iccv iccv2013 iccv2013-138 knowledge-graph by maker-knowledge-mining

138 iccv-2013-Efficient and Robust Large-Scale Rotation Averaging

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Author: Avishek Chatterjee, Venu Madhav Govindu

Abstract: In this paper we address the problem of robust and efficient averaging of relative 3D rotations. Apart from having an interesting geometric structure, robust rotation averaging addresses the need for a good initialization for largescale optimization used in structure-from-motion pipelines. Such pipelines often use unstructured image datasets harvested from the internet thereby requiring an initialization method that is robust to outliers. Our approach works on the Lie group structure of 3D rotations and solves the problem of large-scale robust rotation averaging in two ways. Firstly, we use modern ?1 optimizers to carry out robust averaging of relative rotations that is efficient, scalable and robust to outliers. In addition, we also develop a twostep method that uses the ?1 solution as an initialisation for an iteratively reweighted least squares (IRLS) approach. These methods achieve excellent results on large-scale, real world datasets and significantly outperform existing methods, i.e. the state-of-the-art discrete-continuous optimization method of [3] as well as the Weiszfeld method of [8]. We demonstrate the efficacy of our method on two large- scale real world datasets and also provide the results of the two aforementioned methods for comparison.

Reference: text

Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 in i Abstract In this paper we address the problem of robust and efficient averaging of relative 3D rotations. [sent-4, score-0.368]

2 Apart from having an interesting geometric structure, robust rotation averaging addresses the need for a good initialization for largescale optimization used in structure-from-motion pipelines. [sent-5, score-0.464]

3 Such pipelines often use unstructured image datasets harvested from the internet thereby requiring an initialization method that is robust to outliers. [sent-6, score-0.134]

4 Our approach works on the Lie group structure of 3D rotations and solves the problem of large-scale robust rotation averaging in two ways. [sent-7, score-0.816]

5 1 optimizers to carry out robust averaging of relative rotations that is efficient, scalable and robust to outliers. [sent-9, score-0.778]

6 1 solution as an initialisation for an iteratively reweighted least squares (IRLS) approach. [sent-11, score-0.155]

7 Introduction In this paper we address the problem of robust averaging of 3D relative rotations in the context of structure-frommotion (henceforth SfM) estimation. [sent-17, score-0.684]

8 The canonical SfM solution of nonlinear bundle adjustment that minimizes reprojection error is statistically optimal [15]. [sent-18, score-0.15]

9 Such pipelines need to robustly handle outliers that may exist in unstructured image datasets harvested from the internet. [sent-20, score-0.205]

10 Given the high dimensional optimization involved and the presence of outliers, successful convergence critically depends on both a good initial guess as well as the robustness of the optimization methods used. [sent-21, score-0.125]

11 This is often achieved by incremental bundle adjustment that robustly grows the solution one image at a time instead of carrying out a single batch optimization. [sent-22, score-0.208]

12 Motion Averaging Preliminaries In contrast to bundle adjustment, an alternate approach for global camera motion estimation is to average relative motions. [sent-24, score-0.244]

13 Motion averaging was introduced in [5] and further developed in [6] to use the geometric structure of Lie × × groups. [sent-25, score-0.242]

14 While such a formulation is generic enough to apply to a variety of scenarios, in the context of SfM, motion averaging leverages the observation that in a set of N images, there exist as many as NC2 = pairs for which the relative motions can be estimated. [sent-26, score-0.348]

15 We can represent the relationships between all the cameras by means of a graph G = {V, E} known as the viewgraph where each vertex in VG represents a camera athned an edge (hi, w wjh) r∈e eEa implies txh iant tVhe r emproetisoenn bse atw caeemne cameras ni eadngde j can b ∈e eEst i mmpalteieds. [sent-27, score-0.198]

16 1 t Iant the following we represent 3D rotations by the 3 3 ortthhoen foorlmlowal mngat wrixe, Repr, ei. [sent-28, score-0.316]

17 th Ief absolute 3D rotation of the k-th camera as Rk, then the relative rotation between cameras iand j, Rij, can be written in terms of the global motions of cameras iand j,i. [sent-32, score-0.613]

18 N(N2−1) Rij = RjRi−1, ∀{i, j} ∈ E (1) All 3 3 rotation matrices form a closed group known as tAhlel Special Orthogonal group mS aO c(l3o)s wdh gicrohu apl kson hwans the differentiable properties of a Riemannian manifold, i. [sent-34, score-0.281]

