nips nips2006 nips2006-153 knowledge-graph by maker-knowledge-mining

153 nips-2006-Online Clustering of Moving Hyperplanes

Source: pdf

Author: René Vidal

Abstract: We propose a recursive algorithm for clustering trajectories lying in multiple moving hyperplanes. Starting from a given or random initial condition, we use normalized gradient descent to update the coefﬁcients of a time varying polynomial whose degree is the number of hyperplanes and whose derivatives at a trajectory give an estimate of the vector normal to the hyperplane containing that trajectory. As time proceeds, the estimates of the hyperplane normals are shown to track their true values in a stable fashion. The segmentation of the trajectories is then obtained by clustering their associated normal vectors. The ﬁnal result is a simple recursive algorithm for segmenting a variable number of moving hyperplanes. We test our algorithm on the segmentation of dynamic scenes containing rigid motions and dynamic textures, e.g., a bird ﬂoating on water. Our method not only segments the bird motion from the surrounding water motion, but also determines patterns of motion in the scene (e.g., periodic motion) directly from the temporal evolution of the estimated polynomial coefﬁcients. Our experiments also show that our method can deal with appearing and disappearing motions in the scene.

Reference: text

Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 edu Abstract We propose a recursive algorithm for clustering trajectories lying in multiple moving hyperplanes. [sent-5, score-0.71]

2 As time proceeds, the estimates of the hyperplane normals are shown to track their true values in a stable fashion. [sent-7, score-0.329]

3 The segmentation of the trajectories is then obtained by clustering their associated normal vectors. [sent-8, score-0.519]

4 The ﬁnal result is a simple recursive algorithm for segmenting a variable number of moving hyperplanes. [sent-9, score-0.534]

5 We test our algorithm on the segmentation of dynamic scenes containing rigid motions and dynamic textures, e. [sent-10, score-0.451]

6 Our method not only segments the bird motion from the surrounding water motion, but also determines patterns of motion in the scene (e. [sent-13, score-0.548]

7 , periodic motion) directly from the temporal evolution of the estimated polynomial coefﬁcients. [sent-15, score-0.265]

8 A natural extension i=1 of PCA is subspace clustering, which refers to the problem of ﬁtting a union of n ≥ 1 linear subspaces {Sj ⊂ RD }n of unknown dimensions dj = dim(Sj ), 0 < dj < D, to N points j=1 X = {xi ∈ RD }N drawn from ∪n Sj , without knowing which points belong to which subspace. [sent-18, score-0.211]

9 Existing methods randomly choose a basis for each subspace, and then iterate between data segmentation and standard PCA. [sent-21, score-0.208]

10 Then, a set of polynomials is ﬁtted to the projected data points and a basis for each one of the projected subspaces is obtained from the derivatives of these polynomials at the data points. [sent-25, score-0.223]

11 the subspace bases and the segmentation of the data are obtained after all the data points have been collected. [sent-28, score-0.329]

12 In addition, existing methods are designed for clustering data lying in a collection of static subspaces, i. [sent-29, score-0.218]

13 , dynamic texture segmentation, one typically applies them to a moving time window, under the assumption that the subspaces are static within that window. [sent-34, score-0.403]

14 A major disadvantage of this approach is that it does not incorporate temporal coherence, because the segmentation and the bases at time t + 1 are obtained independently from those at time t. [sent-35, score-0.34]

15 In this paper, we propose a computationally simple and temporally coherent online algorithm for clustering point trajectories lying in a variable number of moving hyperplanes. [sent-37, score-0.477]

16 We model a union of n moving hyperplanes in RD , Sj (t) = {x ∈ RD : b⊤ (t)x = 0}, j = 1, . [sent-38, score-0.612]

17 , n, where b(t) ∈ RD , j as the zero set of a polynomial with time varying coefﬁcients. [sent-41, score-0.272]

18 Starting from an initial polynomial at time t, we compute an update of the polynomial coefﬁcients using normalized gradient descent. [sent-42, score-0.551]

19 The hyperplane normals are then estimated from the derivatives of the new polynomial at each trajectory. [sent-43, score-0.518]

20 The segmentation of the trajectories is obtained by clustering their associated normal vectors. [sent-44, score-0.519]

21 As time proceeds, new data are added, and the estimates of the polynomial coefﬁcients are more accurate, because they are based on more observations. [sent-45, score-0.27]

22 This not only makes the segmentation of the data more accurate, but also allows us to handle a variable number of hyperplanes. [sent-46, score-0.233]

23 We test our approach on the challenging problem of segmenting dynamic textures from rigid motions in video. [sent-47, score-0.327]

24 2 Recursive estimation of a single hyperplane In this section, we review the normalized gradient algorithm for estimating a single hyperplane. [sent-48, score-0.285]

