nips nips2009 nips2009-61 knowledge-graph by maker-knowledge-mining

61 nips-2009-Convex Relaxation of Mixture Regression with Efficient Algorithms

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Author: Novi Quadrianto, John Lim, Dale Schuurmans, Tibério S. Caetano

Abstract: We develop a convex relaxation of maximum a posteriori estimation of a mixture of regression models. Although our relaxation involves a semideﬁnite matrix variable, we reformulate the problem to eliminate the need for general semideﬁnite programming. In particular, we provide two reformulations that admit fast algorithms. The ﬁrst is a max-min spectral reformulation exploiting quasi-Newton descent. The second is a min-min reformulation consisting of fast alternating steps of closed-form updates. We evaluate the methods against Expectation-Maximization in a real problem of motion segmentation from video data. 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 ca Abstract We develop a convex relaxation of maximum a posteriori estimation of a mixture of regression models. [sent-7, score-0.673]

2 Although our relaxation involves a semideﬁnite matrix variable, we reformulate the problem to eliminate the need for general semideﬁnite programming. [sent-8, score-0.265]

3 In particular, we provide two reformulations that admit fast algorithms. [sent-9, score-0.122]

4 The second is a min-min reformulation consisting of fast alternating steps of closed-form updates. [sent-11, score-0.162]

5 We evaluate the methods against Expectation-Maximization in a real problem of motion segmentation from video data. [sent-12, score-0.289]

6 In practice, however, data is often better modeled by a mixture of regressors, as demonstrated by the prominence of mixture regression in a number of application areas. [sent-14, score-0.486]

7 Gaffney and Smyth [1], for example, use mixture regression to cluster trajectories, i. [sent-15, score-0.318]

8 Each trajectory is believed to have been generated from one of a number of components, where each component is associated with a regression model. [sent-18, score-0.193]

9 [2] have employed an identical mixture regression model in the context of planning: regression functions are strategies for a given planning problem. [sent-20, score-0.507]

10 Elsewhere, the mixture of regressors model has been shown to be useful in addressing covariate shift, i. [sent-21, score-0.284]

11 Storkey and Sugiyama [3] model the covariate shift process in a mixture regression setting by assuming a shift in the mixing proportions of the components. [sent-24, score-0.448]

12 In each of these problems, one must estimate k distinct latent regression functions; that is, estimate functions whose values correspond to the mean of response variables, under the assumption that the response variable is generated by a mixture of k components. [sent-25, score-0.416]

13 This estimation problem can be easily tackled if it is known to which component each response variable belongs (yielding k independent regression problems). [sent-26, score-0.199]

14 A commonly adopted approach for maximum-likelihood estimation with latent variables (in this case, component membership for each response variable) is Expectation-Maximization (EM) [4]. [sent-28, score-0.145]

15 We are not aware of any approach to maximum likelihood estimation of a mixture of regression models that is not based on the non-convex marginal likelihood objective of EM. [sent-30, score-0.354]

16 In this paper we present a convex relaxation of maximum a posteriori estimation of a mixture of regression models. [sent-31, score-0.673]

17 Recently, convex relaxations have gained considerable attention in machine learning (c. [sent-32, score-0.234]

18 By exploiting convex duality, we reformulate a relaxation of mixture regression as a semideﬁnite program. [sent-35, score-0.727]

19 To achieve a scalable approach, however, we propose two reformulations that admit fast algorithms. [sent-36, score-0.122]

20 We present experimental results comparing our methods against EM, both in synthetic problems and real computer vision problems, and show some beneﬁts of a convex approach over a local solution method. [sent-39, score-0.226]

21 1 Related work Goldfeld and Quandt [7] introduced a mixture regression model with two components called switching regressions. [sent-40, score-0.38]

22 The problem is re-cast into a single composite regression equation by introducing a switching variable. [sent-41, score-0.212]

23 A consistent estimator is then produced by a continuous relaxation of this switching variable. [sent-42, score-0.237]

24 Sp¨ th [9] introduced a problem called clusterwise linear regression, consisting of ﬁnding a a k-partition of the data such that a least squares regression criterion within those partitions becomes a minimum. [sent-44, score-0.237]

25 [10, 11, 1]) and extended to various situations including the use of variable length trajectory data and to non-parametric regression models. [sent-48, score-0.193]

26 In the extreme, each individual could have its speciﬁc regression model but coupled at higher level with a mixture on regression parameters [12]. [sent-49, score-0.468]

27 The Hierarchical Mixtures of Experts [13] model also shares some similarity to mixture regression in that gating networks which contain mixtures of generalized linear models are deﬁned. [sent-51, score-0.371]

28 We use t to denote the sample size, n to denote the dimensionality of the data and k to denote the number of mixture components. [sent-54, score-0.168]

29 Λ(a) denotes a diagonal matrix whose diagonal is equal to vector a, and diag(A) is a vector equal to the diagonal of matrix A. [sent-55, score-0.186]

30 We are given a matrix of regressors X ∈ Rt×n and a vector of regressands y ∈ Rt×1 where the response variable y is generated by a mixture of k components, but we do not know which component of the mixture generates each response yi . [sent-57, score-0.589]

31 We therefore use the matrix Π ∈ {0, 1}t×k , Π1 = 1, to denote the hidden assignment of mixture labels to each observation: Πij = 1 iff observation i has mixture label j. [sent-58, score-0.372]

32 observation i as a row vector), πi to denote the ith row of Π and yi to denote the ith element of y. [sent-61, score-0.135]

33 Also, one may want to constrain the size of the largest mixture component. [sent-68, score-0.168]

34 Combining these assumptions and adopting matrix notation we obtain the optimization problem: minimize the negative log-posterior of the entire sample A(ψi , w) − min Π,w i 1 T 1 α y Ψw + 2 y T y + wT w , where 2 σ 2σ 2 (3) 1 T T 1 w ψi ψi w + log(2πσ 2 ). [sent-70, score-0.189]

35 min (5) 2 Π,w 2σ σ 2 Although marginally convex on w, this objective is not jointly convex on w and Π (and involves non-convex constraints on Π owing to its discreteness). [sent-75, score-0.512]

36 Instead, in the following we develop a convex relaxation for problem (5). [sent-78, score-0.355]

37 2 3 Semideﬁnite Relaxation To obtain a convex relaxation we proceed in three steps. [sent-79, score-0.355]

38 Solving for A∗ (c) we obtain A∗ (c) = 1 σ 2 cT c, and 2 therefore by deﬁnition of Fenchel duality A(Ψw) = maxc cT Ψw − 1 σ 2 cT c. [sent-86, score-0.163]

39 2 A second Lemma is required to further establish the relaxation: Lemma 2 The following set inclusion holds {ΠΠT : Π ∈ {0, 1}t×k , Π1 = 1, max(diag(ΠT Π)) ≤ γt} ⊆ {M : M ∈ R t×t , tr M = t, γtI M 0}. [sent-87, score-0.255]

40 Finally, note that (ΠΠT )Π = Π(ΠT Π) where ΠT Π is a diagonal matrix and therefore its diagonal elements are the eigenvalues of ΠΠT and in particular max(diag(ΠT Π)) ≤ γt means that the largest possible eigenvalue of ΠΠT is γt, which implies γtI ΠΠT . [sent-93, score-0.203]

41 Theorem 3 The following convex optimization problem T 1 y 1 y max − σ 2 cT c − − c M XX T −c M :tr M =t,γtI M 0 c 2 2α σ 2 σ2 is a relaxation of (5) only in the sense that domain (6) is replaced by domain (7). [sent-97, score-0.504]

42 min (8) Proof We ﬁrst use Lemma 1 in order to rewrite the objective (5) and obtain min Π,w 1 1 α max cT Ψw − σ 2 cT c − 2 y T Ψw + wT w . [sent-98, score-0.251]

43 α σ2 Substituting (11) in the objective of (10) results in 1 1 min max − σ 2 cT c − c Π 2 2α y −c σ2 T ΨΨT y −c σ2 . [sent-100, score-0.178]

44 Therefore the minimization over Π effectively takes place over ΠΠT (since X is observed), and we have that (12) can be rewritten as T 1 1 y y min max − σ 2 cT c − − c ΠΠT XX T −c . [sent-102, score-0.142]

45 (13) c 2 2α σ 2 σ2 ΠΠT So far no relaxation has taken place. [sent-103, score-0.175]

