assumption is equivalent to assuming that the camera as an aspect ratio of one and zero skew, and holds for most - if not all - digital cameras. Our proposed method works by first deriving 9 linear constraints on the projective camera matrix from the box, leaving a 3 dimensional subspace in which the projective camera matrix can lye. A single solution in this 3D subspace is then found via a method by Triggs in 1999, which uses the squared pixel assumption to set up a 4th degree polynomial to which the solution is the desired model. This approach is, however, numerically challenging, and we use several means to combat this issue. Lastly the solution is refined in an iterative manner, i.e. using bundle adjustment.
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| 2. |
- Byröd, Martin, et al.
(författare)
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A Column-Pivoting Based Strategy for Monomial Ordering in Numerical Gröbner Basis Calculations
- 2008
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Ingår i: Lecture Notes in Computer Science. - Berlin, Heidelberg : Springer Berlin Heidelberg. - 1611-3349 .- 0302-9743. ; 5305, s. 130-143
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Konferensbidrag (refereegranskat)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 QR-factorization 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 SVD-based method. Moreover, we get typically get similar or better reduction of the largest errors.
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| 3. |
- Byröd, Martin, et al.
(författare)
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Fast and Robust Numerical Solutions to Minimal Problems for Cameras with Radial Distortion
- 2008
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Ingår i: [Host publication title missing]. ; , s. 2586-2593
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Konferensbidrag (refereegranskat)abstract
- A number of minimal problems of structure from motion for cameras with radial distortion have recently been studied and solved in some cases. These problems are known to be numerically very challenging and in several cases there exist no known practical algorithm yielding solutions in floating point arithmetic. We make some crucial observations concerning the floating point implementation of Gröbner basis computations and use these new insights to formulate fast and stable algorithms for two minimal problems with radial distortion previously solved in exact rational arithmetic only: (i) simultaneous estimation of essential matrix and a common radial distortion parameter for two partially calibrated views and six image point correspondences and (ii) estimation of fundamental matrix and two different radial distortion parameters for two uncalibrated views and nine image point correspondences. We demonstrate on simulated and real experiments that these two problems can be efficiently solved in floating point arithmetic.
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| 4. |
- Byröd, Martin, et al.
(författare)
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Fast and Stable Polynomial Equation Solving and Its Application to Computer Vision
- 2009
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Ingår i: International Journal of Computer Vision. - : Springer Science and Business Media LLC. - 1573-1405 .- 0920-5691. ; 84:3, s. 237-256
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Tidskriftsartikel (refereegranskat)abstract
- This paper presents several new results on techniques for solving systems of polynomial equations in computer vision. 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. In this paper we derive a generalization of the Gröbner basis method for polynomial equation solving, which improves overall numerical stability. We show how the action matrix can be computed in the general setting of an arbitrary linear basis for ℂ[x]/I. In particular, two improvements on the stability of the computations are made by studying how the linear basis for ℂ[x]/I should be selected. The first of these strategies utilizes QR factorization with column pivoting and the second is based on singular value decomposition (SVD). Moreover, it is shown how to improve stability further by an adaptive scheme for truncation of the Gröbner basis. These new techniques are studied on some of the latest reported uses of Gröbner basis methods in computer vision and we demonstrate dramatically improved numerical stability making it possible to solve a larger class of problems than previously possible.
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| 5. |
- Byröd, Martin, et al.
(författare)
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Fast Optimal Three View Triangulation
- 2007
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Ingår i: Lecture Notes in Computer Science. - Berlin, Heidelberg : Springer Berlin Heidelberg. - 9783540763895 ; 4844, s. 549-559
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Konferensbidrag (refereegranskat)abstract
- We consider the problem of $L_2$-optimal triangulation from three separate views. Triangulation is an important part of numerous computer vision systems. Under gaussian noise, minimizing the $L_2$ norm of the reprojection error gives a statistically optimal estimate. This has been solved for two views. However, for three or more views, it is not clear how this should be done. A previously proposed, but computationally impractical, method draws on Gr{"o}bner basis techniques to solve for the complete set of stationary points of the cost function. We show how this method can be modified to become significantly more stable and hence given a fast implementation in standard IEEE double precision. We evaluate the precision and speed of the new method on both synthetic and real data. The algorithm has been implemented in a freely available software package which can be downloaded from the Internet.
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| 6. |
- Byröd, Martin, et al.
(författare)
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Fast Optimal Three View Triangulation
- 2008
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Ingår i: ; , s. 95-98
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Konferensbidrag (övrigt vetenskapligt/konstnärligt)abstract
- We consider the problem of L2-optimal triangulation from three separate views. Triangulation is an important part of numerous computer vision systems. Under gaussian noise, minimizing the L2 norm of the reprojection error gives a statistically optimal estimate. This has been solved for two views. However, for three or more views, it is not clear how this should be done. A previously proposed, but computationally impractical, method draws on Gröbner basis techniques to solve for the complete set of stationary points of the cost function. We show how this method can be modified to become significantly more stable and hence given a fast implementation in standard IEEE double precision. We evaluate the precision and speed of the new method on both synthetic and real data. The algorithm has been implemented in a freely available software package which can be downloaded from the Internet.