25 We consider both static and moving hyperplanes, and analyze the stability of the algorithm in each case. [sent-49, score-0.26]

26 8, page 77 of [5], the following normalized gradient recursive identiﬁer ˆ (b(t)⊤ x(t) − y(t)) ˆ ˆ b(t + 1) = b(t) − µ x(t), (2) 1 + µ x(t) 2 ˆ where µ > 0 is a ﬁxed parameter, is such that b(t) → b exponentially if the regressors {x(t)} are persistently exciting, i. [sent-56, score-0.581]

27 As shown in [6], if the regressors {x(t)} are persistently exciting and the sequence {b(t+1)−b(t)} is L2 -stable, i. [sent-63, score-0.311]

28 sup b(t+1)−b(t) 2 < ∞, then the normalized t≥1 ˆ ˆ gradient recursive identiﬁer (2) produces an estimate b(t) of b(t) such that {b(t)− b(t)} is L2 -stable. [sent-65, score-0.38]

29 Let {x(t)} be a set of measurements lying in the moving hyperˆ plane S(t) = {x ∈ RD : b⊤ (t)x = 0}. [sent-67, score-0.241]

30 Notice that the main difference between linear regression and hyperplane estimation is that in the latter case the parameter vector b(t) is constrained to lie in the unit sphere SD−1 . [sent-69, score-0.233]

31 Therefore, the update equation for the normalized gradient recursive identiﬁer on the sphere is v(t) ˆ ˆ b(t + 1) = b(t) cos( v(t) ) + sin( v(t) ), (4) v(t) where the negative normalized gradient is computed as ⊤ ˆ (b (t)x(t))x(t) ˆ ˆ⊤ v(t) = −µ ID − b(t)b (t) . [sent-72, score-0.485]

32 1 + µ x(t) 2 (5) Notice that the gradient on the sphere is essentially the same as the Euclidean gradient, except that it ˆ ˆ⊤ needs to be projected onto the subspace orthogonal to ˆ by the matrix ID − b(t)b (t) ∈ RD×(D−1) . [sent-73, score-0.254]

33 Under persistence of excitation condition (6), if b(t) = b the identiﬁer ˆ ˆ (4) is such that b(t) → b exponentially, while if {b(t + 1) − b(t)} is L2 -stable, so is {b(t) − b(t)}. [sent-75, score-0.204]

34 3 Recursive segmentation of a known number of moving hyperplanes In this section, we generalize the recursive identiﬁer (4) and its stability properties to the case of N trajectories {xi (t)}N lying in n hyperplanes {Sj (t)}n . [sent-76, score-1.849]

35 However, as we do not know the segmentation of the data, we do not know which data to use to update each one of the n identiﬁers. [sent-78, score-0.208]

36 In the approach, the n hyperplanes are represented with a single polynomial whose coefﬁcients do not depend on the segmentation of the data. [sent-79, score-0.888]

37 Then there is a vector bj (t) normal to Sj (t) such that b⊤ (t)x(t) = 0. [sent-83, score-0.3]

38 j Thus, the following homogeneous polynomial of degree n in D variables must vanish at x(t): pn (x(t), t) = b⊤ (t)x(t) 1 b⊤ (t)x(t) · · · b⊤ (t)x(t) = 0. [sent-84, score-0.335]

39 2 n (7) This homogeneous polynomial can be written as a linear combination of all the monomials of degree n in x, xI = xn1 xn2 · · · xnD with 0 ≤ nk ≤ n for k = 1, . [sent-85, score-0.238]

40 Since both the normal vectors and the coefﬁcient vector are deﬁned up to scale, we will assume that bj (t) = c(t) = 1, without loss of generality. [sent-104, score-0.328]

41 Thanks to the polynomial equation (8), we now propose a new online hyperplane clustering algorithm that operates on the polynomial coefﬁcients c(t), rather than on the normal vectors {bj (t)}n . [sent-106, score-0.831]

42 Since in practice we do not know the true polynomial coefﬁˆ cients c(t), and we estimate b(t) from ˆ(t), we need to show that both ˆ(t) and b(x(t)) track their c c true values in a stable fashion. [sent-116, score-0.336]

43 Consider the recursive identiﬁer (11)–(13) and assume that the embedded regressors {νn (xi (t))}N are persistently exciting, i. [sent-120, score-0.529]

44 Furthermore, if a trajectory x(t) belongs to the jth ˆ hyperplane, then the corresponding b(x(t)) in (13) is such that bj (t) − ˆ b(x(t)) is L2 -stable. [sent-124, score-0.267]

45 If in ˆ ˆ addition the hyperplanes are static, then c(t) − c(t) → 0 and bj (t) − b(x(t)) → 0 exponentially. [sent-125, score-0.668]