46 Instead, we can reformulate problem (8) to allow for a fast algorithmic approach, without the introduction of any additional relaxation. [sent-107, score-0.129]

47 Then q tr M V T = max M :tr(M )=q,I M 0 tr M V T argmax λi (14) and i=1 T Pq Pq . [sent-120, score-0.579]

48 This will turn out to be critical for the efﬁciency of the optimization procedure, since Theorem 4 describes a purely spectral optimization routine, which is far more efﬁcient (O(t3 )) than standard interior-point methods used for semideﬁnite programming (O(t6 )). [sent-124, score-0.16]

49 The following optimization problem 1 1 max max − σ 2 cT c − c 2 2α M :tr M =t,γtI M 0 tr(M (XX T (¯ − c)(¯ − c)T )) y y (16) is equivalent to optimization problem (8). [sent-126, score-0.298]

50 Proof By Sion’s minimax theorem [14], minM and maxc in (8) can be interchanged max c min M :tr M =t,γtI M 0 1 1 T (¯ − c) M y − σ 2 cT c − 2 2α XX T (¯ − c) y (17) XX T (¯ − c) . [sent-127, score-0.3]

51 y (18) which, by distributivity of the (min, +) semi-ring, is equivalent to 1 1 max − σ 2 cT c + min c 2 2α M :tr M =t,γtI T M 0 − (¯ − c) M y Now, deﬁne K := XX T . [sent-128, score-0.2]

52 The objective of the minimization in (18) can then be written as − (¯ − c)T (M y (Mij Kij ) (¯ − c)(¯ − c)T y y =− K) (¯ − c)(¯ − c)T y y K)(¯ − c) = − tr (M y ij ij = − tr(M (K Mij Kij (¯ − c)(¯ − c)T y y =− (19) ij (20) ij (¯ − c)(¯ − c)T )) = − tr(M (XX T y y (¯ − c)(¯ − c)T )). [sent-129, score-0.603]

53 The following optimization problem 1 t max − σ 2 cT c − c 2 2αq ¯ tr(M (XX T max ¯ ¯ ¯ M :tr M =q,I M is equivalent to optimization problem (16). [sent-136, score-0.298]

54 Instead of solving (8), which demands O(t6 ) operations, we instead solve (22), which has as inner optimization a max eigenvalue problem, demanding only O(t3 ) operations. [sent-146, score-0.311]

55 In the next section we describe an algorithm to jointly optimize for M and c in (22), which will essentially consist of alternating the efﬁcient spectral solution over M with a subgradient optimization over c. [sent-147, score-0.213]

56 1 Max-Min Algorithm Algorithm 1 describes how we solve optimization problem (22). [sent-149, score-0.149]

57 We then need to recover M ∗ such that (c∗ , M ∗ ) is a saddle point (note that problem (22) is concave in c T and convex in M ). [sent-155, score-0.294]

58 If M ∗ = Pq Pq is such that Pq is unique, then we are done and the labeling solution of mixture membership is M ∗ (subject to roundoff). [sent-157, score-0.264]

59 pl ], l > q, where each of the additional l − q eigenvectors has an associated eigenvalue which is equal to the eigenvalue of some of the previous q eigenvectors. [sent-165, score-0.274]

60 We then have that at the saddle point there must exist a diagonal matrix Λ such that M ∗ = Pl ΛPlT , subject to Λ 0 and tr Λ = q (if this were not the case there would be an ascent direction in c∗ , contradicting the hypothesis that c∗ is optimal). [sent-166, score-0.364]

61 Speciﬁcally, work in this area lead to convex optimization problems expressed jointly over two matrix variables where each step is an alternating min-min descent that can be executed in closed-form or by a very fast algorithm. [sent-177, score-0.419]

62 Although it is not immediately apparent that this algorithmic strategy is applicable to the problem at hand, with some further reformulation of (8) we discover that in fact the same min-min algorithmic approach can be applied to our mixture of regression problem. [sent-178, score-0.48]

63 Theorem 8 The following optimization problem 1 T 1 α y diag(XAT ) + 2 diag(XAT )T diag(XAT ) + min min tr(AT M −1 A) σ2 2σ 2γt {M :I M 0,tr M =1/γ} A (25) is equivalent to optimization problem (8). [sent-179, score-0.306]

64 1 min min max − Min-Min Algorithm The problem (25) is jointly convex in A and M [14] and Algorithm 2 describes how to solve it. [sent-183, score-0.507]

65 Remark 9 In practice, similarly to [17], a regularizer on M is added to avoid singularity, resulting in the following regularized objective function, 1 1 min min 2 y T diag(XAT ) + 2 diag(XAT )T diag(XAT ) A σ 2σ {M :I M 0,tr M =1/γ} α + tr(AT M −1 A) + tr(M −1 ). [sent-186, score-0.182]

66 (29) 2γt The problem is still jointly convex in M and A. [sent-187, score-0.223]

67 6 6 Experiments Our primary objective in formulating this convex approach to mixture regression is to tackle a difﬁcult problem in video analysis (see below). [sent-188, score-0.58]

68 We generated 30 synthetic data points according to yi = (πi ⊗ xi )w + i , with xi ∈ R, i ∼ N (0, 1) and w ∈ U (0, 1). [sent-190, score-0.195]

69 The response variable yi is assumed to be generated from a mixture of 5 components. [sent-191, score-0.274]

70 We compared the quality of the relaxation in (22) to EM. [sent-192, score-0.175]

71 The visualization of the recovered membership for one of the runs is given in Figure 1. [sent-201, score-0.169]

72 This demonstrates that the relaxation can retain much of the structure of the problem. [sent-202, score-0.175]

73 1 Vision Experiment In a dynamic scene, various static and moving objects are viewed by a possibly moving observer. [sent-204, score-0.187]

74 For example, consider a moving, hand-held camera ﬁlming a scene of several cars driving down the road. [sent-205, score-0.236]

75 Each car has a separate motion, and even the static objects, such as trees, appear to move in the video due to the self-motion of the camera. [sent-206, score-0.16]

76 The task of segmenting each object according to its motion, estimating the parameters of each motion, and recovering the structure of the scene is known as the multibody structure and motion problem. [sent-207, score-0.344]

77 From epipolar geometry, given a pair of corresponding points pi and qi from two images (pi , qi ∈ T R3×1 ), we have the epipolar equation qi F pi = 0. [sent-213, score-0.74]

78 The fundamental matrix F encapsulates information about the translation and rotation relative to the scene points between the positions of the camera where the two images were captured, as well as the camera calibration parameters such as its focal length. [sent-214, score-0.305]

79 In a static scene, where only the camera is moving, there is only one fundamental matrix, which arises from the camera self-motion. [sent-215, score-0.265]

80 However, if some of the scene points are moving independently under multiple different motions, there are several fundamental matrices. [sent-216, score-0.179]

81 If there are k motion groups, the epipolar equation can be expressed in term of the multibody fundamental k T matrix [21], i. [sent-217, score-0.427]

82 An alternative approach, which we follow here, is by Li [22], who casts the k T problem as a mixture of fundamental matrices, i. [sent-221, score-0.214]

83 qi ( j=1 πij Fj )pi = 0 where the membership variable πij = 1 when image point i belongs to motion group j, and zero otherwise. [sent-223, score-0.416]

84 Furthermore, T since qi F pi = 0 is bilinear in the image points, we can rewrite it to be xT wj = 0, with the column i x x x π vectors xi = [qi px qi py qi pπ . [sent-224, score-0.614]

85 The weight vector wj for motion group j can be i recovered easily if the indicator variable πij is known. [sent-230, score-0.253]

86 We are interested in assessing the effectiveness of EM-based and convex relaxation-based methods for this multibody structure and motion problem. [sent-231, score-0.449]

87 6 GHz machine, the elapsed time (in seconds) for two cranes dataset is 16. [sent-238, score-0.145]

88 Rounding for the convex versions was done by k-means, which introduces some differences in the ﬁnal results for both algorithms. [sent-242, score-0.18]

89 Visualizations of the motion segmentation on two cranes, three cars, and cars2 07 datasets are given in Figure 2 (for kanatani2 and articulated please refer to Appendix). [sent-244, score-0.289]

90 7 Conclusion The mixture regression problem is pervasive in many applications and known approaches for parameter estimation rely on variants of EM, which naturally have issues with local minima. [sent-245, score-0.318]

91 In this paper we introduced a semideﬁnite relaxation for the mixture regression problem, thus obtaining a convex formulation which does not suffer from local minima. [sent-246, score-0.673]

