nips nips2002 nips2002-137 knowledge-graph by maker-knowledge-mining

137 nips-2002-Location Estimation with a Differential Update Network

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Author: Ali Rahimi, Trevor Darrell

Abstract: Given a set of hidden variables with an a-priori Markov structure, we derive an online algorithm which approximately updates the posterior as pairwise measurements between the hidden variables become available. The update is performed using Assumed Density Filtering: to incorporate each pairwise measurement, we compute the optimal Markov structure which represents the true posterior and use it as a prior for incorporating the next measurement. We demonstrate the resulting algorithm by calculating globally consistent trajectories of a robot as it navigates along a 2D trajectory. To update a trajectory of length t, the update takes O(t). When all conditional distributions are linear-Gaussian, the algorithm can be thought of as a Kalman Filter which simpliﬁes the state covariance matrix after incorporating each measurement.

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 edu Abstract Given a set of hidden variables with an a-priori Markov structure, we derive an online algorithm which approximately updates the posterior as pairwise measurements between the hidden variables become available. [sent-2, score-0.576]

2 The update is performed using Assumed Density Filtering: to incorporate each pairwise measurement, we compute the optimal Markov structure which represents the true posterior and use it as a prior for incorporating the next measurement. [sent-3, score-0.352]

3 We demonstrate the resulting algorithm by calculating globally consistent trajectories of a robot as it navigates along a 2D trajectory. [sent-4, score-0.293]

4 To update a trajectory of length t, the update takes O(t). [sent-5, score-0.409]

5 When all conditional distributions are linear-Gaussian, the algorithm can be thought of as a Kalman Filter which simpliﬁes the state covariance matrix after incorporating each measurement. [sent-6, score-0.128]

6 Given a sequence of pairwise measurements between the elements of the chain (for example, their differences, corrupted by noise) we are asked to reﬁne our estimate of their values online, as these pairwise measurements become available. [sent-8, score-0.664]

7 We use this mechanism to recover the trajectory of a robot given noisy measurements of its movement between points in its trajecotry. [sent-10, score-0.548]

8 These pairwise displacements are thought of as noise corrupted measurements between the true but unknown poses to be recovered. [sent-11, score-0.529]

9 The recovered trajectories are consistent in the sense that when the camera returns to an already visited position, its estimated pose is consistent with the pose recovered on the earlier visit. [sent-12, score-0.86]

10 Pose change measurements between two points on the trajectory are obtained by bringing images of the environment acquired at each pose into registration with each other. [sent-13, score-0.919]

11 The required transformation to affect the registration is the pose change measurement. [sent-14, score-0.443]

12 There is a rich literature on computing pose changes from a pair of scans from an optical sensor: 2D [5, 6] and 3D transformations [7, 8, 9] from monocular cameras, or 3D transformations from range imagery [10, 11, 12] are a few examples. [sent-15, score-0.377]

13 These have been used by [1, 2] in 3D model acquisition and by [3, 4] in robot navigation. [sent-16, score-0.149]

14 The trajectory of the robot is deﬁned as the unknown pose from which each frame was acquired, and is maintained in a state vector which is updated as pose changes are measured. [sent-17, score-1.053]

15 Figure 1: Independence structure of a differential update network. [sent-18, score-0.231]

16 An alternative method estimates the pose of the robot with respect to ﬁxed features in the world. [sent-19, score-0.45]

17 These methods represent the world as a set of features, such as corners, lines, and other geometric shapes in 3D [13, 14, 15] and match features between a scan at the current pose and the acquired world representation. [sent-20, score-0.373]

18 However, measurements are still pairwise, since they depend on a feature and the poses of the camera. [sent-21, score-0.334]

19 Because both the feature list and the poses are maintained in the state vector, the differential Update Framework can be applied to both scan-based methods and feature-based methods. [sent-22, score-0.247]

20 Our algorithm incorporates each pose change measurement by updating the pose associated with every frame encountered. [sent-23, score-0.821]

21 To insure that each update can happen in time linear to the length of the trajectory, the correlation structure of the state vector is approximated with a simpler Markov chain after each measurement. [sent-24, score-0.228]

22 In addition, we focus on frame-based trajectory estimation due to the ready availability of pose change estimators, and to avoid the complexity of maintaining an explicit feature map. [sent-28, score-0.552]

23 Sections 3 and 4 sketch existing batch and online methods for obtaining globally consistent trajectories. [sent-30, score-0.27]

24 Section 5 derives the update rules for our algorithm, which is then applied to a 2D trajectory estimation in section 6. [sent-31, score-0.328]

25 We assume the hidden variables xt have a Markov structure with known transition densities: T p(xt |xt−1 ). [sent-33, score-0.708]

26 p(X) = t=1 Pairwise measurements appear on the chain one by one. [sent-34, score-0.243]

27 Conditioned on the hidden variables, these measurements are assumed to be independent: t p(ys |xs , xt ), p(Y |X) = (s,t)∈M where M is the set of pairs of hidden variables which have been measured. [sent-35, score-0.968]

