cvpr cvpr2013 cvpr2013-301 knowledge-graph by maker-knowledge-mining
Source: pdf
Author: Xinchu Shi, Haibin Ling, Junling Xing, Weiming Hu
Abstract: In this paper we formulate multi-target tracking (MTT) as a rank-1 tensor approximation problem and propose an ?1 norm tensor power iteration solution. In particular, a high order tensor is constructed based on trajectories in the time window, with each tensor element as the affinity of the corresponding trajectory candidate. The local assignment variables are the ?1 normalized vectors, which are used to approximate the rank-1 tensor. Our approach provides a flexible and effective formulation where both pairwise and high-order association energies can be used expediently. We also show the close relation between our formulation and the multi-dimensional assignment (MDA) model. To solve the optimization in the rank-1 tensor approximation, we propose an algorithm that iteratively powers the intermediate solution followed by an ?1 normalization. Aside from effectively capturing high-order motion information, the proposed solver runs efficiently with proved convergence. The experimental validations are conducted on two challenging datasets and our method demonstrates promising performances on both.
Reference: text
sentIndex sentText sentNum sentScore
1 edu Abstract In this paper we formulate multi-target tracking (MTT) as a rank-1 tensor approximation problem and propose an ? [sent-5, score-0.856]
2 In particular, a high order tensor is constructed based on trajectories in the time window, with each tensor element as the affinity of the corresponding trajectory candidate. [sent-7, score-1.731]
3 Our approach provides a flexible and effective formulation where both pairwise and high-order association energies can be used expediently. [sent-10, score-0.49]
4 To solve the optimization in the rank-1 tensor approximation, we propose an algorithm that iteratively powers the intermediate solution followed by an ? [sent-12, score-0.714]
5 The former tracks multiple targets with observations till the current frame, while the latter collects a batch of evidences within a time span and treat tracking as a multi-frame multiple target association1 problem. [sent-19, score-0.369]
6 Sequential tracking is suitable for online tasks [6, 24], but sometimes meets problems when dealing with target occlusions. [sent-20, score-0.161]
7 Relation of tensor and association, with a 4-frame as- sociation as an example. [sent-23, score-0.662]
8 Each trajectory has a corresponding tensor item computed from the trajectory affinity. [sent-27, score-1.093]
9 However, the integer optimization in MDA is NP-hard for three or higher dimensional association in general. [sent-38, score-0.476]
10 Some alternative works evade the global association by using hierarchical strategies [15, 7], the optimum local associations are achieved first and used to obtain longer tracks later. [sent-39, score-0.57]
11 In this work, we propose a tensor based approach for multi-frame multi-target association. [sent-42, score-0.662]
12 First, we construct a high-order tensor from all trajectory candidates over a time span, as illustrated in Figure 1. [sent-43, score-0.879]
13 Then, we show that the rank1 approximation of this tensor has the same energy formulation as the multi-dimensional assignment. [sent-44, score-0.769]
14 1 tensor power iteration with row/column unit norm is introduced to solve the approximation problem. [sent-46, score-1.105]
15 To validate the proposed method, we apply it to multi-target tracking using two challenging datasets, one containing wide area motion sequences and the other containing public area surveillance videos. [sent-51, score-0.197]
16 Related work Study of data association has a long history, with early research focusing on radar target tracking [3], where Multiple Hypothesis Tracking (MHT) [21] is the classic method. [sent-58, score-0.588]
17 With a batch of observations, MHT finds all possible association combinations and selects the most likely association set as the optimal solution. [sent-59, score-0.918]
18 Multiple target association across multiple frames can be formulated as the multiple dimensional assignment problem. [sent-61, score-0.629]
19 iK ei sa f1i nwithyen o fth teh trajectory yis { tirue and 0} otherwise; ij denotes the observation index in j-th frame. [sent-107, score-0.189]
20 Two-frame association is a special case of MDA, and exact solutions with polynomial time such as Hungarian algorithm are available. [sent-108, score-0.485]
21 However, the solution is NP-hard when the association is computed over three or more frames. [sent-109, score-0.453]
22 When the cost of the trajectory is decomposed as the product of pairwise terms, MDA can be formulated as a network flow problem, which can be solved by using linear programming [16], shortest path algorithms [5], etc. [sent-112, score-0.372]
23 Such network flow formulation, while having global optimal solutions with polynomial time com- plexity, is limited to use pairwise affinity and misses high order kinematic information. [sent-113, score-0.33]
24 Other approaches for solving MDA include greedy search [23, 25] and hierarchical target association [15, 7]. [sent-118, score-0.48]
25 Our approach shares a similar procedure, but we use the tensor framework and propose an analytical iterative solution. [sent-120, score-0.662]
