iccv iccv2013 iccv2013-115 knowledge-graph by maker-knowledge-mining
Source: pdf
Author: Laurent Kneip, Simon Lynen
Abstract: This work makes use of a novel, recently proposed epipolar constraint for computing the relative pose between two calibrated images. By enforcing the coplanarity of epipolar plane normal vectors, it constrains the three degrees of freedom of the relative rotation between two camera views directly—independently of the translation. The present paper shows how the approach can be extended to n points, and translated into an efficient eigenvalue minimization over the three rotational degrees of freedom. Each iteration in the non-linear optimization has constant execution time, independently of the number of features. Two global optimization approaches are proposed. The first one consists of an efficient Levenberg-Marquardt scheme with randomized initial value, which already leads to stable and accurate results. The second scheme consists of a globally optimal branch-and-bound algorithm based on a bound on the eigenvalue variation derived from symmetric eigenvalue-perturbation theory. Analysis of the cost function reveals insights into the nature of a specific relative pose problem, and outlines the complexity under different conditions. The algorithm shows state-of-the-art performance w.r.t. essential-matrix based solutions, and a frameto-frame application to a video sequence immediately leads to an alternative, real-time visual odometry solution. Note: All algorithms in this paper are made available in the OpenGV library. Please visit http : / / l aurent kne ip .github . i / opengv o
Reference: text
sentIndex sentText sentNum sentScore
1 au Abstract This work makes use of a novel, recently proposed epipolar constraint for computing the relative pose between two calibrated images. [sent-4, score-0.449]
2 By enforcing the coplanarity of epipolar plane normal vectors, it constrains the three degrees of freedom of the relative rotation between two camera views directly—independently of the translation. [sent-5, score-1.011]
3 The present paper shows how the approach can be extended to n points, and translated into an efficient eigenvalue minimization over the three rotational degrees of freedom. [sent-6, score-0.337]
4 Each iteration in the non-linear optimization has constant execution time, independently of the number of features. [sent-7, score-0.225]
5 The second scheme consists of a globally optimal branch-and-bound algorithm based on a bound on the eigenvalue variation derived from symmetric eigenvalue-perturbation theory. [sent-10, score-0.63]
6 Analysis of the cost function reveals insights into the nature of a specific relative pose problem, and outlines the complexity under different conditions. [sent-11, score-0.328]
7 Introduction The computation of the relative pose between two planar projections of a scene is certainly one of the most studied problems in geometric vision and—more generally— structure from motion. [sent-20, score-0.263]
8 ch Most common solutions to the calibrated relative pose problem make use of the essential matrix parametrization, which can result in a linear solution, depending on the number of employed correspondences. [sent-25, score-0.583]
9 However, there remains a number of problems linked to the indirect essential matrix parametrization: • Mixing of parameters: The parameters of the essentMiailx imngatr oixf daor nmoet represent pmaroatmioent-ererslat oefd t hvear eiasbselensdirectly, but rather functions of these variables. [sent-27, score-0.247]
10 Haolw soelvveerr,s even froalrl a single essential matrix the decomposition still leads to multiple solutions. [sent-30, score-0.203]
11 The functional mapping between the motion variables—namely rotation and translation— and the essential matrix is not uniquely invertible. [sent-31, score-0.55]
12 • Zero translation: Although most essential-matrix bZaersoed tsroalnustliaontios implicitly solve correctly for rotation in the zero-translation and noise-free situation, the constraint as such deteriorates. [sent-39, score-0.288]
13 The present paper sheds a new light on the relative pose problem by employing a different epipolar constraint not suffering from any of the above mentioned problems. [sent-43, score-0.392]
14 Instead of 22335522 solving for all the motion parameters indirectly, we show how the relative pose problem can be turned into an eigenvalue minimization directly over the three parameters of the frame-to-frame rotation. [sent-44, score-0.508]
15 The relative translation results implicitly in form of the corresponding eigenvector. [sent-45, score-0.272]
16 The result is an elegant and simple non-linear optimization over three parameters only, and the cost function typically shows only a small number of geometrically meaningful local minima besides the globally optimal solution. [sent-46, score-0.311]
17 Unlike the essential matrix constraint, the cost function maintains a basin of attraction in the zero-translation situation, and the algorithm can be solved with constant iteration time independently of the number of features. [sent-47, score-0.445]