92 0000 (a) Ground Truth (b) EM (c) Convex Relaxation Figure 1: Recovered membership on synthetic data with EM and convex relaxation. [sent-265, score-0.322]

93 30 data points are generated according to yi = (πi ⊗ xi )w + i , with xi ∈ R, i ∼ N (0, 1) and w ∈ U (0, 1). [sent-266, score-0.149]

94 In two cranes (ﬁrst row), EM produces more segmentation errors at the left crane. [sent-269, score-0.22]

95 In three cars (second row), the max-min method gives the least segmentation error (at the front side of the middle car) and EM produces more segmentation errors at the front side of the left car. [sent-270, score-0.237]

96 The contrast of EM and convex methods is apparent for cars2 07 (third row): the convex methods segment correctly the static grass ﬁeld object, while EM makes mistakes. [sent-271, score-0.431]

97 Further, the min-min method can almost perfectly segment the car in the middle of the scene from the static tree background. [sent-272, score-0.189]

98 Maximum likelihood from incomplete data via the em algorithm. [sent-297, score-0.273]

99 Maximum likelihood estimates of the parameters of a mixture of two regression lines. [sent-324, score-0.318]

100 A convex formulation for learning shared structures from multiple tasks. [sent-388, score-0.18]

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The phase shift spatio-temporal RF is modeled with a complex valued function f ( x, y, t ) = g ( x, y ) ⋅ h ( t , Φ ) , where the spatial and temporal RFs are denoted by g ( x, y ) and h ( t , Φ ) respectively, g ( x, y ) = N ( x, y | 0, C ) exp ( jΩ x x + jΩ y y ) h ( t , Φ ) = hreal ( t ) + exp ( jΦ ) himag ( t ) (1) and C is the covariance matrix of the spatial Gaussian envelope and Φ is the phase tuning of the motion energy neuron. The real and imaginary profiles of the temporal receptive field are Gamma modulated sinusoidal functions with quadrature phases, hreal ( t ) = G ( t | α ,τ ) cos ( Ωt t ) (2) himag ( t ) = G ( t | α ,τ ) sin ( Ωt t ) The envelopes for complex exponentials are functions of Gaussian and Gamma distributions, N ( x, y | 0, C ) = ⎛ x2 y2 exp ⎜ − 2 − 2 ⎜ 2σ x 2σ y 2πσ xσ y ⎝ 1 ⎞ ⎟ ⎟ ⎠ (3) hreal (t ) g ( x, y ) himag (t ) g ( x, y ) (·)2 (·)2 M M M (·)2 (·)2 M M M 2 (·) 2 (·) Vreal V (Φ ) e jΦ Vimag (a) Ev ( Φ max ) (·)2 wc ( x, y ) e jΦmin Ev ( (b) M 0 ) w ( x, y ) c M Ev ( Φ min ) M EΔv ( Θ ) ∫∫∫ K x , y ,Φ e j0 e jΦmin ws ( x, y ) Ks c e ∫∫∫ jΘ ws ( x, y ) e j0 x , y ,Φ ws ( x, y ) wc ( x, y ) M e jΦ max e jΦmax (·)2 M (·)2 M M (·)2 M 2 (·) M M (·)2 (c) Figure 1. (a) shows the diagram of the phase shift motion energy model adapted from [15]. (b) draws the spatiotemporal representation of the phase shift motion energy neuron with the real and imaginary receptive field demonstrated by the two left pictures. (c) illustrates the construction of differential motion opponency with a phase difference Θ from two populations of phase shift motion energy neurons in two spatial areas c and s. To avoid clutter, the space location (x, y) is not explicitly shown in phase tuned motion energies. G (t | α ,τ ) = 1 ⎛ t t α −1 exp ⎜ − Γ(α )τ α ⎝ τ ⎞ ⎟ u (t ) ⎠ (4) where Γ (α ) is the gamma function and u ( t ) is the unit step function. The parameters α and τ determine the temporal RF size. As derived in [15], the motion energy at location (x, y) can be computed by E v ( x, y, Φ ) = S + P cos ( Ψ − Φ ) (5) where S = Vreal 2 + Vimag 2 * P = 2 VrealVimag ( * Ψ = arg VrealVimag (6) ) and complex valued responses in real and imaginary paths are obtained as, Vreal ( x, y, t ) = ∫∫∫ g (ξ , ζ ) h (η ) I ( x − ξ , y − ζ , t − η ) dξ dζ dη real ξ ,ζ ,η Vimag ( x, y, t ) = ∫∫∫ g (ξ , ζ ) h (η ) I ( x − ξ , y − ζ , t − η ) dξ dζ dη ξ ζ η (7) imag , , The superscript * represents the complex conjugation and the phase shift parameter Φ controls the spatio-temporal orientation tuning. To avoid clutter, the spatial location variables x and y for S, P, Ψ, Vreal and Vimag are not explicitly shown in Eq. (5) and (6). Figure 1(b) demonstrates the even and odd profiles of the spatio-temporal RF tuned to a particular phase shift. Θ 0 Θ 0 (a) (b) Figure 2. Two types of differential motion opponency constructions of (a) center-surrounding interaction and (b) left-right interaction. Among cells in area MT with surrounding modulations, 25% of cells are with the antagonistic RF structure in the top row and another 50% of cells have the integrative RF structure as shown in the bottom row. 3 Extending Phase Op p on ency Mechanism to D i f f e r e nt i a l Motion Based on the above phase shift motion energy model, the local image velocity at each spatial location can be represented by a phase shift which leads to the peak response across a population of motion energy neurons. Across regions of different motions, there are clear discontinuities on the estimated velocity map. The motion discontinuities can be detected by edge detectors on the velocity map to segment different motions. However, this algorithm for motion discontinuities detection can’t discriminate between the object motion and uniform motions in contextual backgrounds. Here we propose a phase mechanism to detect differential motions inspired by the disparity energy model and adopt the center-surround inhibition mechanism to pop out the object motion from contextual background motions. The motion differences between different spatial locations can be modeled in the similar way as the disparity model. The motion energies from two neighboring locations are considered as the retinal images to the left and right eyes. Thus, we can construct a differential motion opponency by placing two populations of phase shift motion energy neurons at different spatial locations and the energy EΔv ( Θ ) of the opponency is the squared modulus of the averaged phase shift motion energies over space and phase, E Δv ( Θ ) = ∫∫∫ E ( x, y, Φ ) ⋅ w ( x, y, Φ | Θ ) dxdyd Φ v 2 (8) where w ( x, y, Θ ) is the profile for differential motion opponency and Δv is the velocity difference between the two spatial regions defined by the kernel w ( x, y, Θ ) . Since w ( x, y, Θ ) is intended to implement the functional role of spatial interactions, it is desired to be a separable function in space and phase domain and can be modeled by phase tuned summation of two spatial kernels, w ( x, y, Φ | Θ ) = wc ( x, y ) e jΦ + e jΘ+ jΦ ws ( x, y ) (9) where wc ( x, y ) and ws ( x, y ) are Gaussian kernels of different spatial sizes σ c and σ s , and Θ is the phase difference representing velocity difference between two spatial regions c and s. Substituting Eq. (9) into Eq. (8), the differential motion energy can be reformulated as EΔv ( Θ ) = K c + e jΘ K s 2 (10) 3 3 3 2 2 2 1 1 1 0 0 -1 -1 -2 -2 -2 -3 -3 -3 -3 -3 1 Right Velocity Right Velocity 0.98 0.96 0.94 0.92 0 0.9 0.88 -1 0.86 0.84 0.82 -2 -1 0 1 Left Velocity 2 3 -2 -1 0 1 Left Velocity 2 3 0.8 (a) (b) Figure 3. (a) Phase map and (b) peak magnitude map are obtained from stimuli of two patches of random dots moving with different velocities. The two patches of stimuli are statistically independent but share the same spatial properties: dot size of 2 pixels, dot density of 10% and dot coherence level of 100%. The phase tuned population of motion energy neurons are applied to each patch of random dots with RF parameters: Ωt = 2π/8, Ωt = 2π/16, σx = 5 and τ = 5.5. For each combination of velocities from left and right patches, averaged phase shifts over space and time are computed and so do the magnitudes of peak responses. The unit for velocities is pixels per frame. where Kc = ∫∫∫ E ( x, y, Φ ) exp ( jΦ ) w ( x, y ) dxdyd Φ v,c c x , y ,Φ Ks = ∫∫∫ E ( x, y, Φ ) exp ( jΦ ) w ( x, y ) dxdyd Φ v,s (11) s x, y ,Φ Ev ,c ( x, y, Φ ) and Ev , s ( x, y, Φ ) are phase shift motion energies