28 To apply this network to robot localization, let X = {xt }t=1. [sent-36, score-0.183]

29 T be the trajectory of the robot up to time T , with each xt denoting its pose at time t. [sent-38, score-1.229]

30 These poses can be represented using any parametrization of pose, for example as 3D rotations and translation, 2D translations (which is what we use in section 6, or even non-rigid deformations such as afﬁne. [sent-39, score-0.124]

31 (1) t As the robot moves, the pose change estimator computes the motion y s of the robot from two scans of the environment. [sent-41, score-0.768]

32 Given the true poses, we assume that these measurements t are independent of each other even when they share a common scan. [sent-42, score-0.21]

33 We model each y s as being drawn from a Gaussian centered around xt − xs : t t p(ys |xs , xt ) = N (ys |xt − xs , Λy|xx) (2) t ys The online global estimation problem requires us to update p(X|Y ) as each in Y becomes available. [sent-43, score-1.984]

34 The following section reviews a batch solution for computing p(X|Y ) using this model. [sent-44, score-0.134]

35 Section 4 discusses a recursive approach with a similar running time as the batch version. [sent-45, score-0.194]

36 Section 5 presents our approach, which performs these updates much faster by simplifying the output of the recursive solution after incorporating each measurement. [sent-46, score-0.097]

37 Because the pose dynamics are Markovian, the inverse covariance Λ −1 is tri-diagonal. [sent-48, score-0.351]

38 According X t to equation (2), the observations are drawn from ys = As,t X + ωs,t = xt − xs + ωs,t , with ωs,t white and Gaussian with covariance λs,t . [sent-49, score-1.115]

39 X Y |X (6) If there are M measurements and T hidden variables, this computation will take O(T 2 M ) if performed naively. [sent-51, score-0.266]

40 Note that if M > T , as is the case in the robot mapping problem, the alternate equations (5) and (6) can be used to obtain a running time of O(T 3 ). [sent-52, score-0.149]

41 4 Online Linear Gaussian Solution Lu and Milios [3] proposed a recursive update for updating the trajectory X|Y old after t obtaining a new measurement ys . [sent-53, score-0.847]

42 Because each measurement is independent of past measurements given the X’s, the update is: Bayes t t (7) p(X|Y old , ys ) ∝ p(ys |X)p(X|Y old ). [sent-54, score-0.97]

43 t 2 Using equations (3) and (4) to perform this update for one ys takes O(T ). [sent-55, score-0.382]

44 After integrating M measurements, this yields the same ﬁnal cost as the batch update. [sent-56, score-0.134]

45 Unfortunately, as measurements are incorporated, Λ −1 X|new becomes denser due to the accumulation of the rank 1 terms in equation (9), rendering this approach less effective. [sent-60, score-0.291]

46 5 Approximate Online Solution To implement this idea in the general case, we resort to Assumed Density Filtering (ADF) [16]: we approximate p(X|Y old ) with a simpler distribution q(X|Y old ). [sent-63, score-0.323]

47 To incorporate a t new measurement ys , we apply the update p(X|Y new ) Bayes ∝ t p(ys |xs , xt )q(X|Y old ). [sent-64, score-1.214]

48 new (10) old This new p(X|Y ) has a more complicated independence structure than q(X|Y ), so incorporating subsequent measurements would require more work and the resulting posterior would be even hairier. [sent-65, score-0.519]

49 So we approximate it again with a q(X|Y new ) that has a simpler independence structure. [sent-66, score-0.106]

50 Subsequent measurements can again be incorporated easily using this new q. [sent-67, score-0.235]

51 2 shows how to incorporate a pairwise measurement on the resulting Markov chain using equation (10). [sent-72, score-0.263]

52 1 Simplifying the independence structure We would like to approximate an arbitrary distribution which factors according to p(X) = t pt (xt |Pa[xt ]), using one which factors into q(X) = t qt (xt |Qa[xt ]). [sent-74, score-0.262]

53 Here, Pa[xt ] are the parents of node xt in the graph prescribed by p(X), and Qa[xt ][xt ] = xt−1 are the parents of node xt as prescribed by q(X). [sent-75, score-1.31]

54 The objective is to minimize: q ∗ = arg min KL q pt qt = p(X) ln x p(X) . [sent-76, score-0.151]

55 qt (xt |Qa[xt ]) i (11) After some manipulation, it can be shown that: ∗ qt = p(xt |Qa[xt ]). [sent-77, score-0.194]

56 (12) This says that the best conditional qt is built up from the corresponding pt by marginalizing out the conditions that were removed in the graph. [sent-78, score-0.202]

57 2 Computing posterior transitions on a graph with a single loop This result suggests a simpliﬁcation to the update of equation (10). [sent-81, score-0.28]

58 Because the ultimate goal is to compute q(X|Y new ), not p(X|Y new ), we only need to compute the posterior transitions p(xt |xt−1 , Y new ). [sent-82, score-0.163]

59 We propose computing these transitions in three steps, one for the transitions to the left of xs , another for the loop, and the third for transitions to the right of x t . [sent-84, score-0.321]

