iccv iccv2013 iccv2013-323 knowledge-graph by maker-knowledge-mining
Source: pdf
Author: Yubin Kuang, Kalle Åström
Abstract: In this paper, we study the geometry problems of estimating camera pose with unknown focal length using combination of geometric primitives. We consider points, lines and also rich features such as quivers, i.e. points with one or more directions. We formulate the problems as polynomial systems where the constraints for different primitives are handled in a unified way. We develop efficient polynomial solvers for each of the derived cases with different combinations of primitives. The availability of these solvers enables robust pose estimation with unknown focal length for wider classes of features. Such rich features allow for fewer feature correspondences and generate larger inlier sets with higher probability. We demonstrate in synthetic experiments that our solvers are fast and numerically stable. For real images, we show that our solvers can be used in RANSAC loops to provide good initial solutions.
Reference: text
sentIndex sentText sentNum sentScore
1 s e Abstract In this paper, we study the geometry problems of estimating camera pose with unknown focal length using combination of geometric primitives. [sent-3, score-0.928]
2 We formulate the problems as polynomial systems where the constraints for different primitives are handled in a unified way. [sent-7, score-0.328]
3 We develop efficient polynomial solvers for each of the derived cases with different combinations of primitives. [sent-8, score-0.769]
4 The availability of these solvers enables robust pose estimation with unknown focal length for wider classes of features. [sent-9, score-1.304]
5 We demonstrate in synthetic experiments that our solvers are fast and numerically stable. [sent-11, score-0.635]
6 For real images, we show that our solvers can be used in RANSAC loops to provide good initial solutions. [sent-12, score-0.52]
7 The minimal case of pose estimation using 3 points was studied in [10] and several other formulations are compared and reviewed in [12]. [sent-15, score-0.366]
8 For line-to-line correspondences, solutions are derived for minimal of 3 lines in [8, 7]. [sent-16, score-0.317]
9 Recently, the minimal cases using combination of points and lines are solved in [27]. [sent-17, score-0.381]
10 In [9] a solver is derived for a minimal problem of 2 points and their corresponding tangent directions (equivalently any direction vector through each of the points). [sent-18, score-0.476]
11 For camera pose estimation with unknown focal length, the planar case was studied and solved in [1]. [sent-21, score-0.888]
12 Efficient and numerically stable solvers are developed in [4]. [sent-23, score-0.579]
13 2D-2D and 2D-3D correspondence, [19] investigated several minimal cases for pose estimation with unknown focal length. [sent-44, score-0.867]
14 Additionally, for camera with unknown radial distortion and unknown focal length, the 4-point minimal case is solved in [18, 5]. [sent-45, score-0.92]
15 Many other works focus on solving the over-constrained problem of estimating camera pose with more than three points [15, 25, 24] or lines [25]. [sent-46, score-0.432]
16 Very recently, the approach in [24] was extended to handle unknown focal length [26]. [sent-47, score-0.66]
17 Minimal solvers are the key component of the preprocessing steps for such overconstrained solvers to robustly remove outliers. [sent-49, score-0.978]
18 To be able to utilize correspondences of geometric primitives like points, directions and lines is of great interest to applications e. [sent-50, score-0.428]
19 In typical scenarios of vision-based localization, focal length of the camera is the only unknown that is most difficult to determine accurately (EXIF-tag could provide erroneous estimate) and can render large errors in the pose estimation. [sent-54, score-0.954]
20 All previous methods for pose estimation with unknown focal length use point correspondences. [sent-55, score-0.861]
21 The contribution of this paper is to enable a wider class of geometric features (combinations of points, lines and n-quivers, Figure 1) for simultaneous pose estimation and focal length calibration. [sent-56, score-0.887]
22 We then develop efficient polynomial solvers for several new minimal cases and a slightly over-determined cases using 4 lines. [sent-60, score-0.893]