at location (x, y) and with phase shift Φ. Utilizing the results in Eq. (5) and (6), Eq. (10) and (11) generate similar results, E Δv ( Θ ) = Sopp + Popp cos ( Θopp − Θ ) (12) where Sopp = K c 2 + Ks Popp = 2 K c K s* 2 (13) Θopp = arg ( K c K s* ) According to above derivations, by varying the phase shift Θ between –π and π, the relative motion energy of the differential motion opponency can be modeled as population responses across a population of phase tuned motion opponencies. The response is completely specified by three parameters Sopp , Popp and Θopp . The schematic diagram of this opponency is illustrated in Figure 1(c). The differential motion opponency is constituted by three stages. At the first stage, a population of phase shift motion energy neurons is applied to be selective to different velocities. At the second stage, motion energies from the first stage are weighted by kernels tuned to different spatial locations and phase shifts respectively for both spatial regions and two single differential motion signals in region c and region s are achieved by integrating responses from these two regions over space and phase tuning. Finally, the differential motion energy is computed by the squared modulus of the summation of the integrated motion signal in region c and phase shifted motion signal in region s. The subscripts c and s represent two interacted spatial regions which are not limited to the center and surround regions. The opponency could also be constructed by the neighboring left and right Inhibitive interaction, Θ = π/2 Excitatory interaction, Θ =0 Inhibitory 2 1.6 Responses 1.6 Responses Excitatory 2 1.2 0.8 1.2 0.8 0.4 0.4 0 0 pi/2 pi 3pi/2 Surrouding Direction 0 0 2pi (a) Model by Petkov et al. [8] pi/2 pi 3pi/2 Surrouding Direction (b) Model by Heeger et al. [7] Inhibitory 2 2pi Inhibitory 2 1.6 1.6 Responses Responses 1.2 0.8 1.2 0.8 0.4 0.4 0 0 0 pi/2 pi Surrouding Direction 3pi/2 2pi 0 pi/2 pi Surrouding Direction 3pi/2 2pi (c) (d) Figure 4. Demonstrations of center-surround differential motion opponency, where (a) show the excitation of opposite directions outside the CRF and (b) show the inhibition by surrounding motions in same directions. The center-surround inhibition models by Petkov, et al. [8] and Heeger, et al. [7] are shown in (c) and (d). Responses above 1 indicate enhancement and responses below 1 indicate suppressions. spatial regions. Figure 2 shows two types of structures for the differential motion opponency. In [17], the authors demonstrates that among cells in area MT with surrounding modulations, 25% of cells are with the antagonistic RF structure as shown in Figure 2(a) and another 50% of cells have the integrative RF structure as shown in Figure 2(b). The velocity difference tuning of the opponency is determined by the phase shift parameter Θ combined with parameters of spatial and temporal frequencies for motion energy neurons. The larger phase shift magnitude prefers the bigger velocity difference. This phase tuning of velocity difference is consistent with the phase tuning of motion energy neurons. Figure 3 shows the phase map obtained by using random dots stimuli with different velocities on two spatial patches (left and right patches with sizes of 128 pixels 128 pixels). Along the diagonal line, velocities from left and right patches are equal to each other and therefore phase estimates are zeros along this line. Deviated from the diagonal line to upper-left and lower-right, the phase magnitudes increase while positive phases indicate larger left velocities and negative phases indicate larger right velocities. The phase tuning can give a good classification of velocity differences. 4 V a l i d a t i o n o f D i f f e r e n t i a l M o t i o n O pp o n e n c y Out derivation and analysis above show that the phase shift between two neighboring spatial regions is a good indicator for motion difference between these two regions. In this section, we validate the proposed differential motion opponency by two sets of experiments, which show effects of both surrounding directions and speeds on the center motion. Inhibitory 2 1.6 1.2 1.2 Responses 1.6 Responses Inhibitory 2 0.8 0.4 0.4 0 -2 0.8 0 -1.5 -1 -0.5 0 0.5 Center Speed 1 1.5 2 -2 -1.5 -1 -0.5 0 0.5 Center Speed 1 1.5 2 (a) (b) Figure 5. The insensitivity of the proposed opponency model to absolute center and surrounding velocities is demonstrated in (a), where responses are enhanced for all center velocities from -2 to 2 pixels per frame. In (b), the model by Heeger, et al. [7] only shows enhancement when the center speed matches the preferred speed of 1.2 pixel per frame. Similarly, responses above 1 indicate enhancement and below 1 indicate suppressions. In both curves, the velocity differences between center and surrounding regions are maintained as a constant of 3 pixels per frame. Physiological experiments [2][3] have demonstrated that the neuronal activities in the classical receptive field are suppressed by responses outside the CRF to stimuli with similar motions including both directions and speeds on the center and surrounding regions. On the contrary, visual stimuli of opposite directions or quite different speeds outside the CRF enhance the responses in the CRF. In their experiments, they used a set of stimuli of random dots moving at different velocities, where there are small patches of moving random dots on the center. We tested the properties of the proposed opponency model for motion difference measurement by using similar random dots stimuli. The random dots on background move with different speeds and in different direction but have the same statistical parameters: dot size of 2 pixels, dot density of 10% and motion coherence level of 100%. The small random dots patches are placed on the center of background stimuli to stimulate the neurons in the CRF. These small patches share the same statistical parameters with background random dots but move with a constant velocity of 1 pixel per frame. Figure 4 shows results for the enhanced and suppressed responses in the CRF with varying surrounding directions. The phase shift motion energy neurons had the same spatial and temporal frequencies and the same receptive field sizes, and were selective to vertical orientations. The preferred spatial frequency was 2π/16 radian per pixel and the temporal frequency was 2π/16 radian per frame. The sizes of RF in horizontal and vertical directions were respectively 5 pixels and 10 pixels, corresponding to a spatial bandwidth of 1.96 octaves. The time constant τ was 5.5 frames which resulted in a temporal bandwidth of 1.96 octaves. As shown in Figure 4 (a) and (b), the surrounding motion of opposite direction gives the largest response to the motion in the CRF for the inhibitory interaction and the smallest response for the excitatory interaction. Results demonstrated in Figure 4 are consistent with physiological results reported in [3]. In Born’s paper, inhibitory cells show response enhancement and excitatory cells show response suppression when surrounding motions are in opposite directions. The 3-dB bandwidth for the surrounding moving direction is about 135 degrees for the physiological experiments while the bandwidth is about 180 degrees for the simulation results in our proposed model. Models proposed by Petkov, et al. [8] and Heeger, et al. [7] also show clear inhibition between opposite motions. The Petkov’s model achieves the surrounding suppression for each point in ( x, y, t ) space by the subtraction between responses from that point and its surroundings and followed by a half-wave rectification, + % Ev ,θ ( x, y, t ) = Ev ,θ ( x, y, t ) − α ⋅ Sv ,θ ( x, y, t ) (14) where Ev ,θ ( x, y, t ) is the motion energy at location (x,y) and time