iccv iccv2013 iccv2013-353 knowledge-graph by maker-knowledge-mining
Source: pdf
Author: Yinqiang Zheng, Yubin Kuang, Shigeki Sugimoto, Kalle Åström, Masatoshi Okutomi
Abstract: In this paper, we revisit the classical perspective-n-point (PnP) problem, and propose the first non-iterative O(n) solution that is fast, generally applicable and globally optimal. Our basic idea is to formulate the PnP problem into a functional minimization problem and retrieve all its stationary points by using the Gr¨ obner basis technique. The novelty lies in a non-unit quaternion representation to parameterize the rotation and a simple but elegant formulation of the PnP problem into an unconstrained optimization problem. Interestingly, the polynomial system arising from its first-order optimality condition assumes two-fold symmetry, a nice property that can be utilized to improve speed and numerical stability of a Gr¨ obner basis solver. Experiment results have demonstrated that, in terms of accuracy, our proposed solution is definitely better than the state-ofthe-art O(n) methods, and even comparable with the reprojection error minimization method.
Reference: text
sentIndex sentText sentNum sentScore
1 se Abstract In this paper, we revisit the classical perspective-n-point (PnP) problem, and propose the first non-iterative O(n) solution that is fast, generally applicable and globally optimal. [sent-10, score-0.118]
2 Our basic idea is to formulate the PnP problem into a functional minimization problem and retrieve all its stationary points by using the Gr¨ obner basis technique. [sent-11, score-0.342]
3 The novelty lies in a non-unit quaternion representation to parameterize the rotation and a simple but elegant formulation of the PnP problem into an unconstrained optimization problem. [sent-12, score-0.383]
4 Interestingly, the polynomial system arising from its first-order optimality condition assumes two-fold symmetry, a nice property that can be utilized to improve speed and numerical stability of a Gr¨ obner basis solver. [sent-13, score-0.659]
5 Experiment results have demonstrated that, in terms of accuracy, our proposed solution is definitely better than the state-ofthe-art O(n) methods, and even comparable with the reprojection error minimization method. [sent-14, score-0.338]
6 Introduction Given n (n ≥ 3) 3D reference points in the object framewoGrki vaennd nth (enir ≥ corresponding c2eD p projections, tboj edcette frrammineethe orientation and position of a fully calibrated perspective camera is known as the perspective-n-point (PnP) problem [9]. [sent-16, score-0.12]
7 Unfortunately, to the best of our knowledge, there does not exist a fast (preferably, real-time) and globally optimal solution, which is accurate and applicable to a PnP problem with any point number n (n ≥ 3), any 3eD to point configuration tahn dan arbitrary camera pose. [sent-19, score-0.218]
8 To properly account for the latest progress, we would like to roughly categorize them into two groups - the multi-stage methods and the direct minimization methods. [sent-25, score-0.113]
9 Typically, the multi-stage methods first estimate the coordinates of some (or all) points in the camera framework, and transform the PnP problem into the 3D-3D absolute pose problem, for which there exist closed-form solution- s [2 1]. [sent-26, score-0.125]
10 Without estimating point coordinates, the well-known direct linear transformation (DLT) method [9] is also a multi-stage method, since it first determines the projection matrix and then extracts the calibration parameters and the camera pose. [sent-36, score-0.143]
11 In contrast, the direct minimization methods are characteristic of minimizing a properly defined error function, either in the image space or the object space, while taking into consideration all nonlinear constraints. [sent-41, score-0.19]
12 It is widely known that minimizing the reprojection error is the best criterion, which leads to a challenging nonconvex fractional programming problem. [sent-42, score-0.189]
13 As a trade off, some direct minimization methods [6, 16] minimize instead certain algebraic error functions via local optimization techniques. [sent-46, score-0.256]
14 [6] offered an alternating minimization method to minimize an algebraic error defined in the image space. [sent-49, score-0.202]
15 The above direct minimization methods [6, 16, 17, 20] share another shortage that they return only a single solution, which might not correspond to the true camera pose in case of multiple solutions. [sent-55, score-0.235]