23 We verify our solvers on both synthetic and real images to demonstrate their efficiency and usability in RANSAC. [sent-61, score-0.639]
24 × (1) 4 which can be (2) The rotation matrix R encodes orientational part of the camera pose specifying in which direction the camera is pointing and t relates to the camera position. [sent-66, score-0.545]
25 In this paper, we thereafter assume that the calibration matrix only involves the unknown focal length f. [sent-69, score-0.692]
26 The K matrix can be equivalently written as K =⎣⎡010 10 w00⎦⎤, (3) where w = 1/f and f is focal length of the camera. [sent-70, score-0.594]
27 We know that the problem of determining camera pose with unknown focal length has in total 7 degrees of freedom (3 in rotation, 3 in translation and 1in f). [sent-71, score-0.943]
28 If the 3D line L is represented as a 3D point X and the direction of the line D, one can obtain two equations for the two points in the following form based on (1): lTPX = 0 lTP(X + kD) = 0, (5) where k is an arbitrary constant. [sent-83, score-0.494]
29 Number of constraints enforced by 2D-3D correspondences of different geometric primitive for camera pose estimation. [sent-93, score-0.436]
30 Useful Cases With 2D-3D correspondences of points, lines and nquivers, one can form several novel minimal cases by searching for combination such that 2mp + 2ml + (n + 2)mq = 7, where mp, ml , mq are the number of point, line, n-quiver correspondences, respectively. [sent-96, score-0.421]
31 Two Points and One 1-Quiver (P2Q1) : Given three points and one direction passing through one of the points, we can form 6 equations based on (4) and 1equation based on (6). [sent-98, score-0.329]
32 One 1-Quiver and One 2-Quiver (Q1Q2) : For two points, where one line passing through one of the point, and two lines passing through the other point are known, we can form 4 point equations (4) and 3 equations with respect to the directions (6). [sent-100, score-0.792]
33 Thus, the problem of camera pose with unknown focal length is overdetermined with 4 lines. [sent-103, score-0.881]
34 In a similar manner, other minimal cases include the setups: (i) one point, one line and one 1-quiver (ii) one 3quiver and one point (iii) two lines and one 1-quiver which can be solved in similar manner as the presented solvers. [sent-105, score-0.423]
35 Once the rotational part is recovered, the focal length can easily be calculated using the ratios between the norms of the third and the first two rows of R. [sent-112, score-0.588]
36 [4] formulate the P4P problem with unknown focal length using the invariance of the ratios of distances between the 3D points under rigid transformation. [sent-115, score-0.819]
37 To start with, we can use the three points to form 2 independent distance ratio equations involving three unknowns (two relative stretch ratios α1,α2 and f) as in [4]. [sent-120, score-0.408]
38 However, the resulting equations consists at least one equation of degree 6 (after substitution and simplification) which makes the resulting − polynomial system very difficult to solve. [sent-124, score-0.39]
39 While the use of geometric invariance might yield polynomial system with fewer solutions for previous problems, it is not straightforward to see that such property is preserved for other primitives like directions with unknown focal length. [sent-125, score-1.055]
40 In this paper, we first parameterize the rotation matrix R with quaternion and construct equations directly based on (4), (5) and (6). [sent-126, score-0.34]
41 For polynomial systems with small number of unknowns, Gr¨ obner basis methods are generally fast and numerically stable. [sent-153, score-0.347]
42 Solving polynomial systems can also be seen as solving polynomial eigenvalue problems [13, 23]. [sent-154, score-0.418]
43 Experiments In this section, we study the performance of our solvers on both synthetic and real data. [sent-169, score-0.603]
44 The camera was calibrated except for the focal length. [sent-176, score-0.548]
45 1 Stability and Number of Solutions We evaluate first the solvers on noise-free data to check the numerical stability of the solvers and distribution of number of valid solutions. [sent-179, score-1.111]