t for a given preferred speed v and orientation θ, Sv ,θ ( x, y, t ) is the average motion energy in the surrounding of point (x, y, t), % Ev ,θ ( x, y, t ) is the suppressed motion energy and the factor α controls the inhibition strength. The inhibition term is computed by weighted motion energy Sv ,θ ( x, y, t ) = Ev ,θ ( x, y, t ) ∗ wv ,θ ( x, y, t ) (15) where wv ,θ ( x, y, t ) is the surround weighting function. The Heeger’s model constructs the center-surround motion opponent by computing the weighted sum of responses from motion selective cells, Rv ,θ ( t ) = ∑ β ( x, y ) ⎡ Ev ,θ ( x, y, t ) − E− v ,θ ( x, y, t ) ⎤ ⎣ ⎦ (16) x, y where β ( x, y ) is a center-surround weighting function and the motion energy at each point should be normalized across all cells with different tuning properties. As shown in Figure 4 (c) and (d) for results of Petkov’s and Heeger’s models, we replace the conventional frequency tuned motion energy neuron with our proposed phase tuned neuron. The model by Petkov, et al. [8] is generally suppressive and only reproduces less suppression for opposite motions, which is inconsistent with results from [3]. The model by Heeger, et al. [7] has similar properties with our proposed model with respect to both excitatory and inhibitory interactions. To evaluate the sensitivity of the proposed opponency model to velocity differences, we did simulations by using similar stimuli with the above experiment in Figure 4 but maintaining a constant velocity difference of 3 pixels per frame between the center and surrounding random dot patches. As shown in Figure 5, by varying the velocities of random dots on the center region, we found that responses by the proposed model are always enhanced independent upon absolute velocities of center stimuli, but responses by the Heeger’s model achieve the enhancement at a center velocity of 1.2 pixels per frame and maintain suppressed at other speeds. 5 D i s c u s s i on We proposed a new biologically plausible model of the differential motion opponency to model the spatial interaction property of motion energy neurons. The proposed opponency model is motivated by the phase tuning mechanism of disparity energy neurons which infers the disparity information from the phase difference between complex valued responses to left and right retinal images. Hence, the two neighboring spatial areas can be considered as left and right images and the motion difference between these two spatial regions is detected by the phase difference between the complex valued responses at these two regions. Our experimental results demonstrate a consistent conclusion with physiological experiments that motions of opposite directions and different speeds outside the CRF can show both inhibitive and excitatory effects on the CRF responses. The inhibitive interaction helps to segment the moving object from backgrounds when fed back to low-level features such as edges, orientations and color information. Except providing a unifying phase mechanism in understanding neurons with different functional roles at different brain areas, the proposed opponency model could possibly provide a way to understand the motion integration and motion segmentation. Integration and segmentation are two opposite motion perception tasks but co-exist to constitute two fundamental types of motion processing. Segmentation is achieved by discriminating motion signals from different objects, which is thought to be due to the antagonistic interaction between center and surrounding RFs. Integration is obtained by utilizing the enhancing function of surrounding areas to CRF areas. Both types of processing have been found in motion related areas including area MT and MST. Tadin, et al. [18] have found that motion segmentation dominants at high stimulus contrast and gives the way to motion integration at low stimulus contrast. Huang, et al. [19] suggests that the surrounding modulation is adaptive according to the visual stimulus such as contrasts and noise levels. Since our proposed opponency model determines the functional role of neurons by only the phase shift parameter, this makes the proposed model to be an ideal candidate model for the adaptive surrounding modulation with the phase tuning between two spatial regions. References [1]. K. Nakayama and J. M. Loomis, “Optical velocity patterns, velocity-sensitive neurons, and space perception: A hypothesis,” Perception, vol. 3, 63-80, 1974. [2]. K. Tanaka, K. Hikosaka, H. Saito, M. Yukie, Y. Fukada and E. Iwai, “Analysis of local and wide-field movements in the superior temporal visual areas of the macaque monkey,” Journal of Neuroscience, vol. 6, pp. 134-144, 1986. [3]. R. T. Born and R. B. H. Tootell, “Segregation of global and local motion processing in primate middle temporal visual area,” Nature, vol. 357, pp. 497-499, 1992. [4]. J. Allman, F. Miezin and E. McGuinness, “Stimulus specific responses from beyond the classical receptive field: Neurophysiological mechanisms for local-global comparisions in visual neurons,” Annual Review Neuroscience, vol. 8, pp. 407-430, 1985. [5]. V. A. F. Lamme, “The neurophysiology of figure-ground segregation in primary visual cortex,” Journal of Neuroscience, vol. 15, pp. 1605-1615, 1995. [6]. D. C. Bradley and R. A. Andersen, “Center-surround antagonism based on disparity in primate area MT,” Journal of Neuroscience, vol. 18, pp. 7552-65, 1998. [7]. D. J. Heeger, A. D. Jepson and E. P. Simoncelli, “Recovering observer translation with center-surround operators,” Proc IEEE Workshop on Visual Motion, pp. 95-100, Oct 1991. [8]. N. Petkov and E. Subramanian, “Motion detection, noise reduction, texture suppression, and contour enhancement by spatiotemporal Gabor filters with surround inhibition,” Biological Cybernetics, vol. 97, pp. 423-439, 2007. [9]. M. Escobar and P. Kornprobst, “Action recognition with a Bio-inspired feedforward motion processing model: the richness of center-surround interactions,” ECCV '08: Proceedings of the 10th European Conference on Computer Vision, pp. 186-199, Marseille, France, 2008. [10]. B. J. Frost and K. Nakayama, “Single visual neurons code opposing motion independent of direction,” Science, vol. 200, pp. 744-745, 1983. [11]. A. Cao and P. H. Schiller, “Neural responses to relative speed in the primary visual cortex of rhesus monkey,” Visual Neuroscience, vol. 20, pp. 77-84, 2003. [12]. B. K. Dellen, J. W. Clark and R. Wessel, “Computing relative motion with complex cells,” Visual Neuroscience, vol. 22, pp. 225-236, 2005. [13]. I. Ohzawa, G. C. Deangelis and R. D. Freeman, “Encoding of binocular disparity by complex cells in the cat’s visual cortex,” Journal of Neurophysiology, vol. 77, pp. 2879-2909, 1997. [14]. D. J. Fleet, H. Wagner and D. J. Heeger, “Neural Encoding of binocular disparity: energy model, position shifts and phase shifts,” Vision Research, vol. 26, pp. 1839-1857, 1996. [15]. Y. C. Meng and B. E. Shi, “Normalized Phase Shift Motion Energy Neuron Populations for Image Velocity Estimation,” International Joint Conference on Neural Network, Atlanta, GA, June 14-19, 2009. [16]. E. H. Adelson and J. R. Bergen, “Spatiotemporal energy models for the perception of motion,” J. Opt. Soc. Am. A Opt. Image Sci. Vis., vol. 2, pp. 284-299, 1985. [17]. D. K. Xiao, S. Raiguel, V. Marcar, J. Koenderink and G. A. Orban, “The spatial distribution of the antagonistic surround of MT/V5,” Cereb Cortex, vol. 7, pp. 662-677, 1997. [18]. D. Tadin, J. S. Lappin, L. A. Gilroy and R. Blake, “Perceptual consequences of centre-surround antagonism in visual motion processing,” Nature, vol. 424, pp. 312-315, 2003. [19]. X. Huang, T. D. Albright and G. R. Stoner, “Adaptive surround modulation in cortical area MT,” Neuron, vol. 53, pp. 761-770, 2007.

