iccv iccv2013 iccv2013-314 knowledge-graph by maker-knowledge-mining
Source: pdf
Author: Zhuwen Li, Jiaming Guo, Loong-Fah Cheong, Steven Zhiying Zhou
Abstract: This paper addresses real-world challenges in the motion segmentation problem, including perspective effects, missing data, and unknown number of motions. It first formulates the 3-D motion segmentation from two perspective views as a subspace clustering problem, utilizing the epipolar constraint of an image pair. It then combines the point correspondence information across multiple image frames via a collaborative clustering step, in which tight integration is achieved via a mixed norm optimization scheme. For model selection, wepropose an over-segment and merge approach, where the merging step is based on the property of the ?1-norm ofthe mutual sparse representation oftwo oversegmented groups. The resulting algorithm can deal with incomplete trajectories and perspective effects substantially better than state-of-the-art two-frame and multi-frame methods. Experiments on a 62-clip dataset show the significant superiority of the proposed idea in both segmentation accuracy and model selection.
Reference: text
sentIndex sentText sentNum sentScore
1 j iaming Abstract This paper addresses real-world challenges in the motion segmentation problem, including perspective effects, missing data, and unknown number of motions. [sent-3, score-0.689]
2 It first formulates the 3-D motion segmentation from two perspective views as a subspace clustering problem, utilizing the epipolar constraint of an image pair. [sent-4, score-0.83]
3 It then combines the point correspondence information across multiple image frames via a collaborative clustering step, in which tight integration is achieved via a mixed norm optimization scheme. [sent-5, score-0.316]
4 The resulting algorithm can deal with incomplete trajectories and perspective effects substantially better than state-of-the-art two-frame and multi-frame methods. [sent-8, score-0.438]
5 Introduction , Previous approaches to the 3D motion segmentation problem can be roughly separated into the multi-frame and the two-frame methods. [sent-11, score-0.328]
6 Multi-frame methods have been studied mostly under the affine assumption, because under this assumption the trajectories of a rigid motion across multiple frames lie in an affine subspace with a dimension of no more than 3, or a linear subspace with a dimension of at most 4. [sent-12, score-0.937]
7 One can then solve the problem using either a factorization or a subspace separation framework [2, 7, 9, 10, 11, 14, 19, 22, 27, 28, 3 1, 34]. [sent-13, score-0.285]
8 Two-view methods are usually based on the epipolar geometry, and are thus capable of handling perspective effects. [sent-14, score-0.247]
9 The motion model fitting and selection are carried out by either statistical methods [13, 16, 24, 29] or algebraic methods [23, 32, 33]. [sent-15, score-0.27]
10 The multi-frame methods have been better developed, partly due to the elegance of its formulation and partly due to the release of the Hopkins155 database [30], which contains largely clips with little perspective effects. [sent-16, score-0.464]
11 Motion segmentation results of two sequences with strong perspective effects using SSC. [sent-20, score-0.457]
12 Firstly, multi-frame affine methods suffer from their inability to deal with perspective effects, while this presents no problem in the two-frame method; it becomes a significant consideration when using shorter lenses for shooting outdoor sequences. [sent-24, score-0.279]
13 Figure 1 shows the results of two sequences with perspective effects from Hopkins155; these results are produced by the state-of-the-art clustering algorithm sparse subspace clustering (SSC) [9]. [sent-25, score-0.715]
14 Secondly, multi-frame affine methods generally require the trajectories to have full-length. [sent-27, score-0.216]
15 If one simply filters out the trajectories which are absent in some frames, the density of the trajectories is likely to be significantly decreased, resulting in lack of coverage of many parts of the sequence. [sent-28, score-0.266]
16 Clearly, two-frame methods suffer to a much lesser extent from the missing entry issue. [sent-32, score-0.231]
17 (a) 60 trajectories obtained with the full-length requirement, and (b) 524 trajectories without the full-length requirement. [sent-34, score-0.266]
18 argue that matrix completion techniques can help to fill in the missing entries [5]. [sent-36, score-0.434]
19 Figure 2(c) shows the data matrix of the “delivery van” data, which has about 50% missing entries and is non-uniformly distributed. [sent-39, score-0.342]
20 Thirdly, the number of motion groups is usually assumed to be known a priori for multi-frame affine methods. [sent-41, score-0.429]