46 [Sketch only] When the hyperplanes are static, the exponential convergence of c(t) to c follows with minor modiﬁcations from Theorem 2. [sent-127, score-0.468]

47 , bn are different, the polynomial c⊤ νn (x) has no repeated factor. [sent-133, score-0.212]

48 Consider now the case in which the hyperplanes are moving. [sent-138, score-0.468]

49 Since SD−1 is compact, the sequences {bj (t + 1) − bj (t)}n are trivially L2 -stable, hence so is the sequence j=1 ˆ ˆ {c(t + 1) − c(t)}. [sent-139, score-0.243]

50 Theorem 1 provides us with a method for computing an ˆ estimate b(xi (t)) for the normal to the hyperplane passing through each one of the N trajectories {xi (t) ∈ RD }N at each time instant. [sent-142, score-0.533]

51 We do so by using a recursive version of the K-means algorithm, adapted to vectors on the unit sphere. [sent-144, score-0.286]

52 Essentially, at each t, we seek the normal vectors ˆ bj (t) ∈ SD−1 and the membership of wij (t) ∈ {0, 1} of trajectory i to hyperplane j that maximize N ˆ f ({wij (t)}, {bj (t)}) = n ˆ⊤ ˆ wij (t)(bj (t)b(xi (t)))2 . [sent-145, score-0.724]

53 In order to obtain temporally coherent estimates of the normal vectors, we use the estimates at time t to initialize the iterations at time t + 1. [sent-148, score-0.216]

54 c j=1 i=1 For each t ≥ 1 1: Update the coefﬁcients of the polynomial pn (x(t), t) = ˆ(t)⊤ νn (x(t)) using the recursive procedure ˆ c v(t) sin( v(t) ), v(t) P ` ´ N (ˆ⊤ (t)νn (xi (t)))νn (xi (t))/N i=1 c ˆ c v(t) = −µ IMn (D) − c(t)ˆ⊤ (t) . [sent-150, score-0.567]

55 P 1 + µ N νn (xi (t)) 2 /N i=1 ˆ c(t + 1) = ˆ(t) cos( v(t) ) + c 2: Solve for the normal vectors from the derivatives of pn at the given trajectories ˆ ⊤ Dνn (xi (t))ˆ(t) c ˆ b(xi (t)) = ⊤ (x (t))ˆ(t) Dνn i c i = 1, . [sent-151, score-0.407]

56 , n ˆ ˆ (c) Iterate (a) and (b) until convergence of wij (t), and then set bj (t + 1) = bj (t). [sent-167, score-0.452]

57 4 Recursive segmentation of a variable number of moving hyperplanes In the previous section, we proposed a recursive algorithm for segmenting n moving hyperplanes under the assumption that n is known and constant in time. [sent-168, score-1.822]

58 However, in many practical situations the number of hyperplanes may be unknown and time varying. [sent-169, score-0.502]

59 For example, the number of moving objects in a video sequence may change due to objects entering or leaving the camera ﬁeld of view. [sent-170, score-0.298]

60 In this section, we consider the problem of segmenting a variable number of moving hyperplanes. [sent-171, score-0.276]

61 We denote by n(t) ∈ N the number of hyperplanes at time t and assume we are given an upper bound n ≥ n(t). [sent-172, score-0.502]

62 We show that if we apply Algorithm 1 with the number of hyperplanes set to n, then we can still recover the correct segmentation of the scene, even if n(t) < n. [sent-173, score-0.676]

63 Since the condition on the right hand side of (15) holds trivially when the regressors xi (t) are bounded, the only important condition is the one on the left hand side. [sent-175, score-0.339]

64 Notice that the condition on the left hand side implies that the spatial-temporal covariance matrix of the embedded regressors must be of rank Mn (D) − 1 in any time window of size S for some integer S. [sent-176, score-0.299]

65 In this case, at each time instant we draw data from all n hyperplanes and the data is rich enough to estimate all n hyperplanes at each time instant. [sent-179, score-1.109]

66 As time proceeds, however, the data must be persistently drawn from at least n hyperplanes in order for (18) to hold. [sent-187, score-0.606]

67 This can be achieved either by having n different static hyperplanes and persistently drawing data from all of them, or by having less than n moving hyperplanes whose motion is rich enough so that (18) holds. [sent-188, score-1.364]

68 In summary, as long as the embedded regressors satisfy condition (15) for some upper bound n on the number of hyperplanes, the recursive identiﬁer (11)-(13) will still provide L2 -stable estimates of the parameters, even if the number of hyperplanes is unknown and variable, and n(t) < n for all t. [sent-189, score-0.962]

69 Since the true segmentation is known, we compute the vectors {bj (t)} normal to each i=1 plane, and use them to generate the vector of coefﬁcients c(t). [sent-195, score-0.336]