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A recent benchmark [18] indicated that Local Subspace Afﬁnity (LSA) [21] gave the best performance in terms of classiﬁcation accuracy, although their result was subsequently surpassed by [5, 10]. However, we argue that most of the previous approaches do not simultaneously fulﬁl the qualities desirable of motion segmentation algorithms. Most notably, although some of the approaches have the means to estimate the number of motions, they are generally unreliable in this respect and require manual input of this parameter. In fact this prior knowledge was given to all the methods compared in [18]2 . Secondly, most of the methods (e.g. [19, 5]) do not explicitly deal with outliers. They will almost always breakdown when given corrupted data. These deﬁciencies reduce the usefulness of available motion segmentation algorithms in practical circumstances. In this paper we attempt to bridge the gap between experimental performance and practical usability. Our previous work [2] indicates that robust multi-structure model ﬁtting can be achieved effectively with statistical learning. Here we extend this concept to motion subspace clustering. Drawing inspiration from robust statistical model ﬁtting [4], we estimate random hypotheses of motion subspaces in the data. However, instead of ranking these hypotheses we encapsulate them in a novel Mercer kernel. The kernel can function reliably despite overwhelming sampling imbalance, and it permits the application of non-linear dimensionality reduction techniques to effectively identify and reject outlying trajectories. This is then followed by Kernel PCA [11] to maximize the separation between groups and spectral clustering [13] to recover the number of motions and clustering. Experiments on the Hopkins 155 benchmark dataset [18] show that our method is superior to other approaches in terms of the qualities described above, including computational efﬁciency. 1.1 Brief review of afﬁne model multi-body motion segmentation Let {tf p ∈ R2 }f =1,...,F be the set of 2D coordinates of P trajectories tracked across F frames. In p=1,...,P multi-body motion segmentation the tf p ’s correspond to points on the surface of rigid objects which are moving. The goal is to separate the trajectories into groups corresponding to the motion they belong to. In other words, if we arrange the coordinates in the following data matrix   t11 · · · t1P  . .  ∈ R2F ×P , .. .  T= . (1) . . . tF 1 . . . tF P the goal is to ﬁnd the permutation Γ ∈ RP ×P such that the columns of T · Γ are arranged according to the respective motions they belong to. It turns out that under afﬁne projection [1, 16] trajectories from the same motion lie on a distinct subspace in R2F , and each of these motion subspaces is of dimensions 2, 3 or 4. Thus motion segmentation can be accomplished via clustering subspaces in R2F . See [1, 16] for more details. Realistically actual motion sequences might contain trajectories which do not correspond to valid objects or motions. These trajectories behave as outliers in the data and, if not taken into account, can be seriously detrimental to subspace clustering algorithms. 2 The Ordered Residual Kernel (ORK) First, we take a statistical model ﬁtting point of view to motion segmentation. Let {xi }i=1,...,N be the set of N samples on which we want to perform model ﬁtting. We randomly draw p-subsets from the data and use it to ﬁt a hypothesis of the model, where p is the number of parameters that deﬁne the model. In motion segmentation, the xi ’s are the columns of matrix T, and p = 4 since the model is a four-dimensional subspace3 . Assume that M of such random hypotheses are drawn. i i For each data point xi compute its absolute residual set ri = {r1 , . . . , rM } as measured to the M hypotheses. For motion segmentation, the residual is the orthogonal distance to a hypothesis 2 As conﬁrmed through private contact with the authors of [18]. Ideally we should also consider degenerate motions with subspace dimensions 2 or 3, but previous work [18] using RANSAC [4] and our results suggest this is not a pressing issue for the Hopkins 155 dataset. 3 2 i i subspace. We sort the elements in ri to obtain the sorted residual set ˜i = {rλi , . . . , rλi }, where r 1 M i i the permutation {λi , . . . , λi } is obtained such that rλi ≤ · · · ≤ rλi . Deﬁne the following 1 M 1 M ˜ θi := {λi , . . . , λi } 1 M (2) ˜ as the sorted hypothesis set of point xi , i.e. θi depicts the order in which xi becomes the inlier of the M hypotheses as a ﬁctitious inlier threshold is increased from 0 to ∞. We deﬁne the Ordered Residual Kernel (ORK) between two data points as 1 kr (xi1 , xi2 ) := ˜ Z M/h t ˜ ˜ zt · k∩ (θi1 , θi2 ), (3) t=1 M/h where zt = 1 are the harmonic series and Z = t=1 zt is the (M/h)-th harmonic number. t Without lost of generality assume that M is wholly divisible by h. Step size h is used to obtain the Difference of Intersection Kernel (DOIK) 1 ˜1:α t ˜ ˜ ˜1:α ˜1:α ˜1:α k∩ (θi1 , θi2 ) := (|θi1 t ∩ θi2 t | − |θi1 t−1 ∩ θi2 t−1 |) (4) h ˜a:b where αt = t · h and αt−1 = (t − 1) · h. Symbol θi indicates the set formed by the a-th to ˜i . Since the contents of the sorted hypotheses set are merely permutations of the b-th elements of θ {1 . . . M }, i.e. there are no repeating elements, 0 ≤ kr (xi1 , xi2 ) ≤ 1. ˜ (5) Note that kr is independent of the type of model to be ﬁtted, thus it is applicable to generic statistical ˜ model ﬁtting problems. However, we concentrate on motion subspaces in this paper. Let τ be a ﬁctitious inlier threshold. The kernel kr captures the intuition that, if τ is low, two ˜ points arising from the same subspace will have high normalized intersection since they share many common hypotheses which correspond to that subspace. If τ is high, implausible hypotheses ﬁtted on outliers start to dominate and decrease the normalized intersection. Step size h allows us to quantify the rate of change of intersection if τ is increased from 0 to ∞, and since zt is decreasing, kr will evaluate to a high value for two points from the same subspace. In contrast, kr is always low ˜ ˜ for points not from the same subspace or that are outliers. Proof of satisfying Mercer’s condition. Let D be a ﬁxed domain, and P(D) be the power set of D, i.e. the set of all subsets of D. Let S ⊆ P(D), and p, q ∈ S. If µ is a measure on D, then k∩ (p, q) = µ(p ∩ q), (6) called the intersection kernel, is provably a valid Mercer kernel [12]. The DOIK can be rewritten as t ˜ ˜ k∩ (θi1 , θi2 ) = 1 ˜(αt−1 +1):αt ˜(αt−1 +1):αt (|θ ∩ θi2 | h i1 ˜1:(α ) ˜(α +1):αt | + |θ (αt−1 +1):αt ∩ θ 1:(αt−1 ) |). ˜ ˜ +|θi1 t−1 ∩ θi2 t−1 i1 i2 (7) If we let D = {1 . . . M } be the set of all possible hypothesis indices and µ be uniform on D, each term in Eq. (7) is simply an intersection kernel multiplied by |D|/h. Since multiplying a kernel with a positive constant and adding two kernels respectively produce valid Mercer kernels [12], the DOIK and ORK are also valid Mercer kernels.• Parameter h in kr depends on the number of random hypotheses M , i.e. step size h can be set as a ˜ ratio of M . The value of M can be determined based on the size of the p-subset and the size of the data N (e.g. [23, 15]), and thus h is not contingent on knowledge of the true inlier noise scale or threshold. Moreover, our experiments in Sec. 4 show that segmentation performance is relatively insensitive to the settings of h and M . 2.1 Performance under sampling imbalance Methods based on random sampling (e.g. RANSAC [4]) are usually affected by unbalanced datasets. The probability of simultaneously retrieving p inliers from a particular structure is tiny if points 3 from that structure represent only a small minority in the data. In an unbalanced dataset the “pure” p-subsets in the M randomly drawn samples will be dominated by points from the majority structure in the data. This is a pronounced problem in motion sequences, since there is usually a background “object” whose point trajectories form a large majority in the data. In fact, for motion sequences from the Hopkins 155 dataset [18] with typically about 300 points per sequence, M has to be raised to about 20,000 before a pure p-subset from the non-background objects is sampled. However, ORK can function reliably despite serious sampling imbalance. This is because points from the same subspace are roughly equi-distance to the sampled hypotheses in their vicinity, even though these hypotheses might not pass through that subspace. Moreover, since zt in Eq. (3) is decreasing only residuals/hypotheses in the vicinity of a point are heavily weighted in the intersection. Fig. 1(a) illustrates this condition. Results in Sec. 4 show that ORK excelled even with M = 1, 000. (a) Data in R2F . (b) Data in RKHS Fkr . ˜ Figure 1: (a) ORK under sampling imbalance. (b) Data in RKHS induced by ORK. 3 Multi-Body Motion Segmentation using ORK In this section, we describe how ORK is used for multi-body motion segmentation. 