18 Despite being an iterative approach, the proposed algorithm therefore turns out to be a valid al- ternative to existing calibrated relative pose algorithms, directly applicable to frame-to-frame motion estimation. [sent-48, score-0.437]
19 Related work: The first known solution to the relative pose problem dates back to 1913 and has been presented by Kruppa [9]. [sent-49, score-0.27]
20 While this parametrization still leads to a multivariate polynomial equation system with 11 solutions, more recent advances have shown that the actual number of solutions in the minimal case equals to 10. [sent-50, score-0.434]
21 This is for instance the case in the minimal solution of Nist e´r [16], who derives a tenth-order polynomial that is subsequently solved using Sturm’s root-bracketing approach. [sent-51, score-0.242]
22 [18], [10], and [11] represent alternative solutions to the problem using the Gr¨ obner basis, polynomial eigenvalue, or hidden variable resultant technique, respectively. [sent-52, score-0.212]
23 The solutions show minor differences in terms of accuracy and computational complexity, however all make use of the essential matrix parametrization, and thus suffer from the previously mentioned problems. [sent-53, score-0.308]
24 The linear solver by Longuet-Higgins [13]—later on defended by Hartley [4]—, is one of the first essential matrix based solutions. [sent-54, score-0.309]
25 It delivers a unique essential matrix but suffers from the previously mentioned planar degeneracy. [sent-55, score-0.289]
26 Although most algorithms are still able to find the rotation in the noise-free zero-translation situation, the inlier computation still depends on the translation, which becomes unobservable. [sent-56, score-0.34]
27 [12] compute the rotation independently ofthe translation but depend on a special distri- × bution of the feature correspondences, i. [sent-60, score-0.486]
28 Although their minimal solution depends on model selection, they show the general ability to compute the rotation independently of the translation. [sent-65, score-0.491]
29 In terms of global optimality, the present paper has analogies with the work of Hartley and Kahl [5], who also devise a branch-and-bound solution in rotation space. [sent-68, score-0.34]
30 Although only minimizing an algebraic error, our solution is more efficient: instead of finding the translation for each rotation via second-order cone programming, we only have to compare the closed-form bound on the eigenvalue variation of a 3 3 matrix. [sent-69, score-0.959]
31 We thus obtain also a continuous equivalent to [19] and [21], which represent minimal and non-minimal continuous epipolar solutions. [sent-73, score-0.265]
32 The presented work is also related to research conducted around direct optimization on the essential matrix manifold, such as Ma et al. [sent-74, score-0.306]
33 The main difference in our work is the cost function: While all previous works do an optimization over five degrees of free- dom, we prove in this work the existence of a different cost function that allows an efficient optimization over three degrees of freedom only. [sent-77, score-0.376]
34 We compare our results to this approach, and show the benefits of our method compared to essential matrix based parametrizations. [sent-81, score-0.203]
35 Organization of the paper: Section 2 outlines the formulation of the relative pose problem as an eigenvalue minimization problem. [sent-82, score-0.507]
36 Section 3 presents two approaches to find the solution, an efficient Levenberg-Marquardt scheme as well as a bound on the variation of the eigenvalue allowing the design of a branch-and-bound solution. [sent-83, score-0.491]
37 Section 4 shows comparative results of our algorithm including the possibility to identify the location of local minima in the cost function directly from the geometrical conditioning of the problem. [sent-84, score-0.206]
38 Theory This section outlines the main geometrical concept that allows us to constrain the rotation between different view- points independently of the translation, namely the coplanarity of epipolar plane normal vectors. [sent-87, score-0.921]
39 Using this constraint allows us to formulate the relative pose computation as an eigenvalue minimization problem. [sent-88, score-0.449]
40 We furthermore introduce a good parametrization of the rotation matrix for an efficient optimization, and reformulate the problem such that rotations may be validated in constant time, independently of the number of features. [sent-89, score-0.63]
41 Preliminaries We assume to be in the central, calibrated case such that each location in the image plane can be translated into a unique unit bearing vector originating from the camera center. [sent-92, score-0.366]
42 ) denotes a correspondence of bearing vectors pointing at the same 3D world point pi from two distinct view-points, where fi represents the observation from the first view-point, and fi? [sent-94, score-0.348]
43 The relative pose is given by the translation t—expressed in the first frame and denoting the position of the second frame w. [sent-96, score-0.356]
