iccv iccv2013 iccv2013-200 knowledge-graph by maker-knowledge-mining

200 iccv-2013-Higher Order Matching for Consistent Multiple Target Tracking


Source: pdf

Author: Chetan Arora, Amir Globerson

Abstract: This paper addresses the data assignment problem in multi frame multi object tracking in video sequences. Traditional methods employing maximum weight bipartite matching offer limited temporal modeling. It has recently been shown [6, 8, 24] that incorporating higher order temporal constraints improves the assignment solution. Finding maximum weight matching with higher order constraints is however NP-hard and the solutions proposed until now have either been greedy [8] or rely on greedy rounding of the solution obtained from spectral techniques [15]. We propose a novel algorithm to find the approximate solution to data assignment problem with higher order temporal constraints using the method of dual decomposition and the MPLP message passing algorithm [21]. We compare the proposed algorithm with an implementation of [8] and [15] and show that proposed technique provides better solution with a bound on approximation factor for each inferred solution.

Reference: text


Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 l Abstract This paper addresses the data assignment problem in multi frame multi object tracking in video sequences. [sent-5, score-0.377]

2 Traditional methods employing maximum weight bipartite matching offer limited temporal modeling. [sent-6, score-0.232]

3 It has recently been shown [6, 8, 24] that incorporating higher order temporal constraints improves the assignment solution. [sent-7, score-0.323]

4 Finding maximum weight matching with higher order constraints is however NP-hard and the solutions proposed until now have either been greedy [8] or rely on greedy rounding of the solution obtained from spectral techniques [15]. [sent-8, score-0.404]

5 We propose a novel algorithm to find the approximate solution to data assignment problem with higher order temporal constraints using the method of dual decomposition and the MPLP message passing algorithm [21]. [sent-9, score-0.845]

6 We compare the proposed algorithm with an implementation of [8] and [15] and show that proposed technique provides better solution with a bound on approximation factor for each inferred solution. [sent-10, score-0.231]

7 Introduction Popularity of tracking by detection approaches [2] has led to a renewed interest in the data assignment problem in computer vision. [sent-12, score-0.273]

8 This is typically done by associating a score with each such assignment, and finding the assignment with a maximum score. [sent-15, score-0.256]

9 For example, low scores may be given to matching pairs which are visually dissimilar or are detected far from each other. [sent-17, score-0.229]

10 In a crowded scenario such scores fail to disambiguate the correct assignment from other possible assignments. [sent-18, score-0.348]

11 For example, scores which consider the velocity vectors implied by a matching, and constrain those to be physically valid are recommended for such cases. [sent-20, score-0.193]

12 If the scores factor as a sum over individual assignments, then the problem can be solved via network flow algorithms [4, 25] or as a sum of bipartite matchings [23] defined over set of every two consecutive frames. [sent-23, score-0.594]

13 , a score which depends on three frames simultaneously), and the maximization problem becomes NP hard. [sent-27, score-0.224]

14 We refer to such assignment problems with con- straints involving more than 2 frames as higher order assignment/matching problems. [sent-28, score-0.448]

15 Several approximate maximization algorithms have recently been proposed to address NP hardness of higher order assignment problems [6–8, 15]. [sent-29, score-0.38]

16 Leordeanu and Hebert [15] relax the integrality and matching constraints (a detection in one frame must be assigned to exactly one detection each in previous and next frame). [sent-30, score-0.352]

17 Since the solution obtained may not be 177 feasible, they employ a greedy rounding scheme which iteratively removes the conflicting variables to generate a feasible assignment solution. [sent-32, score-0.418]

18 Collins [8] proposed a block ICM based technique for assignment problems with constraints involving two or more frames. [sent-33, score-0.432]

19 It is similar to the iterated conditional modes (ICM) algorithm, but is applied at each step to a block of variables representing possible associations between two consecutive frames. [sent-35, score-0.257]

20 The block-optimal conditional mode at each step is calculated as the solution to a bipartite matching problem. [sent-36, score-0.267]