16 To resolve these drawbacks, Hesch and Roumeliotis [10] developed a direct least square (DLS) method with complexity O(n), in which all stationary points are retrieved by solving the polynomial system derived from its first-order optimal condition via the resultant technique. [sent-56, score-0.419]
17 Unfortunately, they parameterized rotation by using the Cayley representation, which is degenerate in all cases of 180 degree rotations around the x-, y- and z-axis1 . [sent-57, score-0.13]
18 The accuracy deteriorates seriously when the camera pose approaches these singularities. [sent-58, score-0.088]
19 As pointed out in [ 15], in addition to the number of points n, the configuration of the 3D reference points plays 1On the project page, Hesch and Roumeliotis provided a remedy by solving DLS three times under different rotated 3D points. [sent-59, score-0.262]
20 A desirable PnP solution should be able to handle all 3D point configurations, including the ordinary-3D, the planar and the quasi-singular (nearplanar or near-linear) configuration. [sent-63, score-0.198]
21 However, some existing methods, like [ 14, 20], handle the ordinary-3D and the planar configuration separately, thus tend to be inaccurate in the in-between quasi-singular case. [sent-64, score-0.168]
22 Overview of the Proposed Solution In this paper, we propose the first non-iterative O(n) solution that is fast, globally optimal and universally applicable. [sent-68, score-0.149]
23 Our basic idea is to formulate the PnP problem into a minimization problem and retrieve all its stationary points by using the Gr¨ obner basis technique. [sent-69, score-0.342]
24 By using a unusual non-unit quaternion representation to parameterize rotation, we formulate the PnP problem into an unconstrained optimization problem. [sent-71, score-0.284]
25 The polynomial system arising from its first-order optimality condition is simpler than using the standard unit quaternion parameterization. [sent-72, score-0.588]
26 More interestingly, this polynomial system is of odd-degree, thus assuming two-fold symmetry, a nice property that can be utilized to improve speed and numerical stability of a Gr¨ obner basis solver. [sent-73, score-0.467]
27 Being globally optimal, our proposed solution successfully conquers the problem of local optimality (or even divergence) that might upset a local optimization based method. [sent-74, score-0.223]
28 Unlike [10], our solution does not suffer from any degeneracy of camera pose. [sent-76, score-0.188]
29 Experiment results have demonstrated that, in terms of accuracy, our proposed solution is definitely better than the examined state-of-the-art methods. [sent-77, score-0.121]
30 Actually, although our cost function is only algebraically meaningful, its accuracy is even comparable to that of the reprojection error minimization method. [sent-78, score-0.255]
31 Therefore, the proposed solution is universally applicable to any PnP problem, irrespective of the 3D point configuration, the camera pose and the number of points. [sent-81, score-0.27]
32 T,i Given n 3D reference points qi = yi = 1, 2, · · · ,n, in the object reference framew? [sent-90, score-0.135]
33 matrix R and the translation vector t, accounting for camera orientation and position, respectively. [sent-95, score-0.121]
34 Considering that the perspective camera has been calibrated, we simply assume that the projections pi are measured in the normalized homogeneous image coordinate framework. [sent-96, score-0.098]
35 Rotation Parameterization A critical issue is how to parameterize the rotation matrix R, such that the orthonormal constraint RRT = I and the determinant constraint det(R) = 1could be satisfied. [sent-100, score-0.157]
36 There are various parameterization methods for R, such as the Euler angle, rotation axis-angle, Cayley and unit quaternion parameterization. [sent-101, score-0.404]
37 To facilitate global optimization via polynomial system solving, we advocate instead the non-unit quaternion parameterization, which is free of any trigonometric function. [sent-102, score-0.439]
38 Specifically, letting s = a2 + b2 + c2 + d2, the non-unit quaternion parameterization reads R =1s⎢⎡⎢ ⎢ ⎢ ⎢a2 +2b bcd2+− 2c2a2a−cd 2 a2−2 bcbcd2+ −c2 2a db−d2 a2 −2cb d 2− +2 ca 2cb+d2⎥ ⎥ ⎥⎤⎥ ⎥,(2) where a, b, c, d are the four unknown parameters. [sent-103, score-0.35]
39 It⎦ is straightforward to verify that the parameterization in Eq. [sent-104, score-0.109]
40 and At first sight, the above parameterization is unattractive at all. [sent-106, score-0.109]
41 It is possible to exploit this property to resolve the concerns on the non-unit quaternion representation. [sent-116, score-0.196]
42 Although it is only an algebraic error, as will be demonstrated in the experiment section, its accuracy is very close to that of minimizing the reprojection error, i. [sent-156, score-0.147]