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| 7. |
- Byröd, Martin, et al.
(författare)
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Improving numerical accuracy of Grobner basis polynomial equation solvers
- 2007
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Ingår i: Proceedings of the IEEE 11th International Conference on Computer Vision. - 1550-5499. - 9781424416318 ; , s. 449-456
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Konferensbidrag (refereegranskat)abstract
- This paper presents techniques for improving the numerical stability of Grobner basis solvers for polynomial equations. Recently Grobner basis methods have been used succesfully to solve polynomial equations arising in global optimization e.g. three view triangulation and in many important minimal cases of structure from motion. Such methods work extremely well for problems of reasonably low degree, involving a few variables. Currently, the limiting factor in using these methods for larger and more demanding problems is numerical difficulties. In the paper we (i) show how to change basis in the quotient space R[x]/I and propose a strategy for selecting a basis which improves the conditioning of a crucial elimination step, (ii) use this technique to devise a Grobner basis with improved precision and (iii) show how solving for the eigenvalues instead of eigenvectors can be used to improve precision further while retaining the same speed. We study these methods on some of the latest reported uses of Grobner basis methods and demonstrate dramatically improved numerical precision using these new techniques making it possible to solve a larger class of problems than previously.
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| 8. |
- Enqvist, Olof, et al.
(författare)
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Optimal Correspondences from Pairwise Constraints
- 2009
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Ingår i: IEEE International Conference on Computer Vision. - 1550-5499. - 9781424444199 ; , s. 1295-1302
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Konferensbidrag (refereegranskat)abstract
- Correspondence problems are of great importance in computer vision. They appear as subtasks in many applications such as object recognition, merging partial 3D reconstructions and image alignment. Automatically matching features from appearance only is difficult and errors are frequent. Thus, it is necessary to use geometric consistency to remove incorrect correspondences. Typically heuristic methods like RANSAC or EM-like algorithms are used, but they risk getting trapped in local optima and are in no way guaranteed to find the best solution. This paper illustrates how pairwise constraints in combination with graph methods can be used to efficiently find optimal correspondences. These ideas are implemented on two basic geometric problems, 3D-3D registration and 2D-3D registration. The developed scheme can handle large rates of outliers and cope with multiple hypotheses. Despite the combinatorial explosion, the resulting algorithm which has been extensively evaluated on real data, yields competitive running times compared to state of the art
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| 9. |
- Josephson, Klas, et al.
(författare)
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Image-based localization using hybrid feature correspondences
- 2007
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Ingår i: Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition. - 1063-6919. ; , s. 2732-2739
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Konferensbidrag (refereegranskat)abstract
- Where am I and what am I seeing? This is a classical vision problem and this paper presents a solution based on efficient use of a combination of 2D and 3D features. Given a model of a scene, the objective is to find the relative camera location of a new input image. Unlike traditional hypothesize-and-test methods that try to estimate the unknown camera position based on 3D model features only, or alternatively, based on 2D model features only, we show that using a mixture of such features, that is, a hybrid correspondence set, may improve performance. We use minimal cases of structure-from-motion for hypothesis generation in a RANSAC engine. For this purpose, several new and useful minimal cases are derived for calibrated, semi-calibrated and uncalibrated settings. Based on algebraic geometry methods, we show how these minimal hybrid cases can be solved efficiently. The whole approach has been validated on both synthetic and real data, and we demonstrate improvements compared to previous work. © 2007 IEEE.
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| 10. |
- Josephson, Klas
(författare)
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Mathematical Methods for Image Based Localization
- 2010
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- The underlying question in localization is, where am I? In this thesis a purely image based approach is proposed to solve this problem. In order to create a complete image based system, there are many subproblems that have to be addressed. The localization problem can also be solved in other ways, for example, with a GPS. Two advantages with using images compared to GPS are that no open sky is needed and that a higher precision is possible to achieve. The thesis consists of an introductory chapter followed by six papers. In the first paper, enhancements of Gröbner basis techniques to solve systems of polynomial equations are presented. The new strategies improve the numeric stability with several orders of magnitudes, compared to previous state of the art. This framework is then applied in the next three papers to solve several geometrical pose problems relevant for localization. The main difference between the papers is the level of knowledge of the inner calibration of the cameras. The calibration knowledge ranges from completely calibrated cameras to uncalibrated cameras with unknown radial distortion. The fifth paper of the thesis also treats the pose problem, but the method differs from the previous papers. In this paper a method is presented that guarantees a globally optimal solution at the price of computational complexity. To achieve this, the pose problem is reformulated and solved via a minimal vertex cover. The final paper is devoted to large-scale localization. Methods from image retrieval are utilized, and extended, to be able to perform city-scale localization. Moreover geometry is directly incorporated in the retrieval stage.
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