26 In addition, the association ambiguity is retained in our iteration, which reduces the association errors. [sent-121, score-0.916]
27 Tensor formulation In this section, we give a brief introduction about tensor and its rank-1 approximation. [sent-123, score-0.704]
28 A tensor is the high dimensional generalization of a matrix. [sent-124, score-0.662]
29 In the tensor terminology, each dimenasinodn o1f a te ins≤or i sI associated with a mode. [sent-132, score-0.662]
30 Like matrix-vector and matrix-matrix multiplication, tensor has similar operations, we give the following definition. [sent-133, score-0.662]
31 Definition 1 The n-mode product of a tensor S ∈ RDeI1f ×in. [sent-134, score-0.714]
32 Rank-1 tensor approximation Before we introduce Rank-1 tensor approximation, the notation of Rank-1 tensor is given first. [sent-181, score-2.051]
33 If the K order tensor S is computed as the outer product of K vectors tΠen(1s)o, rΠ S(2) i,s . [sent-182, score-0.772]
34 With the ab-othve ( definition, the problem of rank-1 approximation for tensor S is formulated as following: Problem 1 Given a real K order tensor S ∈ RI1×I2×···×IK, find K unit-norm vectors ΠS =∈ {Π(1), Π(2), . [sent-199, score-1.439]
35 (9) To maximize (9), tensor power iteration ([10, 20]) is proposed with a sound convergence proof. [sent-247, score-0.862]
36 Relations to Multi-Dimensional Assignment In this section, we show the rank-1 tensor approximation has the similar optimization formulation with MDA, with an appropriate tensor item definition. [sent-252, score-1.51]
37 Each trajectory (global association) is decomposed as a sequence of edges (twoframe association), which is formulated as = xi1i2. [sent-255, score-0.237]
38 Note the affinity remains depending on the entire trajectory but with a different subscript, i. [sent-266, score-0.344]
39 Also, we note that the affinity values s and the association variables π in (13) are non-negative, thus the equivalence between (9) and (13) is self-evident. [sent-308, score-0.582]
40 Specifically, Π in tensor approximation must have the ? [sent-310, score-0.727]
41 With this extension, the tensor approximation can be considered as the counterpart of MDA in the real-value domain. [sent-315, score-0.727]
42 An example illustrating the relation between rank-1 tensor approximation and MDA is given in Figure 1. [sent-316, score-0.727]
43 It shows that (1) tensor elements correspond to global associations; and (2) vectors approximating the rank-1 tensor are real solutions of local assignment variables. [sent-317, score-1.506]
44 (7), it aims to minimize the element-wise 222333888977 Algorithm 1 Tensor based multi-target association 1:Input. [sent-319, score-0.427]
45 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: t0: Start frame, K:Number of a batch frames Output: target associations while t0 K−1 ≤ M do Collect+ a b−at1ch ≤ o Mf K do frames observations Φ = {O(t0) , O(t0 + 1) , . [sent-321, score-0.298]
46 Compute the trajectory affinities and construct the K−1 order tensor S. [sent-326, score-0.851]
47 t0 ← t0 + K−1 end w←hil te + reconstruction error between the the trajectory tensor and the reconstructed tensor, which is calculated as the outer product of local assignment vectors. [sent-331, score-1.021]
48 In particular, for a trajectory with a high affinity, the optimization tries to make a high-value outer product to match its affinity. [sent-332, score-0.298]
49 Consequently, the higher the affinity a trajectory has, the more likely it will be picked up in the final solution. [sent-333, score-0.344]
50 Tensor Based Multi-Target Association In this section, we introduce the tensor based multi-target tracking approach, and Algorithm 1outlines the framework. [sent-335, score-0.77]
51 Generally, multi-target association is performed with the batch way. [sent-336, score-0.491]
52 When K frame observations are available, association hypotheses (trajectories) are generated first. [sent-337, score-0.491]
53 With all these hypotheses, a tensor is constructed by computing the trajectory affinities. [sent-338, score-0.851]
54 We highlight the tensor-based multi-target association in the following parts and present details about object detection in the experiment part. [sent-341, score-0.427]
55 Further, we use “soft” association variable,s1 t o≤ m kak ≤e tKhe− optimization more f e“asosifbt”le a. [sent-345, score-0.453]
56 1unit norm 1:Input: K − 1 order tensor S ∈ RI1×I2×. [sent-369, score-0.785]
57 gence Note that the original tensor power iteration implied in Theorem 1 is designed for ? [sent-409, score-0.822]
58 The basic idea is to iteratively update the solution by tensor powering followed by an ? [sent-419, score-0.688]
59 The procedure for general rank-1 tensor approximation is presented in Algorithm 2, where “◦” indicates the tHioanda ism parrde product ( eAllegmoerintth-wmis 2e, product). [sent-421, score-0.8]
60 (18) For clear expression, we denote the k-th vector at the nth iteration as Π(k)(n), which has elements Consider the iteration on Π(1) (n), with all other vectors fixed, we has following proposition. [sent-437, score-0.199]
61 1 unit norm iteration algorithm converges to a (local) extreme. [sent-530, score-0.304]
62 Set the association problem (16) as an example, the ? [sent-535, score-0.427]
63 1 unit norm tensor iteration for Π(1) has the formulation as Πˆj(+1)1 = (S⊗2Π(j2) ⊗3Πj(3). [sent-536, score-1.008]
64 Hypothesis generation We follow the traditional approaches to set a bound for association generation. [sent-576, score-0.427]