19 , SO(3) is a Lie group which is the basis for an efficient approach for averaging rotations. [sent-36, score-0.278]

20 An important property of Lie groups is the existence of direct mappings between the Lie algebra and the group and vice-versa. [sent-42, score-0.15]

21 This leads to the Frechet mean or |the intrinsic average μ ∈ SO(3) of a set of rotations o{rR t1h , e· · i ·n , Rinsnic} w avheicrahg ies dμef ∈ine Sd as ? [sent-49, score-0.335]

22 Analogous to estimating the mean rotation, we may use the intrinsic distance to fit global or absolute rotations to a given set of relative rotation observations. [sent-52, score-0.634]

23 If with respect to a given frame of reference, we define the global rotations as Rglobal = {R1, · · · , RN}, using Eqn. [sent-53, score-0.345]

24 2 We further denote the angle representations of all the rotations as ωglobal = [ω1, Consequently, we can write ωij = ωj − ωi = ? [sent-56, score-0.316]

25 Algorithm 1 Lie-Algebraic Relative Rotation Averaging Input: {Rij1,··· ,Rijk} (|E| relative rotations) Output: Rglobal ·= , R{R1 , }· (· ·E , R reNlat}i v(|eV ro| atabtsioonlust)e rotations) Initialisation: Rglobal to, an ·i ,niRtial guess while | |Δωrel | | < ? [sent-85, score-0.147]

26 In this method for averaging relative rotations, for all the edges in E, the discrepancy between the observations Rij and the Ecu,r rtehent deisstcimreapaten cfoyr tehtew reeelanti tvhee r oobtasteirovna as implied by the global estimate, i. [sent-93, score-0.358]

27 Following this averaged estimate (step 3), the individual rotations are updated by mapping the Lie algebraic update back to the rotation group via the exponential mapping. [sent-96, score-0.666]

28 It may be further noted that following the averaging in the Lie algebra, the exponential mapping used to update individual rotations (step 4) ensures that at every point, the estimates are on the rotation manifold, i. [sent-97, score-0.848]

29 this algorithm provides an intrinsic estimate for the global rotation Rglobal . [sent-99, score-0.259]

30 We also note that in practice the rotation for any camera is fixed to I remove the gauge freedom of Rglobal. [sent-100, score-0.206]

31 to This method of averaging relative rotations can solve for the global rotation of all cameras in an efficient manner, see [6] for details. [sent-101, score-0.888]

32 In the context of SfM, given camera calibration information and a sufficient number of point correspondences between images iand j, we can estimate Rij either from the epipolar geometry or bundle adjustment between the two images. [sent-102, score-0.19]

33 In [12], 522 the authors use relative rotation estimates and information from vanishing point matches to estimate Rglobal . [sent-104, score-0.348]

34 In [11], the authors use rotation averaging in a RANSAC frame- work to robustly estimate the global rotations. [sent-105, score-0.558]

35 [4] uses a dual formulation on a quaterion representation for rotation averaging while [9] provides a survey on rotation averaging. [sent-106, score-0.608]

36 Although this method can efficiently average relative motions, as is well known least squares solutions are non-robust and can give highly erroneous results in the presence of even a single outlier. [sent-113, score-0.191]

37 The problem can be mitigated to some extent by ensuring that the individual relative rotation estimates Rij are robustly estimated, e. [sent-115, score-0.381]

38 In other words, in many contexts the presence of outliers cannot be fully avoided or it can be prohibitively expensive to remove them. [sent-119, score-0.114]

39 In effect, modern SfM systems need a rotation averaging scheme that is efficient, scalable and robust to outliers in the relative rotations. [sent-121, score-0.609]

40 Existing Methods In recent years a few methods have been developed to incorporate robustness into the averaging of relative rotations and they can be classified into two approaches. [sent-124, score-0.645]

41 Some methods detect outliers in the set of relative rotations and remove them before carrying out ? [sent-125, score-0.501]

42 In [18], the authors utilize the fact that a loop of transformations should result in the identity transformation if there is no noise or outliers in the loop. [sent-129, score-0.109]

43 The second category of methods robustly average relative rotations without the need to explicitly detect and remove outliers. [sent-133, score-0.455]

44 In [3], the authors use a combination of discrete-continuous optimization (henceforth DISCO) to average relative rotations in a robust manner. [sent-135, score-0.466]

45 j)∈E (6) where the additional prior terms include information from other sensor measurements that provide an approximate estimate for individual rotations in Rglobal . [sent-138, score-0.377]