3.1 Outlier rejection via non-linear dimensionality reduction Denote by Fkr the Reproducing Kernel Hilbert Space (RKHS) induced by kr . Let matrix A = ˜ ˜ [φ(x1 ) . . . φ(xN )] contain the input data after it is mapped to Fkr . The kernel matrix K = AT A is ˜ computed using the kernel function kr as ˜ Kp,q = φ(xp ), φ(xq ) = kr (xp , xq ), p, q ∈ {1 . . . N }. ˜ (8) Since kr is a valid Mercer kernel, K is guaranteed to be positive semi-deﬁnite [12]. Let K = ˜ Q∆QT be the eigenvalue decomposition (EVD) of K. Then the rank-n Kernel Singular Value Decomposition (Kernel SVD) [12] of A is 1 1 An = [AQn (∆n )− 2 ][(∆n ) 2 ][(Qn )T ] ≡ Un Σn (Vn )T . n n (9) n Via the Matlab notation, Q = Q:,1:n and ∆ = ∆1:n,1:n . The left singular vectors U is an orthonormal basis for the n-dimensional principal subspace of the whole dataset in Fkr . Projecting ˜ the data onto the principal subspace yields 1 1 B = [AQn (∆n )− 2 ]T A = (∆n ) 2 (Qn )T , (10) n×N where B = [b1 . . . bN ] ∈ R is the reduced dimension version of A. Directions of the principal subspace are dominated by inlier points, since kr evaluates to a high value generally for them, but ˜ always to a low value for gross outliers. Moreover the kernel ensures that points from the same subspace are mapped to the same cluster and vice versa. Fig. 1(b) illustrates this condition. Fig. 2(a)(left) shows the ﬁrst frame of sequence “Cars10” from the Hopkins 155 dataset [18] with 100 false trajectories of Brownian motion added to the original data (297 points). The corresponing RKHS norm histogram for n = 3 is displayed in Fig. 2(b). The existence of two distinct modes, 4 15 Outlier mode Bin count Inlier mode 10 5 0 (a) (left) Before and (right) after outlier removal. Blue dots are inliers while red dots are added outliers. 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 Vector norm in principal subspace 0.18 0.2 (b) Actual norm histogram of “cars10”. Figure 2: Demonstration of outlier rejection on sequence “cars10” from Hopkins 155. corresponding respectively to inliers and outliers, is evident. We exploit this observation for outlier rejection by discarding data with low norms in the principal subspace. The cut-off threshold ψ can be determined by analyzing the shape of the distribution. For instance we can ﬁt a 1D Gaussian Mixture Model (GMM) with two components and set ψ as the point of equal Mahalanobis distance between the two components. However, our experimentation shows that an effective threshold can be obtained by simply setting ψ as the average value of all the norms, i.e. ψ= 1 N N bi . (11) i=1 This method was applied uniformly on all the sequences in our experiments in Sec. 4. Fig. 2(a)(right) shows an actual result of the method on Fig. 2(a)(left). 3.2 Recovering the number of motions and subspace clustering After outlier rejection, we further take advantage of the mapping induced by ORK for recovering the number of motions and subspace clustering. On the remaining data, we perform Kernel PCA [11] to seek the principal components which maximize the variance of the data in the RKHS, as Fig. 1(b) illustrates. Let {yi }i=1,...,N ′ be the N ′ -point subset of the input data that remains after outlier removal, where N ′ < N . Denote by C = [φ(y1 ) . . . φ(yN ′ )] the data matrix after mapping the data ˜ to Fkr , and by symbol C the result of adjusting C with the empirical mean of {φ(y1 ), . . . , φ(yN ′ )}. ˜ ˜ ˜ ˜ The centered kernel matrix K′ = CT C [11] can be obtained as 1 ˜ K′ = ν T K′ ν, ν = [IN ′ − ′ 1N ′ ,N ′ ], (12) N where K′ = CT C is the uncentered kernel matrix, Is and 1s,s are respectively the s × s identity ˜ ˜ matrix and a matrix of ones. If K′ = RΩRT is the EVD of K′ , then we obtain ﬁrst-m kernel m ˜ principal components P of C as the ﬁrst-m left singular vectors of C , i.e. 1 ˜ Pm = CRm (Ωm )− 2 , (13) where Rm = R:,1:m and Ω1:m,1:m ; see Eq. (9). Projecting the data on the principal components yields 1 D = [d1 . . . dN ′ ] = (Ωm ) 2 (Rm )T , (14) ′ where D ∈ Rm×N . The afﬁne subspace span(Pm ) maximizes the spread of the centered data in the RKHS, and the projection D offers an effective representation for clustering. Fig. 3(a) shows the Kernel PCA projection results for m = 3 on the sequence in Fig. 2(a). The number of clusters in D is recovered via spectral clustering. More speciﬁcally we apply the Normalized Cut (Ncut) [13] algorithm. A fully connected graph is ﬁrst derived from the data, where ′ ′ its weighted adjacency matrix W ∈ RN ×N is obtained as Wp,q = exp(− dp − dq 2 /2δ 2 ), (15) and δ is taken as the average nearest neighbour distance in the Euclidean sense among the vectors in D. The Laplacian matrix [13] is then derived from W and eigendecomposed. Under Ncut, 5 0.1 0.05 0 −0.05 −0.1 0.1 −0.15 0.15 0.08 0.1 0.05 0 −0.05 −0.1 0.06 (a) Kernel PCA and Ncut results. (b) W matrix. (c) Final result for “cars10”. Figure 3: Actual results on the motion sequence in Fig. 2(a)(left). the number of clusters is revealed as the number of eigenvalues of the Laplacian that are zero or numerically insigniﬁcant. With this knowledge, a subsequent k-means step is then performed to cluster the points. Fig. 3(b) shows W for the input data in Fig. 2(a)(left) after outlier removal. It can be seen that strong afﬁnity exists between points from the same cluster, thus allowing accurate clustering. Figs. 3(a) and 3(c) illustrate the ﬁnal clustering result for the data in Fig. 2(a)(left). There are several reasons why spectral clustering under our framework is more successful than previous methods. Firstly, we perform an effective outlier rejection step that removes bad trajectories that can potentially mislead the clustering. Secondly, the mapping induced by ORK deliberately separates the trajectories based on their cluster membership. Finally, we perform Kernel PCA to maximize the variance of the data. Effectively this also improves the separation of clusters, thus facilitating an accurate recovery of the number of clusters and also the subsequent segmentation. This distinguishes our work from previous clustering based methods [21, 5] which tend to operate without maximizing the between-class scatter. Results in Sec. 4 validate our claims. 4 Results Henceforth we indicate the proposed method as “ORK”. We leverage on a recently published benchmark on afﬁne model motion segmentation [18] as a basis of comparison. The benchmark was evaluated on the Hopkins 155 dataset4 which contains 155 sequences with tracked point trajectories. A total of 120 sequences have two motions while 35 have three motions. The sequences contain degenerate and non-degenerate motions, independent and partially dependent motions, articulated motions, nonrigid motions etc. In terms of video content three categories exist: Checkerboard sequences, trafﬁc sequences (moving cars, trucks) and articulated motions (moving faces, people). 4.1 Details on benchmarking Four major algorithms were compared in [18]: Generalized PCA (GPCA) [19], Local Subspace Afﬁnity (LSA) [21], Multi-Stage Learning (MSL) [14] and RANSAC [17]. Here we extend the benchmark with newly reported results from Locally Linear Manifold Clustering (LLMC) [5] and Agglomerative Lossy Compression (ALC) [10, 9]. We also compare our method against Kanatani and Matsunaga’s [8] algorithm (henceforth, the “KM” method) in estimating the number of independent motions in the video sequences. Note that KM per se does not perform motion segmentation. For the sake of objective comparisons we use only implementations available publicly5. Following [18], motion segmentation performance is evaluated in terms of the labelling error of the point trajectories, where each point in a sequence has a ground truth label, i.e. number of mislabeled points . (16) classiﬁcation error = total number of points Unlike [18], we also emphasize on the ability of the methods in recovering the number of motions. However, although the methods compared in [18] (except RANSAC) theoretically have the means to 4 Available at http://www.vision.jhu.edu/data/hopkins155/. For MSL and KM, see http://www.suri.cs.okayama-u.ac.jp/e-program-separate.html/. For GPCA, LSA and RANSAC, refer to the url for the Hopkins 155 dataset. 5 6 do so, their estimation of the number of motions is generally unrealiable and the benchmark results in [18] were obtained by revealing the actual number of motions to the algorithms. A similar initialization exists in [5, 10] where the results were obtained by giving LLMC and ALC this knowledge a priori (for LLMC, this was given at least to the variant LLMC 4m during dimensionality reduction [5], where m is the true number of motions). In the following subsections, where variants exist for the compared algorithms we use results from the best performing variant. In the following the number of random hypotheses M and step size h for ORK are ﬁxed at 1000 and 300 respectively, and unlike the others, ORK is not given knowledge of the number of motions. 