21 It is indeed a strong indication that model selection is actually difficult for motion segmentation. [sent-42, score-0.27]
22 Related to this issue is the fact that the number of motion groups in each clip of the Hopkins155 dataset remains unchanged throughout the frames, which makes it easy to indulge in the aforementioned assumption. [sent-43, score-0.416]
23 In real videos, the number of motion groups may change throughout a clip as moving objects enter or leave the scene. [sent-44, score-0.452]
24 On the other hand, there are clearly scenes where an observation period as short as two gle image pair, we revisit the epipolar constraint of twoperspective-view (TPV), leading to a subspace segmentation problem formulation that segments the null spaces of the appropriate equations. [sent-50, score-0.383]
25 Thus, the idea of subspace separation applies and one can follow the SSC approach in converting the motion segmentation problem into a graph partitioning problem based on an affinity matrix. [sent-51, score-0.627]
26 A more powerful formulation that integrates multiple frames then follows, in which we derive an aggregated affinity matrix from multiple image pairs and seek a joint sparse coefficient recovery across multiple image pairs, i. [sent-53, score-0.406]
27 , the sparse affinity coefficients of a particular trajectory should be consistently distributed across multiple image pairs in the sense that this trajectory should use the same set of other trajectories to express itself across allimage pairs. [sent-55, score-0.511]
28 We first make a rough model estimation by analyzing the Laplacian matrix of the affinity matrix and over-segment the data into groups. [sent-58, score-0.236]
29 Model selection remains very much an open problem in motion segmentation. [sent-84, score-0.27]
30 While the number of zero eigenvalues of the Laplacian matrix can be related to the number of connected components of the affinity matrix, the challenge lies in determining the number of eigenvalues close to zero in a robust manner [19, 26]. [sent-85, score-0.268]
31 This in turn allows us to perform merging of two over-segmented groups in a very robust way. [sent-96, score-0.235]
32 Our work pays the price of a lower trajectory density for a more accurate motion model and a higher quality data input. [sent-101, score-0.29]
33 ⎣⎡ff 132111 f 2312 f f231333⎦⎤∈ R3×3is the fundamental matrix, w⎣hich connects corre⎦spondences under the same rigid motion in two views. [sent-115, score-0.239]
34 xp yp 1 )f = 0, (2) = )T where f ( f11 f12 f13 f21 f22 f23 f31 f32 f33 is the 9 1 vector made up of the entries of F in row-major oisr tdheer. [sent-122, score-0.234]
35 Clearly, those wp under the same rigid motion k form a hyperplane perpendicular to fk, which we refer to as the TPV motion subspace. [sent-124, score-0.68]
36 Thus, in general the set of × wp for points undergoing the same rigid motion k forms a unique hyperplane perpendicular to fk. [sent-128, score-0.526]
37 From equation (3), it can be shown that wp is related to a 9 1371 3 matrix H? [sent-136, score-0.274]
38 −h23 −h13 0 It can be observed from −(4h) tha−t hthose− wp under the aforementioned degenerate configurations fall on the intersection of three hyperplanes, each of which is perpendicular to one column of H? [sent-140, score-0.378]
39 Thus, wp under these degenerate configurations live in a lower dimensional subspace with dimension no more than 6. [sent-145, score-0.438]
40 Fortunately, there are various subspace separation algorithms [9, 19] that can handle subspaces with different dimensions and the above situation should pose no special problem. [sent-146, score-0.281]
41 Sparse subspace clustering The preceding section has reduced the motion segmentation task to that of clustering subspaces of dimension at most 8 in R9 in general. [sent-150, score-0.741]
42 The SSC algorithm can be used directly to perform subspace clustering for the case of single image pair; the case of multiple image pairs requires joint sparsity and will be discussed in Section 3. [sent-152, score-0.312]
43 1 Single image pair We briefly review the SSC algorithm in the context of the TPV motion subspace: each column wp can be represented as a linear combination of the other columns wq ? [sent-157, score-0.519]
44 An alternative way is to accumulate the individual affinity matrices or adopt the multi-view spectral clustering method [36]. [sent-193, score-0.227]
45 In other words, the nonzero entries of C(l) should be sparse and those columns corresponding to the same trajectory across the different C(l) should share the same support set. [sent-198, score-0.246]
46 Notice that the correspondences can be missing in some image pairs, here “missing” means a trajectory is invisible in either one or both of the image pair. [sent-229, score-0.336]