43 In addition, it suffers from no degeneracy of camera pose, and is independent of the 3D point configuration. [sent-167, score-0.147]
44 Relation to Existing Works In the previous section, we have used the non-unit quaternion to parameterize the rotation, and fixed its scale by using the reciprocal of the average depth. [sent-170, score-0.284]
45 Through some basic operations, one can verify that the Cayley parameterization is the same as R(1, b, c, d), that is, fixing the scale of Eq. [sent-182, score-0.109]
46 The major advantage lies in that the polynomial system in [ 10] is simpler to solve, since there remain only three variables. [sent-184, score-0.2]
47 Here, we prefer to solve the polynomial system of the first-order optimality condition of Eq. [sent-189, score-0.317]
48 The investigated solvers include the blind GB solver without utilizing symmetry (Blind GB), the GB solver using two-fold symmetry (Symmetric GB), the Symmetric GB followed by one damped Newton polishing step (Symmetric GB + Polish) and the resultant based solver used in DLS [10]. [sent-207, score-0.725]
49 The horizontal axis shows the log10 value of the absolute error between the ground-truth unitnorm quaternion and the estimated quaternion after normalization, while the vertical axis shows the counts over 5,000 independent runs. [sent-210, score-0.469]
50 (1 1) with respect to a, b, c, d, the first-order optimality condition reads ∂∂af= 0,∂∂bf= 0,∂∂cf= 0,∂∂df= 0, (13) which is composed of four three-degree polynomials with respect to a, b, c, d. [sent-213, score-0.209]
51 A Blind Gr¨ obner Basis Solver Although solving multivariate polynomial systems is challenging in general, the multiview geometry community has achieved much progress by means of the Gr¨ obner basis (GB) technique [3]. [sent-216, score-0.446]
52 [13] even developed an automatic generator of GB solvers, which facilitates the solving of polynomial systems arising from geometric computer vision problems. [sent-218, score-0.292]
53 Finally, the solutions to the original polynomial system are extracted from the eigen-factorization of the action matrix. [sent-220, score-0.245]
54 By using the automatic generator in [13], we have found that the polynomial system in Eq. [sent-222, score-0.256]
55 The size of the elimination template is 575×656, lwuthiiolne sth. [sent-224, score-0.099]
56 One might be interested in solving the polynomial system arising from the first-order optimality condition of Eq. [sent-228, score-0.426]
57 22334477 Due to the unit-norm constraint, a Lagrange multiplier has to be introduced, thus leading to a three-degree polynomial system with respect to five variables. [sent-230, score-0.2]
58 , the elimination template is of size 1523×1603) and thus much selliomweinr. [sent-233, score-0.099]
59 a tTiohnis t vemerpifliaetse th ise o advantages ×of1 our )no ann-dun thiut quaternion parameterization and our unconstrained formulation. [sent-234, score-0.335]
60 (1 1), we have further noted that the polynomial system in Eq. [sent-238, score-0.2]
61 [2] developed general techniques to make use of p-fold symmetry (p = 2 in our problem) arising from some minimal problems. [sent-246, score-0.24]
62 The basic idea is to directly eliminate the trivial all-zero solution and carefully generate new equations such that the symmetry could be preserved. [sent-247, score-0.208]
63 By using symmetry, the size of the elimination template and that of the action matrix could be drastically reduced, which in return improves computational speed and numerical stability. [sent-248, score-0.154]
64 In [2], one has to solve an integer linear system to extract the symmetric solutions from the action matrix, which is very slow. [sent-249, score-0.124]
65 When implementing the two-fold symmetry GB solver for Eq. [sent-250, score-0.204]
66 With an elimination template of size 348×376 and an action matrix of size 40×40, our towfo s-ifzoeld 3 symmetry dG aBn s aoclvtieorn nta mkeastr xabo ouft s i1z8e. [sent-253, score-0.23]
67 1, its numerical stability is also stronger than the blind version. [sent-256, score-0.162]
68 It is worthy of mentioning that the five-variable polynomial system from Eq. [sent-257, score-0.2]
69 (13) is the first fully symmetric system, arising from a non-minimal problem, that we know of. [sent-260, score-0.115]
70 Solution Polishing and Extraction After obtaining all stationary points of Eq. [sent-263, score-0.119]
71 (1 1), we can further improve the numerical stability by polishing them via a single damped Newton step. [sent-264, score-0.3]