4.2 Data without gross outliers We apply ORK on the Hopkins 155 dataset. Since ORK uses random sampling we repeat it 100 times for each sequence and average the results. Table 1 depicts the obtained classiﬁcation error among those from previously proposed methods. ORK (column 9) gives comparable results to the other methods for sequences with 2 motions (mean = 7.83%, median = 0.41%). For sequences with 3 motions, ORK (mean = 12.62%, median = 4.75%) outperforms GPCA and RANSAC, but is slightly less accurate than the others. However, bear in mind that unlike the other methods ORK is not given prior knowledge of the true number of motions and has to estimate this independently. Column Method 1 REF 2 GPCA Mean Median 2.03 0.00 4.59 0.38 Mean Median 5.08 2.40 28.66 28.26 3 4 5 6 LSA MSL RANSAC LLMC Sequences with 2 motions 3.45 4.14 5.56 3.62 0.59 0.00 1.18 0.00 Sequences with 3 motions 9.73 8.23 22.94 8.85 2.33 1.76 22.03 3.19 8 ALC 9 ORK 10 ORK∗ 3.03 0.00 7.83 0.41 1.27 0.00 6.26 1.02 12.62 4.75 2.09 0.05 Table 1: Classiﬁcation error (%) on Hopkins 155 sequences. REF represents the reference/control method which operates based on knowledge of ground truth segmentation. Refer to [18] for details. We also separately investigate the accuracy of ORK in estimating the number of motions, and compare it against KM [8] which was proposed for this purpose. Note that such an experiment was not attempted in [18] since approaches compared therein generally do not perform reliably in estimating the number of motions. The results in Table 2 (columns 1–2) show that for sequences with two motions, KM (80.83%) outperforms ORK (67.37%) by ≈ 15 percentage points. However, for sequences with three motions, ORK (49.66%) vastly outperforms KM (14.29%) by more than doubling the percentage points of accuracy. The overall accuracy of KM (65.81%) is slightly better than ORK (63.37%), but this is mostly because sequences with two motions form the majority in the dataset (120 out of 155). This leads us to conclude that ORK is actually the superior method here. Dataset Column Method 2 motions 3 motions Overall Hopkins 155 1 2 KM ORK 80.83% 67.37% 14.29% 49.66% 65.81% 63.37% Hopkins 155 + Outliers 3 4 KM ORK 00.00% 47.58% 100.00% 50.00% 22.58% 48.13% Table 2: Accuracy in determining the number of motions in a sequence. Note that in the experiment with outliers (columns 3–4), KM returns a constant number of 3 motions for all sequences. We re-evaluate the performance of ORK by considering only results on sequences where the number of motions is estimated correctly by ORK (there are about 98 ≡ 63.37% of such cases). The results are tabulated under ORK∗ (column 10) in Table 1. It can be seen that when ORK estimates the number of motions correctly, it is signiﬁcantly more accurate than the other methods. Finally, we compare the speed of the methods in Table 3. ORK was implemented and run in Matlab on a Dual Core Pentium 3.00GHz machine with 4GB of main memory (this is much less powerful 7 than the 8 Core Xeon 3.66GHz with 32GB memory used in [18] for the other methods in Table 3). The results show that ORK is comparable to LSA, much faster than MSL and ALC, but slower than GPCA and RANSAC. Timing results of LLMC are not available in the literature. Method 2 motions 3 motions GPCA 324ms 738ms LSA 7.584s 15.956s MSL 11h 4m 1d 23h RANSAC 175ms 258ms ALC 10m 32s 10m 32s ORK 4.249s 8.479s Table 3: Average computation time on Hopkins 155 sequences. 4.3 Data with gross outliers We next examine the ability of the proposed method in dealing with gross outliers in motion data. For each sequence in Hopkins 155, we add 100 gross outliers by creating trajectories corresponding to mistracks or spuriously occuring points. These are created by randomly initializing 100 locations in the ﬁrst frame and allowing them to drift throughout the sequence according to Brownian motion. The corrupted sequences are then subjected to the algorithms for motion segmentation. Since only ORK is capable of rejecting outliers, the classiﬁcation error of Eq. (16) is evaluated on the inlier points only. The results in Table 4 illustrate that ORK (column 4) is the most accurate method by a large margin. Despite being given the true number of motions a priori, GPCA, LSA and RANSAC are unable to provide satisfactory segmentation results. Column Method Mean Median Mean Median 1 2 3 4 GPCA LSA RANSAC ORK Sequences with 2 motions 28.66 24.25 30.64 16.50 30.96 26.51 32.36 10.54 Sequences with 3 motions 40.61 30.94 42.24 19.99 41.30 27.68 43.43 8.49 5 ORK∗ 1.62 0.00 2.68 0.09 Table 4: Classiﬁcation error (%) on Hopkins 155 sequences with 100 gross outliers per sequence. In terms of estimating the number of motions, as shown in column 4 in Table 2 the overall accuracy of ORK is reduced to 48.13%. This is contributed mainly by the deterioration in accuracy on sequences with two motions (47.58%), although the accuracy on sequences with three motions are maintained (50.00%). This is not a surprising result, since sequences with three motions generally have more (inlying) point trajectories than sequences with two motions, thus the outlier rates for sequences with three motions are lower (recall that a ﬁxed number of 100 false trajectories are added). On the other hand, the KM method (column 3) is completely overwhelmed by the outliers— for all the sequences with outliers it returned a constant “3” as the number of motions. We again re-evaluate ORK by considering results from sequences (now with gross outliers) where the number of motions is correctly estimated (there are about 75 ≡ 48.13% of such cases). The results tabulated under ORK∗ (column 5) in Table 4 show that the proposed method can accurately segment the point trajectories without being inﬂuenced by the gross outliers. 5 Conclusions In this paper we propose a novel and highly effective approach for multi-body motion segmentation. Our idea is based on encapsulating random hypotheses in a novel Mercel kernel and statistical learning. We evaluated our method on the Hopkins 155 dataset with results showing that the idea is superior other state-of-the-art approaches. It is by far the most accurate in terms of estimating the number of motions, and it excels in segmentation accuracy despite lacking prior knowledge of the number of motions. The proposed idea is also highly robust towards outliers in the input data. Acknowledgements. We are grateful to the authors of [18] especially Ren´ Vidal for discussions e and insights which have been immensely helpful. 8 References [1] T. Boult and L. Brown. Factorization-based segmentation of motions. In IEEE Workshop on Motion Understanding, 1991. [2] T.-J. Chin, H. Wang, and D. Suter. Robust ﬁtting of multiple structures: The statistical learning approach. In ICCV, 2009. [3] J. Costeira and T. Kanade. A multibody factorization method for independently moving objects. IJCV, 29(3):159–179, 1998. [4] M. A. Fischler and R. C. Bolles. Random sample concensus: A paradigm for model ﬁtting with applications to image analysis and automated cartography. Comm. of the ACM, 24:381–395, 1981. [5] A. Goh and R. Vidal. Segmenting motions of different types by unsupervised manifold clustering. In CVPR, 2007. [6] A. Gruber and Y. Weiss. Multibody factorization with uncertainty and missing data using the EM algorithm. In CVPR, 2004. [7] K. Kanatani. Motion segmentation by subspace separation and model selection. In ICCV, 2001. [8] K. Kanatani and C. Matsunaga. Estimating the number of independent motions for multibody segmentation. In ACCV, 2002. [9] Y. Ma, H. Derksen, W. Hong, and J. Wright. Segmentation of multivariate mixed data via lossy coding and compression. TPAMI, 29(9):1546–1562, 2007. [10] S. Rao, R. Tron, Y. Ma, and R. Vidal. Motion segmentation via robust subspace separation in the presence of outlying, incomplete, or corrupted trajectories. In CVPR, 2008. [11] B. Sch¨ lkopf, A. Smola, and K. R. M¨ ller. Nonlinear component analysis as a kernel eigeno u value problem. Neural Computation, 10:1299–1319, 1998. [12] J. Shawe-Taylor and N. Cristianini. Kernel methods for pattern analysis. Cambridge University Press, 2004. [13] J. Shi and J. Malik. Normalized cuts and image segmentation. TPAMI, 22(8):888–905, 2000. [14] Y. Sugaya and K. Kanatani. Geometric structure of degeneracy for multi-body motion segmentation. In Workshop on Statistical Methods in Video Processing, 2004. [15] R. Toldo and A. Fusiello. Robust multiple structures estimation with J-Linkage. In ECCV, 2008. [16] C. Tomasi and T. Kanade. Shape and motion from image streams under orthography. IJCV, 9(2):137–154, 1992. [17] P. Torr. Geometric motion segmentation and model selection. Phil. Trans. Royal Society of London, 356(1740):1321–1340, 1998. [18] R. Tron and R. Vidal. A benchmark for the comparison of 3-D motion segmentation algorithms. In CVPR, 2007. [19] R. Vidal and R. Hartley. Motion segmentation with missing data by PowerFactorization and Generalized PCA. In CVPR, 2004. [20] R. Vidal, Y. Ma, and S. Sastry. Generalized Principal Component Analysis (GPCA). TPAMI, 27(12):1–15, 2005. [21] J. Yan and M. Pollefeys. A general framework for motion segmentation: independent, articulated, rigid, non-rigid, degenerate and non-degenerate. In ECCV, 2006. [22] L. Zelnik-Manor and M. Irani. Degeneracies, dependencies and their implications on multibody and multi-sequence factorization. In CVPR, 2003. [23] W. Zhang and J. Koseck´ . Nonparametric estimation of multiple structures with outliers. In a Dynamical Vision, ICCV 2005 and ECCV 2006 Workshops, 2006. 9

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