iccv iccv2013 iccv2013-17 knowledge-graph by maker-knowledge-mining
Source: pdf
Author: Nianjuan Jiang, Zhaopeng Cui, Ping Tan
Abstract: We present a linear method for global camera pose registration from pairwise relative poses encoded in essential matrices. Our method minimizes an approximate geometric error to enforce the triangular relationship in camera triplets. This formulation does not suffer from the typical ‘unbalanced scale ’ problem in linear methods relying on pairwise translation direction constraints, i.e. an algebraic error; nor the system degeneracy from collinear motion. In the case of three cameras, our method provides a good linear approximation of the trifocal tensor. It can be directly scaled up to register multiple cameras. The results obtained are accurate for point triangulation and can serve as a good initialization for final bundle adjustment. We evaluate the algorithm performance with different types of data and demonstrate its effectiveness. Our system produces good accuracy, robustness, and outperforms some well-known systems on efficiency.
Reference: text
sentIndex sentText sentNum sentScore
1 Our method minimizes an approximate geometric error to enforce the triangular relationship in camera triplets. [sent-2, score-0.274]
2 an algebraic error; nor the system degeneracy from collinear motion. [sent-5, score-0.402]
3 Introduction Structure-from-motion (SfM) methods simultaneously estimate scene structure and camera motion from multiple images. [sent-12, score-0.319]
4 First, relative poses between camera pairs or triplets are computed from matched image feature points, e. [sent-14, score-0.376]
5 Second, all camera poses (including orientations and positions) and scene point coordinates are recovered in a global coordinate system according to these relative poses. [sent-17, score-0.475]
6 Some well-known systems, such as [36, 2], compute camera poses in an incremental fashion, ∗These authors contributed equally to this work. [sent-25, score-0.298]
7 Thus, it is highly desirable that all camera poses are solved simultaneously for efficiency and accuracy. [sent-32, score-0.298]
8 [3] derived a novel linear algorithm that is robust to different camera baseline lengths. [sent-39, score-0.274]
9 Yet it still suffers from the same degeneracy as [13] for collinear cameras (e. [sent-40, score-0.448]
10 Unlike earlier algebraic methods, we compute the camera positions (translations) by minimizing a geometric error the Euclidean distance between the camera centers and the lines collinear with their corresponding baselines. [sent-47, score-0.882]
11 This novel approach generates more precise results, and does not degenerate with collinear camera motion. [sent-48, score-0.506]
12 We want to stress that the robustness with collinear motion is an important advantage, since collinear motion is common (e. [sent-49, score-0.662]
13 Furthermore, our estimation of camera poses does not involve reconstructing any 3D point. [sent-52, score-0.298]
14 Effectively, we first solve the ‘motion’ camera poses, and then solve the ‘structure’ scene points. [sent-53, score-0.329]
15 Once the camera poses are recovered, the scene points can be reconstructed from – – – nearby cameras. [sent-56, score-0.369]
16 In the special case of three cameras, our algorithm effectively computes the trifocal tensor from three essential matrices. [sent-57, score-0.362]
17 In our experiment, we find that our method is more robust than the four-point algorithm [26] which solves trifocal tensor from three calibrated images. [sent-58, score-0.365]
18 Many well-known SfM systems take a sequential [3 1, 36, 2] or hierarchical [11, 22, 17] approach to register cameras incrementally to a global coordinate system from their pairwise relative poses. [sent-72, score-0.282]
19 Factorization based 3D reconstruction was proposed by Tomasi and Kanade [39] to recover all camera poses and 3D points simultaneously under weak perspective projection. [sent-75, score-0.373]
20 Some global methods solve all camera poses together in two steps. [sent-79, score-0.368]
21 Typically, they first compute camera rotations and solve translations in the next step. [sent-80, score-0.407]
22 While global rotations can be computed robustly and accurately by rotation averaging [15], translations are difficult because the input pairwise relative translations are only known up to a scale. [sent-82, score-0.441]
23 The pioneer works [13, 4] solved translations according to linear equations derived from pairwise relative translation directions. [sent-83, score-0.365]
24 These earlier methods suffer from degeneracy of collinear camera motion and unbalanced constraint weighting caused by different camera baseline length. [sent-84, score-0.91]
25 [7] computed homographies to glue individual triplet reconstructions by loop analysis and nonlinear optimization. [sent-87, score-0.321]
26 [35] registered individual pairwise reconstructions by solving their individual global scaling and translation in a robust linear system. [sent-89, score-0.291]
27 However, it still suffers from the degeneracy of collinear camera motion like [13, 4]. [sent-94, score-0.636]