65 Basically, we make an association hypothesis between two object candidates from two consecutive frames only when they are spatially close to each other. [sent-577, score-0.53]
66 For handling this issue, in each frame we include two dummy targets, a source and a sink, to generate the entrance and exit association for each real target. [sent-582, score-0.528]
67 Tensor computation The tensor S is constructed based on the global associatTihoen hypothesis, w coitnhs tthruec trajectory a ofnfin tihteies g as tahle a tseson-sor elements. [sent-585, score-0.885]
68 Given different affinity representations, there are different optimization formulations. [sent-586, score-0.181]
69 When the affinity is computed as the product of pairwise costs, i. [sent-612, score-0.228]
70 o1πrmsl(∗K u− 1la)tπiol(Kn− 1in)etw(2o8rk) flow ([5]), thus network flow is a special case of the proposed tensor framework. [sent-627, score-0.749]
71 To summarize, tensor approximation provides a flexible framework to take advantage of global and local association energy. [sent-628, score-1.188]
72 Detailed representation about the affinity computation is presented in the experiment section. [sent-629, score-0.176]
73 Initialization and termination The initial point is important for tensor iteration. [sent-632, score-0.662]
74 For exam222333889199 ple, when one target has 4 association candidates, the initial value for each candidate is 1/4. [sent-634, score-0.48]
75 The solution given by tensor power iteration is realvalue, which must be discretized to meet the integer and one-to-one mapping constraints in the assignment. [sent-636, score-0.871]
76 To leverage the conflicts between different local association candi- × dates, we treat the real-value solutions as the costs for corresponding association candidates and feed them into a bipartite problem. [sent-637, score-0.916]
77 The affinity of a trajectory is defined as sl1l2. [sent-657, score-0.344]
78 lK−1 , (29) where alt is the affinity of the local association lt and is computed using histogram appearance and bounding box Table 1. [sent-666, score-0.612]
79 , (30) where Ult is the velocity vector of association lt. [sent-696, score-0.427]
80 We use the same affinity (29) in our approach and ICMlike, and alt is used as the cost in Hungarian algorithm. [sent-698, score-0.185]
81 One reason lies in that the association ambiguity is retained in the iteration process in our approach till final decision. [sent-713, score-0.624]
82 In this experiment, we compare the proposed tensorbased association with ICM-like. [sent-723, score-0.427]
83 We use the same affinity in [8] defined as s = E0 − Econt − Ecurv ? [sent-724, score-0.155]
84 5); pi is the target position in frame i; E0 is a large constant to make the affinity positive; Econt penalizes large position jumps between successive point pair; and Ecurv defines the constantvelocity motion model. [sent-747, score-0.306]
85 The large motion offsets of targets in the low frame-rate dense scene cause multiple association possibilities, which confuse the association algorithms. [sent-751, score-0.973]
86 the association ambiguity till the final binarization stage, thus acquires a better result than does ICM-like. [sent-760, score-0.513]
87 The performance gains of our approach in sparse and higher framerate sequences are small, since the results of ICM-like are close to saturation, and there are less association ambiguities for the data too. [sent-761, score-0.455]
88 Tensor based association has a computation complexity of O(fn) in each iteration, n is the length of the table (i. [sent-768, score-0.427]
89 By contrast, ICM-like has a complexity of O(mn), where m is the total number of two-frame association candidates. [sent-771, score-0.427]
90 Because the iteration on each variable in our approach only needs lookup-table operations, while the iteration of ICM-like on each variable needs the global search across the table. [sent-772, score-0.206]
91 Conclusion In this work, we first consider the global trajectory as the high-order tensor item, and formulate the multiple dimensional assignment task as the (row/column) constrained tensor approximation problem. [sent-776, score-1.72]
92 1 unit norm tensor power iteration algorithm is proposed to solve the optimization, and we provide the convergence proof. [sent-778, score-1.08]
93 The two features in our approach, using global trajectory affinity and maintaining the association ambiguity, advance the global association performance. [sent-779, score-1.266]
94 Power iteration for row/column unit norm Given a matrix E ∈ RM×N, constitutes of elements epq(1 ≤ p ≤ M, n1 a ≤ q ≤ N). [sent-790, score-0.377]
95 1 unit norm iteration, the total score converges to the extreme with each row/column unit iteration. [sent-825, score-0.333]
96 Robust object tracking by hierarchical association of detection responses. [sent-925, score-0.535]
97 Markov chain monte carlo data association for multi-target tracking. [sent-937, score-0.472]
98 Multidimensional assignment formulation of data association problems arising from multitarget and multisensor tracking. [sent-947, score-0.583]
99 Gmcp-tracker: Global multiobject tracking using generalized minimum clique graphs. [sent-986, score-0.157]
100 Global data association for multiobject tracking using network flows. [sent-992, score-0.621]
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