70 We also consider the error of the polynomial coefﬁcients and the normal vectors by computing the angles between the estimated and true values. [sent-198, score-0.34]

71 62◦ for the normals, and 4% for the segmentation error. [sent-202, score-0.208]

72 True polynomial coefficients Estimation error of the polynomial (degrees) Estimated polynomial coefficients 1 1 0. [sent-203, score-0.7]

73 We now apply our algorithm to the problem of segmenting video sequences of dynamic textures, i. [sent-213, score-0.25]

74 Since the trajectories of the output of a linear dynamical system live in the so-called observability subspace, the intensity trajectories of pixels associated with a single dynamic texture lie in a subspace. [sent-219, score-0.392]

75 Therefore, the set of all intensity trajectories lie in multiple subspaces, one per dynamic texture. [sent-220, score-0.212]

76 Given γ consecutive frames of a video sequence {I(f )}t =t−γ+1 , we interpret the data as a matrix f W (t) ∈ RN ×3γ , where N is the number of pixels, and 3 corresponds to the three RGB color channels. [sent-221, score-0.23]

77 We obtain a data point xi (t) ∈ RD from image I(t) by projecting the ith row of W (t), w⊤ (t) onto a subspace of dimension D, i. [sent-222, score-0.22]

78 We applied our method to a sequence (110 × 192, 130 frames) containing a bird ﬂoating on water, while rotating around a ﬁx point. [sent-228, score-0.338]

79 The task is to segment the bird’s rigid motion from the water’s dynamic texture, while at the same time tracking the motion of the bird. [sent-229, score-0.303]

80 We chose D = 5 principal components of the γ = 5 ﬁrst frames of the RGB video sequence to project each frame onto a lower dimensional space. [sent-230, score-0.357]

81 Although the convergence is not guaranteed with only 130 frames, it is clear that the polynomial coefﬁcients already capture the periodicity of the motion. [sent-232, score-0.212]

82 As shown in the last row of Figure 2, some coefﬁcients of the polynomial oscillate in time. [sent-233, score-0.212]

83 One can notice that the orientation of the bird is related to the value of the coefﬁcient c8 . [sent-234, score-0.364]

84 If the bird is facing to the right showing her right side, the value of c8 achieves a local maximum. [sent-235, score-0.295]

85 On the contrary if the bird is oriented to the left, the value of c8 achieves a local minimum. [sent-236, score-0.295]

86 One can distinguish three behaviors for the polynomial coefﬁcients: oscillations, pseudooscillations or quasi-linearity. [sent-238, score-0.212]

87 This example shows that the coefﬁcients of the estimated polynomial give useful information about the scene motion. [sent-240, score-0.247]

88 04 50 Time (seconds) 100 0 50 Time (seconds) 100 Figure 2: Segmenting a bird ﬂoating on water. [sent-256, score-0.295]

89 To test the performance of our method on a video sequence with a variable number of motions, we extracted a sub-clip of the bird sequence (55 × 192, 130 frames) in which the camera moves up at 1 pixel/frame until the bird disappears at t = 51. [sent-260, score-0.847]

90 The camera stays stationary from t = 56 to t = 66, and then moves down at 1 pixel/frame, the bird reappears at t = 76. [sent-261, score-0.368]

91 For our method we chose D = 5 principal components of the γ = 5 ﬁrst frames of the RGB video sequence to project each frame onto a ﬁxed lower dimensional space. [sent-264, score-0.357]

92 We set the parameter of the recursive algorithm to µ = 1. [sent-265, score-0.258]

93 Notice that both methods give excellent results during the ﬁrst few frames, when both the bird and the water are present. [sent-267, score-0.361]

94 Nevertheless, notice that the performance of GPCA deteriorates dramatically when the bird disappears, because GPCA overestimates the number of hyperplanes, whereas our method is robust to this change and keeps segmenting the scene correctly, i. [sent-269, score-0.506]

95 When the bird reappears, our method detects the bird correctly from the ﬁrst frame whereas GPCA produces a wrong segmentation for the ﬁrst frames after the bird reappears. [sent-272, score-1.237]

96 In addition the ﬁxed projection and the recursive estimation of the polynomial coefﬁcients make our method much faster than GPCA. [sent-275, score-0.47]

97 Sequence GPCA Our method Figure 3: Segmenting a video sequence with a variable number of dynamic textures. [sent-276, score-0.211]

98 6 Conclusions We have proposed a simple recursive algorithm for segmenting trajectories lying in a variable number of moving hyperplanes. [sent-280, score-0.778]

99 The algorithm updates the coefﬁcients of a polynomial whose derivatives give the normals to the moving hyperplanes as well as the segmentation of the trajectories. [sent-281, score-1.146]

100 We applied our method successfully to the segmentation of videos containing multiple dynamic textures. [sent-282, score-0.273]

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