47 In this case, we fill in with a 09× 1 column vector for the missing data so as to ensure that 9a×ll1 W(l) have the same dimension. [sent-230, score-0.233]
48 More specifically, if a trajectory p is missing in the image pair l, then in the l-th correspondence matrix W(l) , the p-th column = 09× 1. [sent-231, score-0.447]
49 Our rationales for filling in with 09× 1 are t- wp(l) × wofold: 1) when we want to obtain the sparse coding for the p-th point, the optimal solution for the missing data in the l-th image pair is 0P−1 1, not incurring any cost in equation (8), nor biasing the solution for other C(l) in any way. [sent-232, score-0.342]
50 q, the missing data will not be chosen to represent the point q in the l-th image pair since it contributes nothing to the representation of q. [sent-235, score-0.247]
51 This allows us to treat a trajectory with missing data in a uniform manner, without affecting the joint optimization scheme. [sent-236, score-0.29]
52 If a corrupted match is detected in E∗(l) , we will delete it from image pair lbut preserve the correct matches of that trajectory in other image pairs unless all matches of that trajectory are corrupted. [sent-262, score-0.339]
53 Merging via coefficient analysis As the number of motion groups is usually not known a priori in reality, we have to come to grips with the model selection problem. [sent-265, score-0.515]
54 In view of the difficulty of cluster detection, we propose to first over-segment the data based on the number of zero eigenvalues of the Laplacian matrix of the affinity matrix, and then attempt to merge the clusters later via the following model selection scheme. [sent-266, score-0.317]
55 Given a data point q ∈ RD and a group of points {piG}iMi=ve1n nst aac dkaetda as tihnet c qolu ∈mn Rs of the matrix P ∈ RD×M {anpd spanning tehde a subspace mS,n isf o we use Patr itxo represent q, ia. [sent-267, score-0.323]
56 Now consider two groups of points obtained from the over-segmentation step, P ∈ RD×M and Q ∈ RD×N, wovheors-ese cgmoleumntnatsi {np si}teiMp=,1 Pand ∈ { Rqi}iN=1 are e Qxtra ∈cte Rd from subspaces uSmu nasnd { pSv} respectively. [sent-282, score-0.264]
57 =e case for the multiple image pairs in a manner analogous to the collaborative clustering algorithm in (8). [sent-303, score-0.229]
58 Ppu0t ←: S Cetu rorfe mnto stieot no fg groups Pfor k← = C 1u →ren (tK se t− of 1) g rdouo rfo kr =ea 1ch → group pair d doo Compute relationship matrix R according to (13). [sent-326, score-0.358]
59 return Pk enrde tifu end for return Pk return P One might question what if some of the groups are too small or degenerate such that they do not adequately represent the underlying subspace S. [sent-332, score-0.418]
60 Ctruleea trhlya,t sthucehre invariably exist some other groups whose points fully span the subspace S. [sent-334, score-0.361]
61 We choose the first and the last frames of all sequences as the image pair for the testing, which ensures that all correspondences in the scene have sufficient displacements in the image plane. [sent-344, score-0.241]
62 For the sake of comparison, we assume the number of motion groups is known in this experiment, like what many algorithms did. [sent-345, score-0.346]
63 We also list the classification er- rors when applying ALC[22], GPCA[3 1], LSA[34], SSC[9] and LRR[19] to the affine motion subspace for comparison. [sent-346, score-0.489]
64 1Even if a motion group consists of say, just two walls, the degenerate case of the over-segmentation yielding two walls cleanly (and thus not mergeable) seldom arises; instead, the points of the two walls are usually segmented non-exactly by our over-segmentation step. [sent-348, score-0.437]
65 34 The results indicate that segmentation from two properly chosen views is almost as good as segmentation from the multiple views. [sent-368, score-0.26]
66 We believe that this is due to a combination of factors such as the better modeling of perspective effect and the choice of better clustering methods. [sent-370, score-0.249]
67 Results on multiple image pairs We now evaluate the complete algorithm using multiple image pairs without knowing the number of motion groups and with challenges like missing data and perspective effects. [sent-373, score-0.821]
68 Since Hopkins155 has a very unbalanced number of 2-motion and 3motion clips (120 and 35 respectively), we retain only the 50 original seed videos (the other 105 2-motion clips are created by splitting off from the 3-motion clips). [sent-375, score-0.47]
69 More importantly, to evaluate the performance under missing data and perspective effects, we added 12 clips with incomplete trajectories, of which 4 are from [25] and the other 8 are captured by us using a handheld camera with a wide angle lens. [sent-376, score-0.659]