46 For the simulation results in Figure 3, the focal length of the camera was set to around 1000. [sent-180, score-0.666]
47 The numerical errors for all our solvers are fairly low for most of the cases. [sent-181, score-0.672]
48 T−15wLog0preal1tivn orsd−5tfcanlsoite10-qrFycnu2150 i ver,1sy2nt3Nuhmb4eriofc5alsx6uptoine7rm8sf 5000 runs on noise-free data with focal length approximately 1000. [sent-185, score-0.586]
49 Left: Histogram of relative errors for rotation, translation, focal length; Right: Histogram of number solutions with real and positive focal lengths. [sent-186, score-1.062]
50 We can see that for noise-free data, the Gr¨ obner basis solver for (P2Q1) is consistently stable for different focal lengths for both planar and non-planar scenes (Figure 4). [sent-192, score-0.882]
51 Similar numerical behaviors are observed for the solver using lines (P4L, Figure 5). [sent-193, score-0.348]
52 Given that the performance of other solvers are similar, related figures are not shown individually here. [sent-194, score-0.489]
53 Our solvers can also be further optimized for speed using strategies in [22, 21]. [sent-198, score-0.489]
54 2 Noise Sensitivity To study the behaviors of the solvers with noisy measurements, we add noise of different levels both to the image point positions and the angles of the directions. [sent-203, score-0.628]
55 In Figure 6, it is shown that the P2Q1 solver gives fairly good estimates for focal lengths with small noise, and is still able to provide (though not as frequently) reasonably good initial solutions when the noise is around 5 pixels. [sent-204, score-0.781]
56 We have also noticed that the solvers can be sensitive to errors in the direction measurements. [sent-205, score-0.611]
57 We also test the P4L solvers for noisy line measurements by perturbing the intersections between the lines and the x, y axis. [sent-206, score-0.771]
58 From Figure 7, we can see that the P4L solver is capable of recovering the focal length accurately for small perturbation and can become unreliable for large perturbation. [sent-207, score-0.676]
59 Synthetic experiments of P2Q1 on noise-free data with varying focal lengths. [sent-209, score-0.437]
60 Left: Boxplot of relative errors of focal lengths for non-planar points and directions; Right: planar cases. [sent-210, score-0.825]
61 Synthetic experiment of P4L on noise-free data with varying focal lengths. [sent-212, score-0.437]
62 Left: Boxplot of the relative errors of focal lengths for non-planar line configurations Right: planar cases. [sent-213, score-0.821]
63 Left: Relative errors of focal lengths for non-planar lines; Right: planar cases. [sent-230, score-0.717]
64 ity, we demonstrate the performance of the solvers on real image measurements in Section 4. [sent-231, score-0.568]
65 3 RANSAC Experiments To test the advantage of the proposed solvers for different geometric primitives, we simulate data with outliers and Figure 8. [sent-235, score-0.569]
66 Distribution ofinlier proportions for 1000 RANSAC runs for different solvers P4P, P2Q1 and Q1Q2. [sent-236, score-0.52]
67 For a fixed camera with focal length 1000, we generate randomly 1000 scene points as in the previous section, directions through points are also generated randomly. [sent-238, score-0.923]
68 We compare the solvers for two points and one 1-quiver (P2Q1) and one 1-quiver and one 2-quiver (Q1Q2) with the P4P solver in [4]. [sent-241, score-0.691]
69 Here we define the inliers as the image points with reprojection errors less than a predefined threshold. [sent-243, score-0.358]
70 Real Data We took 16 images of seven cardboards placed in a nonplanar configuration with varying focal lengths (Figure 9), using a standard Canon EOS 50D camera. [sent-248, score-0.564]
71 We used these images to verify the applicability of the proposed solvers on real images with point, line and quiver features. [sent-252, score-0.804]
72 The resulting construction of the 3D points and the camera poses as well as the focal lengths after bundle adjust- ment are fairly accurate and thus serves as ground truth. [sent-263, score-0.865]
73 Given the reconstruction of the detected lines and intersection points, we use the proposed solvers to estimate both the camera poses and the focal lengths for each of the image. [sent-264, score-1.294]