46 Ignoring the twist component of rotations, [3] parametrizes rotations as a discrete set of labels on a unit sphere. [sent-141, score-0.36]

47 The resultant averaging problem in the discrete labelling form is solved using discrete loopy belief propagation on a Markov Random Field. [sent-142, score-0.304]

48 not directly lend itself to the averaging of relative rotations. [sent-156, score-0.329]

49 To work around this limitation, [8] updates individual rotations one-at-a-time while holding all other rotation estimates fixed. [sent-157, score-0.558]

50 If we hold all rotations in Rglobal fixed except for Rj, then the Weiszfeld optimization reduces to the ? [sent-158, score-0.316]

51 1 average of the rotations Rj = {RijRi |∀i ∈ N(j)} where N(j) is the sroett otifo vnesr tRices= =co {nRnected| ∀toi v ∈er tNex( j. [sent-159, score-0.316]

52 Thus, in [8], Rglobal is indirectly eelstdim-baasteedd using nne ostfe Rd iteration where the inner loop consists of updating Rj to the Weiszfeld median of Rj and the outer loop is iterated till convergence. [sent-161, score-0.135]

53 Although the methods of DISCO [3] and Weiszfeld [8] solve the problem of robust averaging of relative rotations, they suffer from significant limitations. [sent-163, score-0.368]

54 DISCO [3] can handle large-scale averaging problems but at a significant cost of implementational and computational complexity. [sent-164, score-0.242]

55 More crucially, disregarding the geometric structure of SO(3) and treating averaging as a complicated discrete labelling MRF problem makes DISCO an extremely expensive problem requiring a significant amount of hardware. [sent-169, score-0.284]

56 Since the Weiszfeld method cannot update all rotations in Rglobal simultaneously, it is forced to update the rotation of each camera (i. [sent-172, score-0.568]

57 1 Rotation Averaging Since neither DISCO [3] nor the Weiszfeld method [8] satisfies both the requirements of a computationally efficient and scalable robust averaging scheme, we propose an alternate approach that can efficiently handle large-scale problems. [sent-185, score-0.304]

58 4, we recognise that if a specific relative rotation is an outlier, then an ? [sent-187, score-0.27]

59 2 average in the Lie algebra will result in wrong rotation estimates. [sent-188, score-0.265]

60 The remedy lies in robust Lie algebraic averaging so that at each iteration the averaging step in the Lie algebra is robust. [sent-189, score-0.627]

61 As the Lie algebra is a vector space, our problem of robust averaging in the Lie algebra is analogous to robust estimation for a linear system of equations. [sent-193, score-0.484]

62 Recent work in compressive sensing [2] has shown that we can efficiently and accurately estimate x in the presence of outliers by solving argmxin ||Ax − b||? [sent-200, score-0.121]

63 If we consider the presence of outliers in the observed relative rotations in the Lie algebra at any given iteration, we have Δωrel = AΔωglobal + e. [sent-208, score-0.578]

64 1 robust rotation averaging method (denoted as L1RA) can be stated as Algorithm 1 where step 3 is solved as the minimizer of | |AΔωglobal Δωrel | |? [sent-210, score-0.483]

65 Here Δωglobal ∈ R3|V| and Δωrel ∈ R3|E| since each rotation angle i∈s a 3-vector. [sent-213, score-0.183]

66 1 rotation average estimate in the presence of outliers, we can further improve this solution by treating the problem of robust rotation averaging as one of robust regression or M-estimator modifications of least squares estimation. [sent-221, score-0.818]

67 Since the observations of relative rotations are corrupted by the presence of outliers, we take recourse to robust estimation using the ? [sent-225, score-0.477]

68 1 minimization is accurate enough, the fitting error for individual relative rotations gives us a good estimate of the reliability of the input Rij . [sent-229, score-0.517]

69 We utilise this information to iteratively solve for a robust weighted least squares averaging of the relative rotations. [sent-230, score-0.457]

70 Instead of using the stan524 dard least squares cost function eTe where e = Ax − b, we choose to minimize a robust verseion w hofe trhee e ec o=st Afunxc −tio bn,, i. [sent-240, score-0.108]

71 Since our L1RA method is efficient and provides a good estimate of Rglobal we use its output as the initial guess for robust rotation averaging using the IRLS estimator. [sent-270, score-0.552]

72 We can now state our complete robust rotation averaging algorithm (denoted as L1-IRLS) in terms of the steps in Algorithm 1 as Algorithm 3. [sent-271, score-0.464]