70 The newly captured sequences contain about 100 frames each, some of which experience heavy occlusions, posing significant challenge to the matrix completion task, as we shall see later. [sent-377, score-0.311]
71 Of the resultant 62 motion clips, 26 contain two motions, 36 contain three motions, 12 suffer from missing data, and 9 have strong perspective effects (some of these categories are not mutually exclusive). [sent-378, score-0.671]
72 Classification results on 62-clip dataset MethodALCGPCALBFLRRMSMCORKSSCM-TPV Classification error (%) - clips with perspective effect: 9 clips Mean16. [sent-385, score-0.675]
73 46) Classification (%) clips with missing data: 12 clips Mean25. [sent-401, score-0.668]
74 37) Group number estimation all 62 clips # correct2133293525373346 error - error - error - - second order difference (SOD) method as in LBF. [sent-449, score-0.361]
75 For those algorithms which do not explicitly handle missing data, such as LBF, LRR, ORK and SSC, we recover the data matrix using Chen’s matrix completion approach[5], which in our experience has the best performance among various competing algorithms (such as OptSpace[20], GROUSE[1] and etc. [sent-451, score-0.436]
76 Since the estimated number of motion groups may not be the same as the ground truth number, we exhaustively test all the cluster pairings to obtain the best error rates. [sent-455, score-0.388]
77 Furthermore, to investigate if good model selection results in good segmentation, the error rates obtained by only considering sequences where the number of motions is correctly estimated are shown in the bracket. [sent-456, score-0.291]
78 We also show some qualitative results obtained with the newly captured clips in Figure 4. [sent-457, score-0.235]
79 In the first part, the classification error rates of the 9 clips with strong perspective effects are presented. [sent-459, score-0.558]
80 Although ALC and MSMC also reported good results when the number of motion groups is correctly estimated, perspective effects have a significant detrimental impact on their model selection steps, resulting in substantially higher error rates of ALC and MSMC. [sent-461, score-0.702]
81 GPCA broke down mainly due to the instability of the Power Factorization method used for filling in missing data. [sent-464, score-0.246]
82 Of the only sequence whose motion number is correctly estimated (the “Van” clip, last row of Figure 4), LRR has a very poor classification error rate. [sent-467, score-0.279]
83 MSMC failed in those sequences with complicated objects and backgrounds due to its simple motion model based on homography. [sent-468, score-0.283]
84 These clips are relatively easy, because they have complete trajectories. [sent-471, score-0.235]
85 The average classification error of our method on all 50 clips is 7. [sent-472, score-0.316]
86 56% is clearly the best compared to other stateof-the-art motion segmentation algorithms. [sent-476, score-0.328]
87 These figures also demonstrate that model selection remains a recalcitrant problem, and to achieve real progress in motion segmentation, we must meet this challenge heads-on. [sent-477, score-0.301]
88 It has 46 correct motion number estimation out of 62 clips (next best is 37), and the average classification error of all clips is 7. [sent-479, score-0.749]
89 These overall performances demonstrate that our method is capable of handling the var- ious real challenges in the motion segmentation problem. [sent-482, score-0.359]
90 Conclusions We solve the 3D motion segmentation problem of multiple frames rooted in the epipolar geometry of two perspective views via a collaborative clustering algorithm. [sent-484, score-0.773]
91 Qualitative GPCA LBF LRR results of the real data with missing entries. [sent-487, score-0.229]
92 MSMC ORK SSC M-TPV The segmentation results of the 50-th frames of the sequences are presented. [sent-488, score-0.276]
93 Leveraging on this, we first over-segment the motion groups, and then merge them according to the relationships. [sent-491, score-0.245]
94 Multibody factorization with uncertainty and missing data using the em algorithm. [sent-556, score-0.274]
95 Two-view motion segmentation with model selection and outlier removal by ransac-enhanced dirichlet process mixture models. [sent-566, score-0.454]
96 Track to the future: Spatiotemporal video segmentation with long-range motions cues. [sent-577, score-0.222]
97 Robust algebraic segmentation of mixed rigid-body and planar motions from two views. [sent-629, score-0.279]
98 A benchmark for the comparison of 3-d motion segmentation algorithms. [sent-665, score-0.328]
99 Motion segmentation with missing data by power factorization and generalized pca. [sent-670, score-0.404]
100 A general framework for motion segmentation: Independent, articulated, rigid, non-rigid, degenerate and non-degenerate. [sent-687, score-0.266]
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