28 Other global methods solve all camera poses and 3D scene points at once. [sent-96, score-0.404]
29 In the special case of three cameras, the camera geometry is fully captured by a trifocal tensor. [sent-104, score-0.485]
30 Effectively, our method provides a linear solution for trifocal tensor from three essential matrices (i. [sent-107, score-0.362]
31 An essential matrix Eij between two images i,j provides the relative rotation Rij and the translation direction tij . [sent-114, score-0.366]
32 i sO au 3r goal rist toon recover aaltlr tihxe a nabds toliust ea camera poses itno a global ocaolo isrd itnoa rteec system. [sent-116, score-0.336]
33 tWhee use a rotation matrix Ri and a translation vector ci to denote the orientation and position of the i-th camera (1 ≤ i ≤ N). [sent-117, score-0.697]
34 In real data, these equations mweail n sno etq uhoalldit precisely sacnadl we nne reeda lto d afitand, a esseet of Ri, ci that best satisfy these equations. [sent-122, score-0.303]
35 Secondly, the camera poses should be solved separately from the scene points. [sent-126, score-0.334]
36 There are often much more scene points than cameras so that solving camera poses without scene points will significantly reduce the number of unknowns. [sent-127, score-0.465]
37 We first apply the linear method described in [24] to compute the global camera rotations Ri. [sent-128, score-0.337]
38 Once all rotations are fixed, we then solve all camera centers (ci, 1 ≤ i ≤ N) without reconstructing any 3D point. [sent-131, score-0.384]
39 Translation Registration Given the global camera rotations computed in the previous section, we first transform each tij to the global rotation reference frame as cij = −R? [sent-133, score-0.746]
40 The constraint on camera centers in Equation =(1) − can be written as in [13], cij (cj − ci) = 0. [sent-135, score-0.485]
41 Hhoisw iesv ae lri, equations oobnt aaibnoeudt this way degenerate for collinear camera motion. [sent-139, score-0.57]
42 In fact, Equation (2) minimizes the cross product between cij and the baseline direction cj −ci. [sent-142, score-0.584]
43 The × relative translation cij , cik , and cjk between camera pairs are known. [sent-148, score-1.041]
44 We need to estimate camera centers ci, cj , and ck. [sent-149, score-0.618]
45 Ideally, the three unit vectors cij , cik , and cjk should be coplanar. [sent-150, score-0.661]
46 Here, | |ci − cj | | is the distance between ci and cj . [sent-168, score-0.911]
47 ) = | |cj ck −| | /c| ||c|i/ cj −| −| are effectively the baseline length ) ra =tio ||sc. [sent-173, score-0.56]
48 To linearize it, we observe that sjikj −c | |ci − cj | |cik = | |ci − cj −||c | |Ri (θ? [sent-179, score-0.761]
49 (4) Here, Ri (φ) is the rotation matrix around the axis cij × for an angle φ (counter-clockwise). [sent-182, score-0.345]
50 Thus we obtain th×e following linear equation, cik 2ck − ci − cj = Ri(θi? [sent-183, score-0.816]
51 (5) Note Rj (·) is a rotation matrix around the direction cij cjk. [sent-186, score-0.312]
52 Similarly, we can oonb mtaiant rtihxe following et dwior elictnieoanr equations of camera centers by assuming cik and cjk are free from error respectively, 2cj − ci − ck 2ci −cj −ck Ri(−θi? [sent-187, score-1.231]
53 = − − (7) Solving these three linear equations can determine the camera centers. [sent-192, score-0.293]
54 Note that Equation (5) does not require the orientation cj − ci to be the same as cij . [sent-193, score-0.778]
55 This introduces a rotation ambiguity in the plane defined by the camera centers. [sent-194, score-0.383]
56 We can solve it by computing the average rotation to align cj ci, ck ci and ck cj with the projection of cij , cik and cjk in the− camera plane, respectively, after the initial registration. [sent-195, score-2.228]
57 Calculating baseline length ratios by the sine angles as described earlier is only valid when cij, cik and cjk are not collinear. [sent-197, score-0.539]
58 In order to be robust regardless of the type of camera motion, we compute − − − 483 all baseline length ratios from locally reconstructed scene points. [sent-198, score-0.381]
59 The translation registration does not involve reconstructing any scene point in the global coordinate system. [sent-206, score-0.314]
60 Given a triplet graph (see definition in Section 5), we collect all equations (i. [sent-210, score-0.332]
61 the rotation ambiguity in all triplets share the same rotation axis), there is a global in-plane rotation ambiguity similar to the three-camera case. [sent-220, score-0.502]
62 Therefore, the unknown camera centers are implicitly given different weights depending on the number of constraints containing that particular camera when we solve for Ac = 0. [sent-223, score-0.543]
63 Thus, for every camera i, we count the number of triplet constraints containing its center, denoted by Ki. [sent-224, score-0.453]
64 Each triplet constraint involving camera i, j, k is re-weighted by min(Ki1,Kj,Kk). [sent-225, score-0.453]
65 1) We verify every triplet in the match graph, and remove EGs which participate in no triplet that passes the verification. [sent-243, score-0.504]
66 Specifically, we apply our translation registration to each triplet and calculate the average difference between the relative translation directions before and after the registration. [sent-244, score-0.572]