44 the first one—and the rotation R—transforming vectors from the second into the first frame. [sent-99, score-0.33]
45 Relative pose as an eigenvalue problem The epipolar plane of a correspondence is defined to be the plane that contains the two camera centers as well as the × observed 3D point. [sent-103, score-0.649]
46 The set of epipolar planes hence forms a pencil of planes all intersecting in the line of translation. [sent-104, score-0.25]
47 In other words, the normal vectors of the epipolar planes all need to be coplanar. [sent-105, score-0.334]
48 A normal vector in the pure translation situation is easily given by ni = f fi? [sent-106, score-0.272]
49 (1) The work of Kneip [8] proposes to enforce the coplanarity of triplets of normal vectors in order to come up with a minimal solution for translation independent computation of the relative rotation. [sent-110, score-0.681]
50 An interesting result around the novel epipolar constraint is that—when varying the rotation—virtual and notably non-coplanar epipolar plane normal vectors appear even in the zero-translation situation, which renders the constraint robust against vanishing translation magnitudes. [sent-111, score-0.726]
51 However, the Gr¨ obner basis solver presented in [8] still turns out to be unstable in this case, which is related to numerical instabilities in the fixed sequence of s-polynomials when the number of solutions changes. [sent-112, score-0.294]
52 In the following, we will present a more interesting n-point iterative optimization scheme that remains functional for any parallax. [sent-113, score-0.212]
53 The basic intuition to enforce the coplanarity of n epipolar plane normals consists of treating the set of normal vectors as a point cloud, and canceling the second moment or dilatation in one direction. [sent-114, score-0.52]
54 smallest eigenvalue of M, the final problem parametrization becomes N? [sent-129, score-0.459]
55 Bearing vectors (in red) from two view-points are the known variables, and the relative pose (in blue) is the searched unknown. [sent-134, score-0.26]
56 Note that M is a real symmetric and positive-definite matrix with rank at most 2. [sent-142, score-0.193]
57 The rotation having three degrees of freedom, this represents at best a non-linear optimization over three parameters only—depending on the rotation parametrization. [sent-146, score-0.686]
58 Minimal parametrization ofthe rotation matrix There exists a large number of possible rotation matrix parametrizations. [sent-149, score-0.879]
59 It represents a good choice because—in practice—the frameto-frame rotation does barely exceed 2π about any of the basis’ axes. [sent-152, score-0.288]
60 It therefore only affects the magnitude of the normal vector, and can be omitted in the coplanarity maximization (2). [sent-155, score-0.224]
61 Rendering the iteration time constant An important observation is that the rotation matrix can be factorized inside the expression (fi Rfi? [sent-159, score-0.433]
62 Second, we use eigenvalue perturbation theory in order to derive a bound on the variation of the smallest eigenvalue, finally enabling a globally optimal branch-andbound optimization of the relative rotation. [sent-279, score-0.839]
63 Local minima are avoided by a random variation of the starting point. [sent-296, score-0.244]
64 Branch and bound the variation of λM,min It is almost trivial to derive an absolute bound on the variation of the rotation matrix R based on an absolute bound ? [sent-299, score-1.015]
65 Using this bound, we can derive an absolute bound on the variation of the elements of M. [sent-301, score-0.273]
66 An important rTehseult p efrrtoumrb athtieo eigenvalue perturbation theory—the Weyltheorem as presented in [2]—tells us that the relative perturbation of the eigenvalues of M is then bounded by |λi,pertu|λrbie|d− λi|≤ ? [sent-308, score-0.639]
67 (5) In an aim to bound the spectral norm, we first of all note that the computation of the eigenvalues of a real symmetric positive-definite 3-by-3 matrix can be done very efficiently and in closed-form. [sent-311, score-0.363]
68 M−1/2M∗M−1/2 hence remains a real symmetric matrix (not necessarily positive definite). [sent-314, score-0.237]
69 Another important property from the spectral theorem then tells us that the spectral norm of a real symmetric matrix is given by the absolute value of its largest eigenvalue. [sent-315, score-0.257]
70 This implicitly bounds the largest eigenvalue of M−1/2M∗M−1/2 1M∗ is a real symmetric matrix where the absolute value of each entry is smMaller or equal to the corresponding entry in M. [sent-317, score-0.539]
71 A good bound on the absolute value of the roots of (6) is for example given by the Lagrangian bound |a2 | |a1|1/2 |a0 | 1/3 (conservatthivee approximation). [sent-320, score-0.3]
72 As expected, the bound on the eigenvalue variation converges to zero along with M. [sent-323, score-0.44]
73 It is however not very efficient due to the fact that the bound on the eigenvalue variation turns out to be fairly conservative. [sent-326, score-0.477]