21 Butt and Collins [6] have proposed to solve a series of independent higher order matching problems over frame triplets which are then merged into longer trajectories. [sent-37, score-0.304]

22 Additionally there is no bound on approximation factor of the solution available with any of the discussed approaches [6, 8, 15]. [sent-40, score-0.231]

23 The problem of matching in a arbitrarily long sequence with constraints involving 3 frames can also be formulated as 3-matching problem defined over a T-partite graph (T is the overall number of frames). [sent-41, score-0.283]

24 [10] have suggested a method called COMPOSE for optimizing matching problems with additional scores on pairs of edges. [sent-45, score-0.274]

25 Given the above, our goal was to develop better approximation algorithms for the higher order assignment problem. [sent-50, score-0.301]

26 It turns out that the dual decomposition (DD) framework (see below) is a perfect fit for this problem, and provides several desirable properties. [sent-51, score-0.383]

27 Third, it can be naturally extended to other higher order scores involving three or more frames. [sent-54, score-0.267]

28 4 for a general overview and [14, 21] for applications to inference) is conceptually simple: it takes a complex score function and breaks it down into a sum of scores that can be efficiently optimized. [sent-58, score-0.298]

29 These problems are then modified using messages such that the sum of the separate maximizations yields an upper bound on the true max. [sent-59, score-0.41]

30 Finally, the messages are optimized such that the bound is as tight as possible. [sent-60, score-0.218]

31 [9] have suggested a Lagrangian relaxation scheme that is related to dual decomposition [20]. [sent-64, score-0.416]

32 However, their objective is more involved and the message passing scheme we suggest is considerably simpler than their algorithm. [sent-65, score-0.221]

33 However, the resulting algorithm is different from ours, since it needs to solve a complete flow problem in each iteration, and uses subgradient updates which typically converge more slowly than coordinate descent [e. [sent-67, score-0.205]

34 The paper is structured as follows: we first present the higher order assignment problem in Section 2 followed by a brief review of the DD approach in Section 3. [sent-70, score-0.269]

35 Specifically, we show that the proposed algorithm outperforms state of the art approaches [8, 15], yielding higher scoring assignments on various publicly available datasets [1, 2, 11], while also providing upper and lower bounds on the optimal score. [sent-73, score-0.302]

36 The goal is to find a set of paths from detections in the first frame to those in the last frame. [sent-85, score-0.195]

37 Since the X variables correspond to a set of disjoint paths, they must satisfy the constraint that each detection in frame t is assigned to a single detection in frame t + 1, and vice versa. [sent-102, score-0.323]

38 teNdex bty, we wish to construct a score function that maps each X to a number indicating how likely the proposed assignment is. [sent-124, score-0.256]

39 For example, since we know that objects tend to move in straight lines, it makes sense to give higher scores to X assignments that correspond to such trajectories, as suggested in [8]. [sent-129, score-0.285]

40 i,j,k (1) This can be simplified, by absorbing the local scores into the pairwise ones. [sent-137, score-0.176]

41 a probabilistic model here, but it is possible to and only local costs are considered, then the problem becomes easy since it can be separated into T separate bipartite matching constraints. [sent-151, score-0.232]

42 However, introducing the higher order scores makes the problem considerably more complicated, requiring approximate solution approaches. [sent-152, score-0.254]

43 In what follows we describe a simple and effective scheme for pairwise scores, which can be generalized to other higher order score functions as well. [sent-153, score-0.28]

44 Specifically, we define a set of dual variables δfi (Xi) for each factor f, each variable i ∈ Sf and each value )X fio (e. [sent-177, score-0.453]

45 These dual variables may be thought of as a message from factor f to variable i, indicating a prior on the value Xi. [sent-180, score-0.562]

46 f Next, define the following dual function L(δ) : L(δ) =? [sent-186, score-0.257]

47 One that is particularly simple and effective is to use block coordinate descent on the δ variables. [sent-196, score-0.188]