73 The Notre Dame dataset is available as raw images4 and for the Quad dataset, relative rotations Rij have been provided by the authors of [3]5. [sent-282, score-0.427]

74 In both cases, the results of bundle adjustment are provided with the datasets and serve as ground truth for all experiments in this Section. [sent-283, score-0.137]

75 For the Notre Dame dataset, we estimate relative rotations by running two-frame bundle adjustment on image pairs using the bundler [13] software. [sent-284, score-0.547]

76 In Table 1, we have also indicated the number of cameras (V) and number of relative rotations Rij (E) ofofr c baomthe rdaasta (sVe)ts. [sent-285, score-0.468]

77 For the results in Table 1, both these methods terminate when the maximum change of rotations in Rglobal between iterations is less than a threshold of ? [sent-288, score-0.367]

78 Instead, running the L1RA method for 5 iterations is sufficient to bring the estimate of Rglobal within the basin of convergence of the IRLS step. [sent-291, score-0.109]

79 Note that the median error and computational time for DISCO are reproduced from [3] and datasets provided by the authors and that their results were obtained on a computer that is substantially more powerful than ours. [sent-301, score-0.138]

80 For the Quad dataset, the Weiszfeld method does poorly and has a high median error of 6. [sent-319, score-0.116]

81 Even with this significant difference in computational power, for the Quad dataset our L1-IRLS averaging method is twice as accurate and is an order of magnitude faster. [sent-337, score-0.242]

82 Error Distribution: Although we have used the median error as our basis of comparison in Table 1, for a more detailed analysis of relative performance we need to use statistics instead of a single number. [sent-339, score-0.18]

83 (a) shows the histograms of errors of individual camera rotation estimates for different methods; (b) represents the fraction of errors below a given level, i. [sent-343, score-0.301]

84 Note that the scale for the iterations (x- axis) is logarithmic and the number of iterations does not reflect the time taken for the different methods. [sent-348, score-0.137]

85 The comparative performance in terms of median error is better represented here as we can observe that the Weiszfeld method of [8] does poorly with a widely spread distribution of error values. [sent-357, score-0.15]

86 It will also be noticed that apart from a large spread of errors, the Weiszfeld method also has two minor peaks in the distribution for very high rotation errors. [sent-358, score-0.214]

87 Reconstruction of the Notre Dame dataset using our robust rotation average results. [sent-360, score-0.222]

88 For each method, for a given level of rotation error (θ), we plot the fraction of estimated camera rotations that have an error less than θ. [sent-365, score-0.59]

89 As in precision-recall curves, the better method has a higher curve implying that a larger number of rotation estimates have better accuracy than a given error bound of θ. [sent-366, score-0.262]

90 1(c) we show the convergence behavior of the Weiszfeld method of [8] and our methods by plotting the median error as a function of the iteration number. [sent-379, score-0.123]

91 Note that the scale for the iterations is logarithmic as the Weiszfeld method takes 1271 iterations to converge while our methods converge in far fewer iterations. [sent-380, score-0.181]

92 1(c), the iterations at which all the methods meet the termination criterion are indicated by dots on the respective convergence curves. [sent-382, score-0.135]

93 The comparative behaviour of the convergence curves is demonstrable evidence that the Weiszfeld update is unsuitable for large graphs and we should use a joint update of all rotations as is done by our methods. [sent-387, score-0.392]

94 3D Reconstruction: Finally, as an illustration of the correctness of our robust rotation averaging, we carry out a full 3D reconstruction of the Notre Dame dataset using our rotation estimate. [sent-388, score-0.433]

95 When the rotation is fixed, estimating the 3D structure as well as 3D camera translations can be robustly solved using an SOCP optimization [11], i. [sent-389, score-0.258]

96 For large-scale datasets, even a small improvement in rotation estimation has major implications for the speed and accuracy of downstream processing in the SfM pipeline. [sent-396, score-0.183]

97 We utilise both the geometric structure of the SO(3) Lie group and also carry out a joint update for all rotations simultaneously. [sent-401, score-0.423]

98 1 optimization in the L1RA approach, we are able to efficiently solve the relative rotation averaging problem in the presence of outliers. [sent-403, score-0.547]

99 Conclusion In summary, we have developed a robust method for averaging relative rotations that outperforms the state-ofthe-art DISCO method of [3] as well as the Weiszfeld method of [8]. [sent-410, score-0.684]

100 Distributed pose averaging in camera networks via consensus on SE(3). [sent-516, score-0.285]

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