74 We first look at the reprojection errors of the poses and focal lengths estimated using different solvers and investigate whether the solvers adapt to real image noisy measurements. [sent-267, score-1.753]
75 To measure the reprojection errors, we run different solvers in a RANSAC manner by choosing random minimal measurements. [sent-268, score-0.726]
76 The average reprojection errors of image points for each solver are reported in Table 2. [sent-269, score-0.382]
77 We can see from Table 2 that the errors of all our proposed solvers are similar to the P4P solver. [sent-270, score-0.562]
78 Average reprojection errors (in pixels) of image points with camera poses and focal lengths of the 16 images estimated with different solvers. [sent-276, score-0.936]
79 Statistics of focal length estimation of different solvers, bundle adjustment and exif-tag for the Cardboard dataset. [sent-278, score-0.704]
80 For the inlier threshold of 3 pixels, the number of inliers (among in total 621 measurements) and the average reprojection errors for inliers are reported in Table 3. [sent-281, score-0.428]
81 The slightly inferior performance of Q1Q2 and P4L solvers might be due to the sensitivity of both solvers to measurement errors in the quiver directions and lines. [sent-283, score-1.353]
82 Number of inliers and average reprojection errors (in pixels) ofinliers with 30% synthetic outliers for the cardboard dataset. [sent-289, score-0.45]
83 To evaluate the accuracy of the solvers, we compare the best focal length estimated (the one with maximum number of inliers) for each solver against the output from bundle adjustment as well as those extracted from EXIF-tag (conversion from 35mm film equivalent). [sent-290, score-0.78]
84 The statistics of the estimated focal lengths are shown in Figure 10. [sent-292, score-0.564]
85 It is noted that the focal lengths given by the exif information seems to be very coarse compared to those estimates from image data directly. [sent-293, score-0.564]
86 We can also see that all solvers gives fairly similar estimates to the results from bundle adjustment. [sent-294, score-0.598]
87 Discussions For the simpler calibrated pose estimation problem, we also see the potential ofcombining the simplicity the quaternion parameterization and the stability of Gr¨ obner basis 534 solvers. [sent-296, score-0.484]
88 In [9], the minimal case of equivalently two 1- quivers (the direction is detected as the tangent to curves instead of arbitrary direction) for pose estimation was studied. [sent-297, score-0.446]
89 Conclusions In this paper, we present several novel cases for pose estimation with unknown focal length utilizing combinations of points, lines and quivers. [sent-302, score-1.031]
90 We have shown that these solvers are fast and numerically stable. [sent-306, score-0.552]
91 The availability of such solvers will serve as an important step towards pose estimation with richer features and also shed light on structure from motion problem with line/direction features which are common in urban scenes. [sent-308, score-0.644]
92 The other key direction is to evaluate the application of new solvers to discriminative feature like SIFT to ease the correspondence problem for edges (direction of a quiver and line). [sent-310, score-0.749]
93 In this case, one need to verify whether the solvers are robust against noisy estimation of the gradient directions. [sent-313, score-0.57]
94 To improve the speed and numerical stability of the solvers, it is of interest to resolve the intrinsic symmetry in the quaternion parameterization either by algebraic manipulation or by deriving alternative set of constraints using geometric invariances. [sent-314, score-0.449]
95 1Codes for the proposed solvers are available for download http://www2. [sent-317, score-0.489]
96 A general solution to the p4p problem for camera with unknown focal length. [sent-341, score-0.653]
97 New efficient solution to the absolute pose problem for camera with unknown focal length and radial distortion. [sent-349, score-0.913]
98 Pose estimation with radial distortion and unknown focal length. [sent-445, score-0.619]
99 Numerically stable optimization of polynomial solvers for minimal problems. [sent-465, score-0.84]
100 Exhaustive linearization for robust camera pose and focal length estimation. [sent-500, score-0.776]
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