60 t For every s < τ < t, notice that p(y, xτ −1 , xt )p(xτ |xτ −1 , xt ) = p(y, xτ −1 , xτ , xt ), (13) because according to ﬁgure 5, p(xτ |xτ −1 , xt ) = p(xτ |xτ −1 , xt , y). [sent-89, score-2.95]

61 If we could ﬁnd this joint distribution for all τ , we could ﬁnd p(xτ |xτ −1 , y) by marginalizing out xt and normalizing. [sent-90, score-0.641]

62 We could also ﬁnd p(xτ |y) by marginalizing out both xt and xτ −1 , then normalizing. [sent-91, score-0.641]

63 Finally, we could compute p(y, xτ , xt ) for the next τ in the iteration. [sent-92, score-0.622]

64 So there are two missing pieces: The ﬁrst is p(y, xs , xt ) for starting the recursion. [sent-93, score-0.761]

65 Computing this term is easy, because p(y|xs , xt ) is the given measurement model, and p(xs , xt ) can be obtained easily from the prior by successively applying the total probability theorem. [sent-94, score-1.286]

66 Note that this quantity does not depend on the measurements and could be computed ofﬂine if we wanted to. [sent-96, score-0.21]

68 Because of the backward nature of (15), p(xτ |xτ −1 , xt ) has to be computed using a pass which runs in the opposite direction of the process of (13). [sent-101, score-0.59]

71 The robot took about 3 minutes to trace each path, producing about 6000 frames of data for each experiment. [sent-114, score-0.184]

72 The trajectory was pre-marked on the ﬂoor so we could revisit speciﬁc locations (see the rightmost diagrams of ﬁgures 6(a,b)). [sent-115, score-0.189]

73 The trajectory estimation worked at frame-rate, although it was processed ofﬂine to simplify data acquisition. [sent-117, score-0.218]

74 In these experiments, the pose parameters were (x, y) locations on the ﬂoor. [sent-118, score-0.301]

75 For each new frame, pose changes were computed with respect to at most three base frames. [sent-120, score-0.339]

76 The selection of base frames was based on a measure of appearance between the current frame and all past frames. [sent-121, score-0.153]

77 The pose change estimator was a Lucas-Kanade optical ﬂow tracker [24]. [sent-122, score-0.359]

78 To compute pose displacements, we computed a robust average of the ﬂow vectors using an iterative outlier rejection scheme. [sent-123, score-0.333]

79 We used the number of inlier ﬂow vectors as a crude estimate of the precision of t p(ys |xs , xt ). [sent-124, score-0.59]

80 The middle plots compare our algorithm (blue) against the batch algorithm which uses equations (5) and (6) (black). [sent-126, score-0.134]

81 Although our recovered trajectories don’t coincide exactly with the batch solutions, like the batch solutions, ours are smooth and consistent. [sent-127, score-0.36]

82 Estimating the motion of each frame with respect to only the previous base frame yields an unsmooth trajectory (green). [sent-129, score-0.437]

83 Furthermore, loops can’t be closed correctly (for example, the robot is not found to return to the origin). [sent-130, score-0.263]

84 The red trajectory shows what happens when we assume individual poses are independent. [sent-132, score-0.344]

85 This corresponds to using a diagonal matrix to represent the correlation between the poses (instead of the tri-diagonal inverse covariance matrix our algorithm uses). [sent-133, score-0.202]

86 Notice that the resulting trajectory is not smooth, and loops are not well closed. [sent-134, score-0.251]

87 By taking into account a minimum amount of correlation between frame poses, loops have been closed correctly and the trajectory is correctly found to be smooth. [sent-135, score-0.411]

88 7 Conclusion We have presented a method for approximately computing the posterior distribution of a set of variables for which only pairwise measurements are available. [sent-136, score-0.382]

89 We call the resulting structure a Differential Update Network and showed how to use Assumed Density Filtering to update the posterior as pairwise measurements become available. [sent-137, score-0.492]

90 The two key insights were 1) how to approximate the posterior at each step to minimize KL divergence, and 2) how to compute transition densities on a graph with a single loop in closed form. [sent-138, score-0.197]

91 We showed how to estimate globally consistent trajectories for a camera using this framework. [sent-139, score-0.209]

92 Although the example used pose change measurements between scans of the environment, our framework can be applied to feature-based mapping and localization as well. [sent-141, score-0.626]

93 (a) (b) Figure 3: Left, naive accumulation (green) and projecting trajectory to diagonal covariance (red). [sent-147, score-0.347]

94 Loops are not closed well, and trajectory is not smooth. [sent-148, score-0.241]

95 The zoomed areas show that in both naive approaches, there are large jumps in the trajectory, and the pose estimate is incorrect at revisited locations. [sent-149, score-0.332]

96 Like the batch solution, our solution generates smooth and consistent trajectories. [sent-151, score-0.168]

97 3d pose tracking with linear depth and brightness constraints. [sent-196, score-0.301]

98 Robot pose estimation in unknown environments by matching 2d range scans. [sent-200, score-0.356]

99 An iterative image registration technique with an application to stereo vision. [sent-283, score-0.109]

100 Automatic camera recovery for closed or open image sequences. [sent-287, score-0.117]

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