48 Here we use the MPLP algorithm [21] which fixes all messages except those from a particular f to all variables i. [sent-198, score-0.25]

49 ∈fδi−f(Xi)⎦⎤, ⎦(9) where |f| denotes the number of variables in the factor θf, and we used (Xi) to denote the sum of messages into i that are not from f. [sent-201, score-0.382]

50 3 Eventually, we are interested in an assignment for X. [sent-211, score-0.184]

51 Any such decoded assignment X provides a natural lower bound on θ∗, namely θ(X). [sent-213, score-0.306]

52 Thus, if the upper bound L(δ) and the lower bound coincide, we know we have found the θ∗ value and maximizing assignment. [sent-214, score-0.253]

53 Our functions will combine the matching constraints with the score elements from S(X). [sent-218, score-0.268]

54 For convenience, we define a function st,i (X) that contains the pairwise scores 4 corresponding to the ith detection in the tth frame: st,i(X) = ? [sent-219, score-0.237]

55 4Pairwise score in the formulation refers to the score corresponding to matching a triplet in three adjacent frames. [sent-226, score-0.286]

56 Next, define a function θt,i (X) that has a value of −∞ if the ith detection in the tth (frXa)me th avito hlaatse sa tvhaleu matching constraint. [sent-227, score-0.156]

57 (14), we introduce dual variables for messages between each factor (t, i) and the variables that participate in this factor. [sent-245, score-0.703]

58 Recall that the factor (t, i) depends on the variables Xt,i,j (i. [sent-246, score-0.196]

59 e, matchings between frame t and t + 1) and Xt−1,j,i (i. [sent-247, score-0.178]

60 e Teno fraedctuocre ( nt,o ti)a iaonnda Xt,i,j by δt,i↑j (Xt,i,j) and the message between factor (t, i) and Xt−1,j,i by δt,i↓j (Xt−1,j,i) (see figure 2). [sent-251, score-0.202]

61 The max operation in these updates involves all variables in θt,i, namely 2D variables (assuming D matching pairs in each two consecutive frames). [sent-265, score-0.455]

62 However, we note that θt,i is non-infinite only for O(D2) assignments satisfying the matching constraints, making the MPLP updatestractable. [sent-267, score-0.19]

63 ,i,k(21) The above MPLP updates monotonically decrease L(δ), providing an upper bound on the MAP. [sent-293, score-0.209]

64 To obtain an assignment from δ we consider the singleton scores θtδ,i,j (Xt,i,j) and return a matching that maximizes these. [sent-294, score-0.508]

65 ,i,jθtδ,i,j(Xt,i,j) (22) This can be solved efficiently by solving a maximum weight bipartite matching independently for each consecutive frames t and t + 1. [sent-296, score-0.371]

66 6This corresponds to a matching between k and iin frames t 1, t respecTtihvisel cyo, arnreds p boentwdese tno ia a mnda j hiinn tghe b efrtwameeens t,ta n+d 1i rines pfraemctivesel ty. [sent-298, score-0.181]

67 As mentioned earlier, in this case, the maximization of S(X) simply turns into T separate bipartite matching problems and can therefore be solved efficiently. [sent-307, score-0.388]

68 The maximization maxX θtδ,i (X) here is particularly simple since it breaks down into two separate maximizations (for the previous and next frames). [sent-313, score-0.197]

69 jδt,i↑j(0) Given this simplified form, we can now take the dual of the minimization in Eq. [sent-320, score-0.257]

70 Introduce dual variables μt,i↓j , μt,i↑j , μt,i,j for the three sets of constraints above. [sent-332, score-0.414]

71 8 In deriving the dual we actually obtain that μt,i↑j = μt+1,j↓i = μt,i,j, namely only the μt,i,j variables are needed. [sent-333, score-0.411]

72 The dual then simplifies to (up to factor 2): sm. [sent-334, score-0.35]

73 181 Algorithm 1 HO Matching Algorithm Algorithm 1HO Matching Algorithm Input: Weights Wt,i,j,kspecifying the score for matching detections i,j,k in frames t − 1,t,t + 1. [sent-342, score-0.32]