72 Specifically, assuming that w is a stationary point, we polish w through the updating rule w ← w Δw. [sent-265, score-0.125]
73 w1, − −th Δew polishing strategy can drastically improve the numerical precision, although only one damped Newton step is used. [sent-268, score-0.246]
74 After the polishing step, we only retain those real and physically feasible stationary points with positive definite Hessian, i. [sent-271, score-0.247]
75 a Wnte hcaenvea ioobsse wrivtehd 4 a ≤fe nw ≤ex 5tr,e itm ise cases, in which two widely different stationary points have almost the same objective value, yet the objective value of the correct stationary point is even slightly larger. [sent-280, score-0.237]
76 The authors of DLS [10] provided a remedy to conquer the degeneracy of the Cayley representation by solving DLS three times under differently rotated 3D points. [sent-291, score-0.132]
77 Considering that our objective function is only algebraically meaningful, it might be of great interest to com- it with the reprojection error minimization method. [sent-293, score-0.289]
78 bTehe toretafol rye, d we mrenintim froizme t hhee reprojection error by using ≤th 5e. [sent-296, score-0.158]
79 varying point numbers (1st and 2nd columns, δ=2 pixels) and varying noise levels (3rd and 4th columns, n=10) in case of ordinary-3D, quasi-singular and planar point configuration, shown in (a), (b) and (c), respectively. [sent-332, score-0.157]
80 × × incorporates the symmetric GB solver and the polishing strategy. [sent-333, score-0.241]
81 e Tnchee points are randomly generated in the camera framework. [sent-343, score-0.09]
82 Then, we choose the ground-truth translation ttrue such that the origin of the object framework coincides with the centroid of these 3D points, and rotate these 3D points by using a randomly generated ground-truth rotation matrix Rtrue. [sent-345, score-0.298]
83 eTnhte t htrean dsoltat ipornod error nisd measured by the relative difference between ttrue and t defined as etrans(%) = ||ttrue t||/||t|| 100. [sent-347, score-0.141]
84 We present the average rotation and translation error in the 1st and 2nd column of Fig. [sent-358, score-0.244]
85 At each noise level, we run 500 independent trials and report the average rotation and translation error in the 3rd and 4th column of Fig. [sent-362, score-0.244]
86 However, this does not necessarily mean that any direct minimization method is accurate. [sent-370, score-0.144]
87 2, OPnP has definitely better accuracy than the examined state-of-the-art methods (except DLS+++), irrespective of the point configuration and the point number. [sent-378, score-0.206]
88 Being an algebraic error based method, it is even comparable with the reprojection error based method OPnP+LM. [sent-379, score-0.301]
89 As pointed out by Hartley [7], minimizing a reasonably defined algebraic error might provide accurate results, as long as all constraints are exactly enforced and data are properly normalized. [sent-381, score-0.209]
90 Our OPnP solution exactly handles the challenging rotation constraint and uses centralized 3D points and image projections (Eq. [sent-382, score-0.258]
91 The mean rotation error (left) and the mean translation error (right) w. [sent-392, score-0.383]
92 2 Computational Time The complexity of solving the polynomial system in Eq. [sent-398, score-0.2]
93 We vary the noise level from 0 to 5 pixels and show the mean rotation and translation error over 500 independent trials in Fig. [sent-417, score-0.275]
94 Conclusion We have revisited the PnP problem and proposed the first non-iterative O(n) solution that is fast, general and globally optimal. [sent-427, score-0.118]
95 Our contribution is to parameterize the rotation by 22335500 Figure5. [sent-428, score-0.157]
96 using non-unit quaternion and formulate the PnP problem into an unconstrained optimization problem. [sent-432, score-0.226]
97 Surprisingly, the polynomial system arising from its first-order optimality condition is of odd-degree, thus assuming two-fold symmetry, a nice property that can be utilized to improve speed and numerical stability of a Gr¨ obner basis solver. [sent-433, score-0.659]
98 As verified by experiment results, even with a few point correspondences, our proposed solution is quite accurate, irrespective of the point configuration and the camera pose. [sent-434, score-0.292]
99 A novel [13] [14] [15] [16] [17] [18] [19] [20] [21] parametrization of the perspective-three-point problem for a direct computation of absolute camera position and orientation. [sent-510, score-0.107]
100 Globally optimal O(n) solution to the PnP problem for general camera models. [sent-557, score-0.13]
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