74 Noise is added by assuming a spherical camera with a focal length of 800 pix, extracting the tangential plane of each bearing vector, and adding a uniformly distributed random offset expressed in pixels inside this plane. [sent-340, score-0.349]
75 The translational accuracy is expressed by the angular error of the normalized direction, and the error in rotation is expressed by the norm of the difference between both the estimated and the ground truth rotation vectors. [sent-343, score-0.666]
76 Behavior of the cost function Direct iterative optimization of the relative pose necessarily requires a comparison to standard two-view bundle adjustment, which depends on proper initialization. [sent-346, score-0.476]
77 The biggest difference lies in the dimensionality of the problem: While standard two-view bundle adjustment reduces the geometric error over 5 + 3n degrees of freedom (n being the number of correspondences), our approach minimizes an algebraic error function over 3 variables only. [sent-347, score-0.284]
78 As indicated in Figure 2, the smooth cost function typically shows only a small number of geometrically meaningful local minima in the neighbourhood of the globally optimal rotation. [sent-348, score-0.26]
79 random variation of the starting point), a direct optimization hence becomes possible under the practically valid assumption that the rotation between the view-points is bounded. [sent-351, score-0.522]
80 As illustrated in Figure 3, a purely translational displacement parallel to the image plane can cause a similar disparity than a pure rotation around an orthogonal axis in the image plane, and vice-versa. [sent-353, score-0.458]
81 We obtain a large basin of attraction around the global minimum in the case of omni-directional bearing vector distributions, which is why our algorithm is particularly well suited for omni-directional cameras. [sent-359, score-0.368]
82 Efficiency and noise resilience In this experiment, we use 10 random points without outliers and add different levels of noise. [sent-367, score-0.225]
83 The starting value for our iterative eigenvalue-based solver as well as non-linear optimization is set by a uniform variation of the true rotation 2. [sent-370, score-0.602]
84 Figure 4 shows that our parametrization induces lower errors, and clearly outperforms all other solutions for all tested noise levels. [sent-371, score-0.32]
85 Even though the execution time of our iterative solver depends on the number of iterations in the optimization, we at least note that it is real-time compliant. [sent-374, score-0.242]
86 01, which ensures that minimum for a proper evaluation of noise resilience we spot the global (a) noise [pix] (b) Figure 4. [sent-386, score-0.239]
87 Translation (a) and rotation (b) error for different noise levels and algorithms, each time averaged over 1000 random problems. [sent-387, score-0.366]
88 This also proves the ability of our iterative random variation scheme to find the global minimum. [sent-393, score-0.248]
89 Using more 3For 10 inlier correspondences and differing random displacements for both ground truth and initial transformation, our approach finds the global minimum in roughly 40% of all cases, compared to 20% for two-view bundle adjustment. [sent-398, score-0.277]
90 We compare the relative rotation accuracy of our approach and [16] against ground truth data delivered by a Vicon motion capture system. [sent-405, score-0.481]
91 The dataset starts with moderate translational displace- ments, then contains rotations about all three camera axes, and finally combined rotation and translation. [sent-425, score-0.42]
92 Errors are mainly caused by inhomogeneous bearing vector distributions in the pinhole-camera case, resulting in the previously mentioned ambiguities between rotational and translational motion estimation. [sent-426, score-0.425]
93 As indicated in Figure 7, the relative rotation accuracy of our approach outperforms the approach in [16], with an error in rotation staying below 1◦ per pair of frames. [sent-427, score-0.71]
94 Conclusion The present paper introduced a novel paradigm for solving the relative pose between two calibrated images, which consists ofiterative enforcement ofthe coplanarity ofepipolar plane normal vectors. [sent-431, score-0.579]
95 The cost function shows only a small number of geometrically meaningful local minima besides the globally optimal solution. [sent-433, score-0.26]
96 Rotation over time (a) and relative rotation accuracy (b) of our solution (eig) and [16] (nist) over a real image sequence. [sent-436, score-0.517]
97 Weyl-type relative perturbation bounds for eigensystems of hermitian matrices. [sent-453, score-0.296]
98 Finding the exact rotation between two images independently of the translation. [sent-486, score-0.348]
99 Polynomial Eigenvalue solutions to the 5-pt and 6-pt relative pose problems. [sent-502, score-0.323]
100 An efficient solution to the five-point relative pose problem. [sent-533, score-0.27]
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