67 We further require that at least one good point (with reprojection error smaller than 4 pixels) can be triangulated by the registered triplet cameras. [sent-246, score-0.374]
68 2) Among the edges of the match graph, we extract a subset of ‘reliable edges’ to compute the global camera orientations as described in Section 3. [sent-247, score-0.352]
69 3) We further use these camera orientations to verify the match graph edges, and discard an edge if the geodesic distance [15] between the loop rotation matrix [45] and the identity matrix is greater than δ2. [sent-252, score-0.504]
70 We further extract connected triplet graphs from the match graph, where each triplet is represented by a vertex. [sent-259, score-0.569]
71 A single connected component of the match graph could generate multiple connected triplet graphs, as illustrated in Figure 2. [sent-261, score-0.454]
72 We then apply our method in Section 4 to compute the positions of cameras in each triplet graph respectively. [sent-262, score-0.363]
73 Our method is much more stable in translation estimation for near collinear camera motions. [sent-267, score-0.626]
74 When there are multiple triplet graphs, their reconstructions are merged to obtain the final result. [sent-269, score-0.284]
75 Camera 0 is placed at the world origin and camera 2 is placed at a random location away from camera 0 by 0. [sent-280, score-0.458]
76 The location of camera 1is sampled randomly in the sphere centered at the middle point between camera 0 and 2, and passing through their camera centers. [sent-282, score-0.687]
77 The scene points are generated randomly within the viewing volume of the first camera and the distance between the nearest scene point and the furthest scene point is about 0. [sent-285, score-0.337]
78 The error of camera orientations Rerr is the mean geodesic distance (in degrees) between the estimated and the true camera rotation matrix. [sent-293, score-0.641]
79 Absolute camera location error cerr is the mean Euclidean distance between the estimated and the true camera centers. [sent-295, score-0.548]
80 We compare with the four-point algorithm [26], which is the only practical algorithm to compute trifocal tensor from three calibrated images as far as we know. [sent-298, score-0.365]
81 We also compare with the recent method [3] to demonstrate the robustness of our method on near collinear camera motions. [sent-303, score-0.506]
82 It is clear that our method produces more stable results for near collinear motion. [sent-311, score-0.309]
83 Multi-view Reconstruction We test the performance of our method with some standard benchmark datasets with known ground-truth camera motion to quantitatively evaluate the reconstruction accuracy. [sent-314, score-0.358]
84 On average our method produces error in ci about 0. [sent-329, score-0.316]
85 The absolute camera rotation error Rerr and camera location error cerr are measured in degrees and meters, respectively. [sent-360, score-0.738]
86 Each internet image collection is reconstructed as one single connected triplet graph by our algorithm. [sent-366, score-0.4]
87 We further manually identify the set of common cameras registered by our method and VisualSFM, respectively, for the Notre Dame example, and compute the difference between the estimated camera motion. [sent-374, score-0.356]
88 007 (when the distance between the two farthest camera is 1). [sent-377, score-0.266]
89 Conclusion We present a novel linear solution for the global camera pose registration problem. [sent-389, score-0.344]
90 It is free from the common degeneration of linear methods on collinear motion, and is robust to different baseline lengths between cameras. [sent-391, score-0.322]
91 For the case of three cameras, it produces more accurate results than prior trifocal tensor estimation method on calibrated images. [sent-392, score-0.397]
92 Spectral solution of large-scale extrinsic camera calibration as a graph embedding problem. [sent-428, score-0.273]
93 Exploiting loops in the graph of trifocal tensors for calibrating a network of cameras. [sent-448, score-0.342]
94 Randomized structure from motion based on atomic 3d models from camera triplets. [sent-513, score-0.283]
95 Self-calibration and [31] [32] [33] [34] [35] [36] [37] [38] [39] metric reconstruction inspite of varying and unknown intrinsic camera parameters. [sent-591, score-0.304]
96 Derivation of Equation (3) We first show that the length of the line segments ciA, cjB are approx- siikj sjikj imately | |ci − cj | | and | |ci − cj | | respectively. [sent-706, score-0.886]
97 The three vector|s| cij , cik a|n adn cjk sh|o|culd− b ce |c|lo rsees ptoe coplanar, so the angle ∠Acick is close to zero, and the length of ciA is close to that of cick. [sent-707, score-0.73]
98 ≈ θk because the three vectors cij , cik and cjk are ≈clo θse to coplanar. [sent-713, score-0.661]
99 The 3D coordinate of A is then approximated by ci + siikj | |ci − cj | |cik. [sent-714, score-0.707]
100 Similarly, we sjikj can obtain the coordinate of B as cj + | |ci −cj | |cjk. [sent-715, score-0.468]
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