74 1: 2: 3: 4: 5: 6: 7: while Change in dual is not small enough do for All factors t,iin a random order do Calculate Wt? [sent-345, score-0.286]

75 Second, the LP for each t is in fact precisely the LP formulation of bipartite matchings, which is known to have an integral solution, and return the maximum bipartite matching (e. [sent-354, score-0.404]

76 Finally, we emphasize that our procedure will in practice return the exact matchings in many other cases, where higher order factors are not zero. [sent-363, score-0.219]

77 Since Spectral requires eigenvalue decomposition and scales quadratically with T as opposed to our method and block ICM, we evaluate it only on the short toy problem sequences. [sent-367, score-0.237]

78 The pairwise scores are set as the distance between the detection in the middle frame and the centroid of detections in the first and third frames of the frame triplet (constant velocity assumption). [sent-377, score-0.624]

79 This is one instance of higher order scores, and other scores utilizing appearance based cues could have been used. [sent-378, score-0.219]

80 However, the purpose of experiments is to study the inference capabilities of the various algorithms with higher order matching constraints when the appearance based cues are ambiguous. [sent-379, score-0.234]

81 Indicatory scores consonant with the objective have been used accordingly without compromising the generality of the algorithmic approach. [sent-380, score-0.172]

82 Furthermore, in 5/9 cases, MPLP finds a provably optimal solution (since the upper and lower bounds coincide). [sent-384, score-0.186]

83 Due to our experience in the toy problem and the scalability issues with Spectral approach we compare only to the block ICM approach. [sent-387, score-0.156]

84 The local and pairwise scores have been set similarly as in toy problem case. [sent-388, score-0.231]

85 One possible explanation for the results could be that the technique in [8] iterates through hard assignments as opposed to the message passing style of our method. [sent-390, score-0.244]

86 The assignment differences between MPLP and block ICM have been marked with white rectangles. [sent-395, score-0.285]

87 MPLP performs better than block ICM in the presence of strong matching ambiguities arising due to multiple close detections. [sent-396, score-0.238]

88 Figure 4: Change in primal and dual during MPLP iterations on PSU Seq 3 Figure 4 shows an instance of the upper and lower bounds reported by MPLP. [sent-397, score-0.477]

89 We show primal and dual val- ues after different outer iterations (an outer iteration processes each factor exactly once) of MPLP on a test dataset (PSU Seq 3). [sent-398, score-0.453]

90 As the iterations proceed, the quality/score of primal solutions keep increasing while the upper bound on the optimal primal given by the value of dual keeps tight183 ening. [sent-399, score-0.524]

91 In this problem instance, the bounds do not meet and we cannot conclude that the solution is optimal. [sent-401, score-0.158]

92 Conclusion We presented an approach for optimizing higher order assignment problems that arise in the context of tracking by detection. [sent-406, score-0.372]

93 Our approach relies on the dual decomposition framework which breaks the difficult assignment problem into simpler tractable tasks. [sent-407, score-0.575]

94 We showed the inference capability of the algorithm in the presence of pairwise matching scores arising from detections in 3 consecutive frames. [sent-408, score-0.433]

95 Such scores can successfully capture the constant velocity assumption which is a useful assignment cue in crowded scene when local scores are ambiguous. [sent-409, score-0.541]

96 9 The DD message passing framework is very general, and thus we expect it will be effective for other higher order factors that are introduced into the tracking problem. [sent-413, score-0.292]

97 As new frames arrive, we can perform a small number of message passes for the most recent frames, to obtain upper and lower bounds for the overall sequence. [sent-417, score-0.346]

98 A generalized s-d assignment algorithm for multisensormultitarget state estimation. [sent-477, score-0.184]

99 Multidimensional assignment formulation of data association problems arising from multitarget and multisensor tracking. [sent-542, score-0.341]

100 A tutorial on dual decomposition and Lagrangian relaxation for inference in natural language processing. [sent-552, score-0.382]


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